Annual EAGER Conference And Workshop On the Occasion of J. Murre’s 75–th Birthday on Algebraic Cycles and Motives August 30– September 3, 2004 Leiden University, the Netherlands Edited by Jan Nagel Universit´ e de Lille I
Chris Peters Universit´ e de Grenoble I
Contents
Foreword Part I
page 4 Survey Articles
11
1 The Motivic Vanishing Cycles and the Conservation Conjecture J. Ayoub 13 2 On the Theory of 1-Motives
L. Barbieri-Viale
67
3 Motivic Decomposition for Resolutions of Threefolds de Cataldo and L. Migliorini 4 Correspondences and Transfers
151 M.
D. Ramakrish292
Research Articles
307
8 Beilinson’s Hodge Conjecture with Coefficients and S. Saito 9 On the Splitting of the Bloch-Beilinson Filtration ville 10 K¨ unneth Projectors
219
A. Krishna and V. Srini278
7 Modular Curves, Surfaces and Threefolds nan Part II
114
F. D´ eglise
5 Algebraic Cycles and Singularities of Normal Functions Green and Ph. Griffiths 6 Zero Cycles on Singular Varieties vas
M.
S. Bloch and H. Esnault
M. Asakura 309 A. Beau344 359
11 The Brill-Noether Curve of a Stable Bundle on a Genus Two Curve S. Brivio and A. Verra 378 3
4
Contents
12 On Tannaka duality for vector bundles on p-adic curves Deninger and A. Werner
C.
13 On finite-dimensional motives and Murre’s conjecture Jannsen
U.
14 On the Transcendental Part of the Motive of a Surface Kahn, J.P. Murre and C. Pedrini
B.
15 A note on finite dimensional motives
Kimura, S.-I.
16 Real Regulators on Milnor Complexes, II
J. D. Lewis
399 417 448 508 519
17 Motives for Picard modular surfaces A. Miller, S. M¨ ullerStach, S. Wortmann, Y.-H.Yang, K. Zuo 546 18 The Regulator Map for Complete Intersections
J. Nagel 582
19 Hodge Number Polynomials for Nearby and Vanishing Cohomology C. Peters and J. Steenbrink 594 20 Direct Image of Logarithmic Complexes
M. Saito
21 Mordell-Weil Lattices of Certain Elliptic K3’s 22 Motives from Diffraction
J. Stienstra
609
T. Shioda 624 645
Foreword
This proceedings contains a selection of papers from the EAGER conference ”Algebraic Cycles and Motives” that was held at the Lorentz Center in Leiden at the occasion of the 75th birthday of Prof. J.P. Murre (Aug 30– Sept 3, 2004). The conference attracted many of the leading experts in the field as well as a number of young researchers. As the papers in this volume cover the main research topics and some interesting new developments, they should give a good indication of the present state of the subject. This volume contains sixteen research papers and six survey papers. The theory of algebraic cycles deals with the study of subvarieties of a given projective algebraic variety X, starting with the free group Z p (X) on irreducible subvarieties of X of codimension p. In order to make this very large group manageable, one puts a suitable equivalence relation on it, usually rational equivalence. The resulting Chow group CH p (X) in general still might be very big. If X is a smooth variety, intersection product makes the direct sum of all the Chow groups into a ring, the Chow ring CH ∗ (X). This up to now still mysterious ring can be studied through its relation to cohomology, the first example of which is the cycle class map: every algebraic cycle defines a class in singular, de Rham, or `-adic cohomology. Ultimately this cohomological approach leads to the theory of motives and motivic cohomology developed by A. Grothendieck, M. Levine, M. Nori, V. Suslin and A. Voevodsky, just to mention a few main actors. There were about 60 participants for the conference, coming from Europe, the United States, India and Japan. During the conference there were 22 one hour lectures. On the last day there were three special lectures devoted to the scientific work of Murre, in honour of his 75th birthday. The lectures covered a wide range of topics, such as the study of algebraic cycles using Abel–Jacobi/regulator maps and normal functions, motives (Voevodsky’s triangulated category of mixed motives, finite–dimensional motives), the 5
6
Foreword
conjectures of Bloch–Beilinson and Murre on filtrations on Chow groups and Bloch’s conjecture, and results of a more arithmetic flavour for varieties defined over number fields or local fields. Let us start by discussing the survey papers. The first, a paper of J. Ayoub is devoted to the construction of a motivic version of the vanishing cycle formalism. It is followed by a paper of L.Barbieri Viale who presents an overview of the main results of the theory of mixed motives of level at most one. In a series of recent papers, M. De Cataldo and L. Migliorini have made a detailed study of the topological properties of algebraic maps using the theory of perverse sheaves. Their survey provides an introduction to this work, illustrated by a number of low–dimensional examples. D´eglise’s paper contains a careful exposition of Voevodsky’s theory of sheaves with transfers over a regular base scheme, with detailed proofs. The paper of M. Green and P.Griffiths contains an outline of an ambitious research program that centers around the extension of normal functions over a higher–dimensional base, and its applications to the Hodge conjecture. (The case where the base space is a curve is known by work of F. El Zein and S. Zucker.) A. Krishna and V. Srinivas discuss the theory of zero–cycles on singular varieties and its applications to algebra. The paper of D. Ramakrishnan is a brief survey of results concerning algebraic cycles on Hilbert modular varieties. In discussing the research papers we have grouped according to the main research themes, although in the proceedings they are listed alphabetically according to the name of the authors. One of the leading themes in the theory of algebraic cycles is the study of the conjectural Bloch–Beilinson filtration on Chow groups. In the course of his work on motives, J. Murre found an equivalent and more explicit version of this conjecture which states that the motive of a smooth projective algebraic variety should admit a Chow-K¨ unneth decomposition with a number of specific properties. The paper of B. Kahn-J. Murre-C. Pedrini contains a detailed exposition of these matters with emphasis on the study of the transcendental part of the motive of a surface. The paper of S. Bloch and H. Esnault is devoted to the construction of an algebraic cycle that induces the K¨ unneth projector onto H 1 (U ) for a quasi–projective variety U , and the paper of Miller et al. shows the existence of certain Chow-K¨ unneth projectors for compactified families of abelian threefolds over a certain Picard modular surface studied by Holzapfel. Beauville studies the splitting of the Bloch–Beilinson filtration for certain symplectic projective manifolds. The notion of ”finite–dimensionality” of motives, which recently attracted a lot of attention, is studied in the papers of S.-I. Kimura and U. Jannsen. The
Foreword
7
latter paper uses this notion to verify Murre’s conjectures in a number of examples. Another important theme is the study of algebraic cycles using Hodge theory. The paper of C. Peters and J. Steenbrink deals with the motivic nearby fiber and its relation to the limit mixed Hodge structure of a family of projective varieties. Morihiko Saito constructs the total infinitesimal invariant of a higher Chow cycle, an object that lives in the direct sum of the cohompology of filtered logarithhmic complexes with coefficients. In the papers of M. Asakura–S. Saito and J. Nagel, infitesimal methods are used to study the regulator map on higher Chow groups. M. Asakura and S. Saito use these techniques to verify a conjecture of Beilinson (”Beilinson’s Hodge conjecture with coefficients”) in certain cases. J. Lewis defines a twisted version of Milnor K–theory and a corresponding twisted version of the regulator, which is shown to have a nontrivial image in certain examples. The remaining papers deal with a variety of topics. The papers of C. Deninger–A. Werner and S. Brivio–A. Verra deal with vector bundles. C. Deninger and A. Werner study the category of degree zero vector bundles with ”potentially strongly semistable reduction” on a p–adic curve. S. Brivio and A. Verra investigate the properties of the theta map defined on the moduli space of semistable vector bundles over a curve. T. Shioda studies the structure of the Mordell–Weil lattice of certain elliptic K3 surfaces, and the paper of J. Stienstra studies a potential link between the theory of motives and string theory using diffraction patterns. The conference has been financed by the Lorentz Center, EAGER (European Algebraic Geometry Research Training Network), the KNAW (Royal Netherlands Academy of Arts and Sciences), and the Thomas Stieltjes Instituut. We heartily thank these institutions for their financial support. It is a pleasure to dedicate this volume to Jacob Murre. The study of algebraic cycles and motives has been his life–long passion, and he has made a number of important contributions to the subject. Chris Peters and Jan Nagel, May 2006.
8
Foreword
Program Day
Hour
Speaker
Title
10:00–11:00
P. Griffiths
Monday
11:15–12:15
A. Beauville
Aug 30
13:30–14:30 14:30–15:30
S. M¨ uller–Stach F. D´eglise
16:00–17:00
O. Tommasi
Algebraic cycles and singularities of normal functions When does the Bloch-Beilinson filtration split? Higher Abel-Jacobi maps Cycle modules and triangulated mixed motives Rational cohomology of the moduli space of genus 4 curves
09:30–10:30
H. Esnault
Tuesday
11:15–12:15
U. Jannsen
August 31
13:30–14:30 14:45–15:45
L. Barbieri Viale B. van Geemen
16:15–17:15
L. Migliorini
Wednesday
09:30–10:30 11:15–12:15
K. K¨ unnemann J.-L Colliot-Th´el`ene
September 1
13:30–14:30
C. Deninger
09:30–10:30
S. Saito
11:15–12:15 13:30–14:30 14:45–15:45
T. Shioda D. Ramakrishnan A. Verra
16:15–17:15
J. Ayoub
10:00–11:00 11:30–12:30 13:30–14:30
A. Conte V. Srinivas F Oort
14:45–15:45
S. Bloch
Thursday September 2
Friday September 3
Deligne’s integrality theorem in unequal characteristic and rational points over finite fields Some remarks on finite dimensional motives Motivic Albanese Some remarks on Brauer groups of elliptic fibrations on K3 surfaces Hodge theory of projective maps Extensions in Arakelov geometry Zero-cycles on linear algebraic groups over local fields Vector bundles on p-adic curves and parallel transport Finiteness results for motivic cohomology Finding cycles on certain K3 surfaces Cycles on Hilbert modular fourfolds Moduli of vector bundles on curves and correspondences: the genus two case Conservation of φ and the Bloch conjecture 25 years of joint work with Jaap Murre Zero cycles on singular varieties Geometric aspects of the scientific work of Jaap Murre, I Geometric aspects of the scientific work of Jaap Murre, II
Foreword
Participants Last name and Initial
Institute
Amerik, E.
Univ. of Paris XI, Orsay, France
Andr´e, Y..
E.N.S. Paris, France
Ayoub. J.
Univ. of Paris VI, Ivry sur Seine, France
Barbieri Viale, L.
Univ. of Roma I, Italy
Beauville, A.
Univ. of Nice, France
Biglari, S.
Univ. of Leipzig, Germany
Bloch , S.
Univ.of Chicago, Ill., United States
Colliot-Th´el`ene, J.-L.
Univ. of Paris XI,Orsay, France
Colombo , E.
Univ. of Milano, Italy
Conte , A.
Univ. of Turin, Italy
De Jeu , R.
Univ. of Durham, United Kingdom
D´eglise , F.
Inst. Galil´ee, Villetaneuse, France
Del Angel, P.
CIMAT, Guanajuato, Mexico
Deninger, C.
Univ. M¨ unster, Germany
Edixhoven, B.
Univ. of Leiden, Netherlands
Eriksson, D.
E.N.S. Paris, France
Esnault, H.
Univ. of Essen, Germany
Faber, C.
KTH, Stockholm, Sweden
Gordon, B.
Univ. Maryland, Washington, United States
Griffiths, P.
IAS, Princeton, United States
Grooten, M.
Univ. of Nijmegen, Netherlands
Guletskii, V.
Belar. St. Ar. Univ., Minsk, Belarus
Haran, S.
Technuion, Haifa, Israel
H¨ oring, A.
Univ. of Bayreuth, Germany
Jannsen , U.
Univ. of Regensburg, Germany
Kahn , B.
Univ. of Paris VI, Paris, France
Kimura, S.
Hiroshima Univ., Japan
Kimura, K.
Tsukuba Univ., Japan
Kloosterman, R.
RUG, Groningen, Netherlands
K¨ unnemann, K.
Univ.of Regensburg, Germany
Lemahieu , A.
Univ. of Leuven, Belgium
Lewis, J.
Univ. of Edmonton, Alberta, Canada
9
10
Foreword Looijenga , E.
Univ. of Utrecht, Netherlands
L¨ ubke, M.
Univ. of Leiden, Netherlands
Marchisio, M.
Univ. of Turin, Italy
Migliorini, L.
Univ. of Bologna, Italy
Miller , A.
Univ. of Heidelberg, Germany
M¨ uller-Stach, S.
Univ. of Mainz, Germany
Murre , J.
Univ. of Leiden, Netherlands
Nagel, J.
Univ. de Lille I, Villeneuve d’Asque,France
Nicaise, J.
Univ. of Leuven, Belgium
Oort, F.
Univ. of Utrecht, Netherlands
Pedrini, C.
Univ. of Genova, Italy
Peters, C.
Univ. of Grenoble I,Saint Martin d’H`eres, France
Popovici, D.
Univ. of Paris VII, Orsay, France
Ramakrishnan, D.
Caltech,Pasadena, CA, United States
Ramon Mari, J.
Humbold Univ., Berlin, Germany
Reuvers, E.
Univ. of Nijmegen, Netherlands
Rydh, D.
Univ. of Gothenburg, Sweden
Saito, S.
Grad. school math.,Tokyo, Japan
Schepers, J.
Univ. of Leuven, Belgium
Shioda, T.
Rikkyo Univ.,Tokyo, Japan
Springer, T.
Univ. of Utrecht, Netherlands
Srinivas, V.
Tata Univ., Mumbai, India
Steenbrink, J.
Univ. of Nijmegen, Netherlands
Stienstr, J.
Univ. of Utrecht, Netherlands
Swierstra, R.
Univ. of Utrecht, Netherlands
Tommasi, O.
Univ. of Nijmegen, Netherlands
Van Geemen, B.
Univ. of Milano, Italy
Verra, A.
Univ. of Roma III, Italy
Veys,W.
Univ. of Leuven, Belgium
Part I Survey Articles
1 The Motivic Vanishing Cycles and the Conservation Conjecture Joseph Ayoub
To Jacob Murre for his 75th birthday
1.1 Introduction Let X be a noetherian scheme. Following Morel and Voevodsky (see [24], [25], [28], [33] and [37]), one can associate to X the motivic stable homotopy category SH(X). Objects of SH(X) are T -spectra of simplicial sheaves on the smooth Nisnevich site (Sm /X)Nis , where T is the pointed quotient sheaf A1X /GmX . As in topology, SH(X) is triangulated in a natural way. There is also a tensor product − ⊗X − and an ”internal hom”: HomX on / Y of noetherian SH(X) (see [20] and [33]). Given a morphism f : X ∗ schemes, there is a pair of adjoint functors (f , f∗ ) between SH(X) and SH(Y ). When f is quasi-projective, one can extend the pair (f ∗ , f∗ ) to a quadruple (f ∗ , f∗ , f! , f ! ) (see [3] and [8]). In particular we have for SH(−) the full package of the Grothendieck six operators. It is then natural to ask if we have also the seventh one, that is, if we have a vanishing cycle formalism (analogous to the one in the ´etale case, developed in [9] and [10]). In the third chapter of our PhD thesis [3], we have constructed a vanishing cycles formalism for motives. The goal of this paper is to give a detailed account of that construction, to put it in a historical perspective and to discuss some applications and conjectures. In some sense, it is complementary to [3] as it gives a quick introduction to the theory with emphasis on motivations rather than a systematic treatment. The reader will not find all the details here: some proofs will be omitted or quickly sketched, some results will be stated with some additional assumptions (indeed we will be mainly interested in motives with rational coefficients over characteristic zero schemes). For the full details of the theory, one should consult [3]. Let us mention also 13
14
J. Ayoub
that M. Spitzweck has a theory of limiting motives which is closely related to our motivic vanishing cycles formalism. For more information, see [35]. The paper is organized as follows. First we recall the classical pictures: the ´etale and the Hodge cases. Although this is not achieved here, these classical constructions should be in a precise sense realizations of our motivic construction. In section 1.3 we introduce the notion of a specialization system which encodes some formal properties of the family of nearby cycles functors. We state without proofs some important theorems about specialization systems obtained in [3]. In section 1.4, we give our main construction and prove motivic analogues of some well-known classical results about nearby cycles functors: constructibility, commutation with tensor product and duality, etc. We also construct a monodromy operator on the unipotent part of the nearby cycles which is shown to be nilpotent. Finally, we propose a conservation conjecture which is weaker than the conservation of the classical realizations but strong enough to imply the Schur finiteness of constructible motives†. In the literature, the functors Ψf have two names: they are called ”nearby cycles functors” or ”vanishing cycles functors”. Here we choose to call them the nearby cycles functors. The properties of these functors form what we call the vanishing cycles formalism (as in [9] and [10]).
1.2 The classical pictures We briefly recall the construction of the nearby cycles functors RΨf in ´etale cohomology. We then explain a construction of Rapoport and Zink which was the starting point of our definition of Ψf in the motivic context. After that we shall recall some facts about limits of variations of Hodge structures. A very nice exposition of these matters can be found in [15].
1.2.1 The vanishing cycles formalism in ´ etale cohomology Let us fix a prime number ` and a finite commutative ring Λ such that `ν .Λ = 0 for ν large enough. When dealing with ´etale cohomology, we shall always assume that ` is invertible on our schemes. For a reasonable scheme V , we denote by D+ (V, Λ) the derived category of bounded below complexes of ´etale sheaves on V with values in Λ-modules. Let S be the spectrum of a strictly henselian DVR (discrete valuation † Constructible motives means geometric motives of [40]. They are also the compact objects in the sense Neeman [30] (see remark 1.3.3).
The Motivic Vanishing Cycles and the Conservation Conjecture
15
ring). We denote by η the generic point of S and by s the closed point: η
j
/So
i
s.
We also fix a separable closure η¯ of the point η. From the point of view of ´etale cohomology, the scheme S plays the role of a small disk so that η is a punctured small disk and η¯ is a universal cover of that punctured disk. We will also need the normalization S¯ of S in η¯: η¯
¯ j
/ S¯ o
¯i
s.
/ S be a finite type S-scheme. We consider the commuNow let f : X tative diagram with cartesian squares
Xη fη
η
j
j
/Xo
i
Xs . fs
f
/So
i
s
Following Grothendieck (see [10]), we look also at the diagram Xη¯ fη¯
η¯
¯ j
¯ j
/X ¯o
¯i
Xs
f¯
/ S¯ o
fs ¯i
s
obtained in the same way by base-changing the morphism f . (This is what we will call the ”Grothendieck trick”). We define then the triangulated functor: / D+ (Xs , Λ)
RΨf : D+ (Xη , Λ)
by the formula: RΨf (A) = ¯i∗ R¯j∗ (AXη¯ ) for A ∈ D+ (Xη , Λ). By construction, the functor RΨf comes with an action of the Galois group of η¯/η, but we will not explicitly use this here. The basic properties of these functors concern the relation between RΨg and RΨg◦h (see [9]): / S be an S-scheme and suppose given Proposition 1.2.1. Let g : Y / an S-morphism h : X Y such that f = g ◦ h. We form the commutative diagram
Xη hη
Yη
j
j
/Xo
i
h
/Y o
i
Xs
hs
Ys .
16
J. Ayoub
There exist natural transformations of functors / RΨf h∗η ,
• αh : h∗s RΨg • βh : RΨg Rhη∗
/ Rhs∗ RΨf .
Furthermore, αh is an isomorphism when h is smooth and βh is an isomorphism when h is proper. The most important case, is maybe when g = idS and f = h. Using the easy fact that RΨidS Λ = Λ, we get that: • RΨf Λ = Λ if f is smooth, • RΨidS Rfη∗ Λ = Rfs∗ RΨf Λ if f is proper. The last formula can be rewritten in the following more expressive way: H´∗et (Xη¯, Λ) = H´∗et (Xs , RΨf Λ). In concrete terms, this means that for a proper S-scheme X, the ´etale cohomology of the constant sheaf on the generic geometric fiber Xη¯ is isomorphic to the ´etale cohomology of the special fiber Xs with value in the complex of nearby cycles RΨf Λ. This is a very useful fact, because usually the scheme Xs is simpler than Xη¯ and the complex RΨf Λ can often be computed using local methods.
1.2.2 The Rapoport-Zink construction We keep the notations of the previous paragraph. We now suppose that X is a semi-stable S-scheme i.e. locally for the ´etale topology X is isomorphic to the standard scheme S[t1 , . . . , tn ]/(t1 . . . tr − π) where π is a uniformizer of S and r ≤ n are positive integers. In [32], Rapoport and Zink constructed an important model of the complex RΨf (Λ). Their construction is based on the following two facts: / Λη (1)[1] in D+ (η, Λ) called • There exists a canonical arrow θ : Λη the fundamental class with the property that the composition θ ◦ θ is zero, / i∗ Rj∗ Λ(1)[1] in D+ (Xs , Λ) has a repre• The morphism θ : i∗ Rj∗ Λ
sentative on the level of complexes θ : M• composition M•
/ M• (1)[1]
/ M• (1)[1] such that the / M• (2)[2]
is zero as a map of complexes. Therefore we obtain a double complex RZ •,• = [· · · → 0 → M• (1)[1] → M• (2)[2] → M• (3)[3] → · · · → M• (n)[n] → . . . ]
The Motivic Vanishing Cycles and the Conservation Conjecture
17
where the complex M• (1)[1] is placed in degree zero. Furthermore, following / Tot(RZ •,• ) which is an Rapoport and Zink, we get a map RΨf Λ isomorphism in D+ (Xs , Λ) (see [32] for more details). Here Tot(−) means the simple complex associated to a double complex. In particular, Rapoport and Zink’s result says that the nearby cycles complex RΨf Λ can be constructed using two ingredients: • The complex i∗ Rj∗ Λ, • The fundamental class θ. Our construction of the nearby cycles functor in the motivic context is inspired by this fact. Indeed, the above ingredients are motivic (see 1.4.1 for a definition of the motivic fundamental class). We will construct in paragraph 1.4.2 a motivic analogue of RZ •,• based on these two motivic ingredients and then define the (unipotent) ”motivic nearby cycles” to be the associated total motive. In fact, for technical reasons, we preferred to use a motivic analogue of the dual version of RZ •,• . By the dual of the Rapoport-Zink complex, we mean the bicomplex Q•,• = [· · · → M• (−n)[−n] → · · · → M• (−1)[−1] → M• → 0 → . . . ] where the complex M• is placed in degree zero. It is true that by passing to the total complex, the double complex Q•,• gives in the same way as RZ •,• the nearby cycles complex.
1.2.3 The limit of a variation of Hodge structures Let D be a small analytic disk, 0 a point of D and D? = D − 0. Let f : / D ? be an analytic family of smooth projective varieties. For t ∈ X? ? D , we denote by Xt the fiber f −1 (t) of f . For any integer q, the local system Rq f∗ C = (Rq f∗ Z) ⊗ C on D? with fibers (Rq f∗ C)t = Hq (Xt , C) is the sheaf of . horizontal sections of the Gauss-Manin connection ∇ on Rq f∗ ΩX ? /D? . The . decreasing filtration F k on the de Rham complex ΩX ? /D? given by . F k ΩX ? /D? = [0 → . . . 0 → ΩkX ? /D? → · · · → ΩnX ? /D? ] . . induces a filtration F k Rq f∗ ΩX ? /D? by locally free OD? -submodules on Rq f∗ ΩX ? /D? . For any t ∈ D? , we get by applying the tensor product − ⊗OD? C(t) a filtration F k on Hq (Xt , C) which is the Hodge filtration. The data: • The local system Rq f∗ Z, . • The OD? -module (Rq f∗ Z) ⊗ OD? = Rq f∗ ΩX ? /D? together with the GaussManin connexion,
18
J. Ayoub
• The filtration F k on (Rq f∗ Z) ⊗ OD? satisfy the Griffiths transversality condition and are called a Variation of (pure) Hodge Structures. Let us suppose for simplicity that f extends to a semi-stable proper an/ D . We denote by ω . alytic morphism: X X/D the relative de Rham ? complex with logarithmic poles on Y = X − X , that is, 1 = Ω1X (log (Y ))/Ω1D (log (0)). ωX/D
¯ ? → D? and a We fix a uniformizer t : D → C, a universal cover D ¯ ? . In [36], Steenbrink constructed an isomorphism logarithm log t on D . / RΨf C depending on these choices. From this, he deduced (ωX/D )|Y . a mixed Hodge structure on Hq (Y, (ωX/D )|Y ) which is by definition the limit of the above Variation of Hodge Structures.
1.2.4 The analogy between the situations in ´ etale cohomology and Hodge theory Let V be a smooth projective variety defined over a field k of characteristic ¯ zero. Suppose also given an algebraic closure k/k with Galois group Gk and an embedding σ : k ⊂ C. In the ´etale case, the `-adic cohomology of Vk¯ is equipped with a structure of a continuous Gk -module. In the complex analytic case, the Betti cohomology of V (C) is equipped with a Hodge structure. / C be a flat and proper family of smooth varieties Now let f : X ¯ over k parametrized by an open k-curve C. Then for any k-point t of q C, we have a continuous Galois module† H (Xt , Q` ). These continuous Galois modules can be thought of as a ”Variation of Galois Representations” parametrized by C which is the ´etale analogue of the Variation of Hodge structures (Hq (Xt (C), Q), F k ) that we discussed in the above paragraph. Now let s be a point of the boundary of C and choose a uniformizer near s. As in the Hodge–theoretic case, the variation of Galois modules above has a ”limit” on s which is a ”mixed” Galois module given by the following data: • A monodromy operator N which is nilpotent. This operator induces the monodromy filtration which turns out to be compatible with the weight filtration of Steenbrink’s mixed Hodge structure on the limit cohomology (see [15]), † In general only an open subgroup of Gk acts on the cohomology, unless t factors trough a k-rational point.
The Motivic Vanishing Cycles and the Conservation Conjecture
19
• The grading associated to the monodromy filtration is a continuous Galois module of ”pure” type. As in the analytic case, this limit is defined via the nearby cycles complex. Indeed, choose an extension of f to a projective scheme X 0 over C 0 = C ∪{s}. Let Y be the special fiber of X 0 . The choice of a uniformizer gives us a complex RΨX 0 /C 0 Q` on Y . Then the ”limit” of our ”Variation of Galois representations” is given by H q (Y, RΨX 0 /C 0 Q` ). The monodromy operator N is induced from the representation on RΨX 0 /C 0 Q` of the ´etale fundamental group of the punctured henselian neighbourhood of s in C.
1.3 Specialization systems The goal of this section is to axiomatize some formal properties of the nearby cycles functors that we expect to hold in the motivic context. The result will be the notion of specialization systems. We then state some consequences of these axioms which play an important role in the theory. Before doing that we recall briefly the motivic categories we use.
1.3.1 The motivic categories Let X be a noetherian scheme. In this paper we will use two triangulated categories associated to X: (i) The motivic stable homotopy category SH(X) of Morel and Voevodsky, (ii) The stable category of mixed motives DM(X) of Voevodsky. These categories are respectively obtained by taking the homotopy category (in the sense of Quillen [31]) associated to the two model categories of T = (A1X /GmX )-spectra: (i) The category SpectTs (X) of T -spectra of simplicial sheaves on the smooth Nisnevich site (Sm /X)Nis , (ii) The category SpectTtr (X) of T -spectra of complexes of sheaves with transfers on the smooth Nisnevich site (Sm /X)Nis . Recall that a T -spectrum E is a sequence of objects (En )n∈N connected / Hom(T, En+1 ) . We sometimes denote by by maps of the form En SpectT (X) one of the two categories SpectTs (X) or SpectTtr (X). We do not intend to give the detailed construction of these model categories as this has already been done in several places (cf. [5], [20], [24], [25], [28],
20
J. Ayoub
[33], [37]). For the reader’s convenience, we however give some indications. We focus mainly on the class of weak equivalences; indeed this is enough to define the homotopy category which is obtained by formally inverting the arrows in this class. The weak equivalences in these two categories of T -spectra are called the stable A1 -weak equivalences and are defined in the three steps. We restrict ourself to the case of simplicial sheaves; the case of complexes of sheaves with transfers is completely analogous. Step 1. We first define simplicial weak equivalences for simplicial sheaves. / B• of simplicial sheaves on (Sm /X)Nis is a simplicial weak A map A• equivalence if for any smooth X-scheme U and any point u ∈ U , the map of / B• (Spec(Oh )) is a weak equivalence simplicial sets† A• (Spec(OhU,u )) U,u (i.e. induces isomorphisms on the set of connected components and on the homotopy groups). Step 2. Next we perform a Bousfield localization of the simplicial model /U structure on simplicial sheaves in order to invert the projections A1U for smooth X-schemes U (see [13] for a general existence theorem on localizations and [28] for this particular case). The model structure thus obtained is the A1 -model structure on simplicial sheaves over (Sm /X)Nis . We denote HoA1 (X) the associated homotopy category. Step 3. If A is a pointed simplicial sheaf and E = (En )n is a T -spectrum of simplicial sheaves we define the stable cohomology groups of A with values in E to be the colimit: Colimn homHoA1 (X) (T ∧n ∧ A, En ). We then say that
/ (E 0 )n is a stable A1 -weak equivalence a morphism of spectra (En )n n if it induces isomorphisms on cohomology groups for every simplicial sheaf A.
By inverting stable A1 -weak equivalences in SpectTs (X) and SpectTtr (X) we get respectively the categories SH(X) and DM(X). Let U be a smooth X-scheme. We can associate to U the pointed simplicial sheaf U+ which is ` simplicially constant, represented by U X and pointed by the trivial map ` /U X . Then, we can associate to U+ its infinite T -suspension X ∞ ΣT (U+ ) given in level n by T ∧n ∧ U+ . This provides a covariant functor M : / SH(X) which associates to U its motive M (U ). Similarly we Sm /X can associate to U the complex Ztr (U ), concentrated in degree zero, and then take its infinite suspension given in level n by Ztr (An ×U )/Ztr ((An −0)×U ) ' L
L
T ⊗n ⊗ U . This also gives a covariant functor M : Sm /X † This map of simplicial sets is the stalk of A• the Nisnevich topology.
/
/ DM(X) .
B• at the point u ∈ U with respect to
The Motivic Vanishing Cycles and the Conservation Conjecture
21
The images in SH(X) and DM(X) of the identity X-scheme are respectively denoted by IX and ZX . When there is no confusion we will drop the index X. Remark 1.3.1. Sometimes it is useful to stop in the middle of the above construction and consider the homotopy category HoA1 (X) of step 2. The abelian version with transfers of HoA1 (X) is the category DMeff (X) which is used at the end of the paper. This is the category of effective motives whose objects are complexes of Nisnevich sheaves with transfers and morphisms obtained by inverting A1 -weak equivalences. Remark 1.3.2. One can also consider the categories SHQ (X) and DMQ (X) obtained from SH(X) and DM(X) by killing torsion objects (using a Verdier localization) or equivalently by repeating the above three steps using simplicial sheaves and complexes of sheaves with transfers of Q-vector spaces (instead of sets and abelian groups). It is important to note that the categories SHQ (X) and DMQ (X) are essentially the same at least for X a field. Indeed, an unpublished result of Morel (see however the announcement [27]) claims that SHQ (k) decomposes into DMQ (k)⊕?(k) with ?(k) a ”small part” equivalent to the zero category unless the field k is formally real (i.e., if (−1) is not a sum of squares in k). Remark 1.3.3. The triangulated categories SH(X) and DM(X) have infinite direct sums. It is then possible to speak about compact motives. A motive M is compact if the functor hom(M, −) commutes with infinite direct sums (see [30]). If U is a smooth X-scheme, then its motive M (U ) (in SH(X) or DM(X)) is known to be compact (see for example [33]). Therefore, the triangulated categories with infinite sums SH(X) and DM(X) are compactly generated in the sense of [30]. We shall denote SHct (X) and DMct (X) the triangulated subcategories of SH(X) and DM(X) whose objects are the compact ones. The letters ct stand for constructible and we shall call them the categories of constructible motives (by analogy with the notion of constructible sheaves in ´etale cohomology considered in [2]). The elementary functorial operators f ∗ , f∗ and f# of the categories SH(−) and DM(−) are defined by deriving the usual operators f ∗ , f∗ and f# on the level of sheaves. For HoA1 (−), the details can be found in [28]. It is possible to extend these operators to spectra (see [34]). For DM(−) one can follow the same construction. Details will appear in [6]. The tensor product is obtained by using the category of symmetric spectra. The details for SH(−) can be found in [20]. For DM(−) this will be included in [6]. Using the elementary functorial operators: f ∗ , f# , f∗ and ⊗, it is possible to fully develop the Grothendieck formalism of the six operators (see chapters
22
J. Ayoub
I and II of [3]). For example, assuming resolution of singularities one can prove that all the Grothendieck operators preserve constructible motives. Except for the monodromy triangle, the formalism of motivic vanishing cycles can be developed equally using the categories SH(−) or DM(−). In fact, one can more generally work in the context of a stable homotopical 2-functor. See [3] for a definition of this notion and for the construction of the functors Ψ in this abstract setting.
1.3.2 Definitions and examples Let B be a base scheme. We fix a diagram j
η
/Bo
i
s
with j (resp. i) an open (resp. closed) immersion. We do not suppose that B is the spectrum of a DVR or that s is the complement of η. Every time / B , we form the commutative diagram we are given a B-scheme f : X with cartesian squares Xη fη
η
j
j
/Xo
i
Xs . fs
f
/Bo
i
s.
We recall the following definition from [3], chapter III: Definition 1.3.4. A specialization system sp over (B, j, i) is given by the following data: (i) For a B-scheme f : X
/ B , a triangulated functor:
spf : SH(Xη ) (ii) For a morphism g : Y
/ SH(Xs )
/ X a natural transformation of functors:
αg : gs∗ spf
/ spf ◦g gη∗ .
These data should satisfy the following three axioms: • The natural transformations α? are compatible with the composition of / Y , the morphisms. More precisely, given a third morphism h : Z
The Motivic Vanishing Cycles and the Conservation Conjecture
23
diagram / spf gh (g ◦ h)∗η
(g ◦ h)∗s spf ∼
/ h∗s spf g gη∗
h∗s gs∗ spf
∼
/ spf gh h∗η gη∗
is commutative, • The natural transformation αg is an isomorphism when g is smooth, / gs∗ spf ◦g by the • If we define the natural transformation βg : spf gη∗ composition spf gη∗
/ gs∗ gs∗ spf gη∗
αg
/ gs∗ spf g gη∗ gη∗
/ gs∗ spf g
then βg is an isomorphism when g is projective. Remark 1.3.5. A morphism sp
/ sp0 of specialization systems is a col-
/ sp0 , one for every B-scheme f , lection of natural transformations spf f commuting with the αg , i.e., such that the squares
gs∗ spf
/ spf g gη∗
/ sp0 fη∗ fg
gs sp0f are commutative.
Remark 1.3.6. Let us keep the notations of the Definition 1.3.4. It is possible to construct from α? two natural transformations (see chapter III of [3]) spf ◦g gη!
/ g ! spf s
and
gs! spf ◦g
/ spf gη! .
These natural transformations are important for the study of the action of the duality operators on the motivic nearby cycles functors in paragraph 1.4.5. However, we will not need them for the rest of the paper. Remark 1.3.7. The above definition makes sense for any stable homotopical 2-functor from the category of schemes to the 2-category of triangulated categories (see chapter I of [3]). In particular, one can speak about specialization systems in DM(−), SHQ (−) and of course in D+ (−, Λ). For example, the family of nearby cycles functors Ψ = (Ψf )f ∈Fl(Sch) of the paragraph 1.2.1 is in a natural way a specialization system in D+ (−, Λ) with base (S, j, i).
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J. Ayoub
Example 1.3.8. It is easy to produce examples of specialization systems. The most simple (but still very interesting) example is what we call in chapter III of [3] the canonical specialization system χ. It is defined by χf (A) = i∗ j∗ (A). Example 1.3.9. Given a specialization system sp and an object E ∈ SH(η), we can define a new specialization system by the formula: sp0f (−) = spf (−⊗ fη∗ E). In the same way, given an object F of SH(s), we define a third specialization system by the formula: sp00f (−) = spf (−) ⊗ fs∗ F . 1.3.3 The basic results We state here some (non-trivial) results that follow from the axioms of Definition 1.3.4. For the proofs (which are too long to be included here) the reader can consult chapter III of [3]. For simplicity, we shall stick to the case where B is an affine, smooth and geometrically irreducible curve over a field k of characteristic zero, s a closed point of B and η a non-empty open subscheme of B − s or the generic point of B. We fix a section π ∈ Γ(B, OB ) which we suppose to have a zero of order one on s and to be invertible on η. We then define for n ∈ N, two simple B-schemes: / B the obvious morphism, / B the obvious morphism. • Bn0 = B[t, u, u−1 ]/(tn − u.π) and e0n : Bn0
• Bn = B[t]/(tn − π) and en : Bn
Recall that the unit objects of SH(X) and DM(X) were respectively denoted by I = IX and Z = ZX . We shall also denote by Q = QX the unit object of DMQ (X). The proofs of the following three theorems are in [3], chapter III. Theorem 1.3.10. 1- Let sp be a specialization system over (B, j, i) for SH (resp. for DM). Suppose that for all n ∈ N, the objects: • spen (I) ∈ Ob(SH((Bn )s )) (resp. spen (Z) ∈ Ob(DM((Bn )s ))), • spe0n (I) ∈ Ob(SH((Bn0 )s )) (resp. spe0n (Z) ∈ Ob(SH((Bn0 )s ))), /B, are constructible (see remark 1.3.3). Then for any B-scheme f : X and any constructible object A of SH(Xη ) (resp. DM(Xη )), the object spf (A) is constructible. 2- Let sp be a specialization system over (B, j, i) for DMQ (−). Suppose that for all n ∈ N, the objects spen (Q) ∈ DMQ (s) are constructible. Then for / B , and any constructible object A ∈ DMQ (Xη ), any B-scheme f : X the object spf (A) is constructible.
The Motivic Vanishing Cycles and the Conservation Conjecture
25
The following result will play an important role: / sp0 be a morphism between two speTheorem 1.3.11. 1) Let sp cialization systems over (B, j, i) for SH (resp. DM). Suppose that for every n ∈ N, the induced morphisms:
• spen (I)
/ sp0 (I) (resp. spe (Z) en n
/ sp0 (Z) ), en
• spe0n (I)
/ sp0e0 (I) (resp. spe0 (Z) n n
/ sp0e0 (Z) ), n
are isomorphisms. / B , and any constructible object Then for any B-scheme f : X A of SH(Xη ) (resp. of DM(Xη )) the morphism spf (A)
/ sp0 (A) f
is an isomorphism. When spf and sp0f both commute with infinite sums, the constructibility condition on A can be dropped. 2) If we are working in DMQ (−) the same conclusions hold under the following weaker condition: For every n ∈ N the morphisms / sp0 (Q) are isomorphisms. spen (Q) en Remark 1.3.12. In part 2 of Theorems 1.3.10 and 1.3.11, we cannot replace DMQ by SHQ . Indeed, we use in an essential way the fact that the stable homotopical 2-functor DMQ is separated (like ”separated” for presheaves) (see chapter II of [3]), that is, the functor e∗ is conservative for a finite surjective morphism e. This property for DMQ is easily proved by reducing to a finite field extension and using transfers. It fails for SHQ already for / Spec(R) . However, using Morel’s result [27], the morphism Spec(C) one sees that SHQ is separated when restricted to the category of schemes on which (−1) is a sum of squares. The previous two theorems are deduced using resolution of singularities from the following result: /B Theorem 1.3.13. Let sp be a specialization system over B. Let f : X be a B-scheme. Suppose that X is regular, Xs is a reduced normal crossing divisor in X and fix a smooth branch D ⊂ Xs . We denote by D0 the smooth locus of f contained in D, i.e., D0 is the complement in Xs of the union of all the branches that meet D properly. Let us denote by u the closed immersion D ⊂ Xs and v the open immersion D0 ⊂ D. The obvious morphism / v∗ v ∗ [u∗ spf fη∗ ] . Fur/ v∗ v ∗ induces an isomorphism: [u∗ spf fη∗ ] id thermore, if p is the projection of D0 over s then v ∗ [u∗ spf fη∗ ] ' p∗ spidB .
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J. Ayoub
Remark 1.3.14. The previous theorem says that in good situations, the knowledge of spidB I suffices to determine (up to extension problems) the motive spf I. Remark 1.3.15. If we work in DMQ (−) and over a field of characteristic zero, then we can drop the condition that Xs is reduced in Theorem 1.3.13 / v∗ v ∗ [u∗ spf fη∗ ] . However it and still have an isomorphism: [u∗ spf fη∗ ] is no longer true that v ∗ [u∗ spf fη∗ ] ' p∗ spidB , unless the branch D is of multiplicity one. Example 1.3.16. To help the reader understand the content of Theorem 1.3.13, we use it to make a computation in a familiar situation: • We shall work with ´etale cohomology, that is in the stable homotopical 2-functor D+ (−, Λ), and with the nearby cycles specialization system RΨ. / S to be a semi-stable curve (not necessarily proper) • We take f : X over a henselian discrete valuation ring S. We suppose that Xs has two branches D1 and D2 that meet in a point C = D1 ∩ D2 . We will compute the cohomology sheaves of the complex of nearby cycles RΨf Λ. We have the following commutative diagram: > D1 CC | CC u1 | c1 | CC || CC | ! || c / Xs CB = BB {{ BB {{ B { c2 BB {u {{ 2
D2 .
/ Di the inclusion of the smooth For i ∈ {1, 2} we denote vi : Di − C locus of f in Di . By Theorem 1.3.13 the restriction u∗i RΨf Λ of RΨf Λ to Di is given by Rvi∗ Λ. As vi is the complement of a closed point in a smooth curve over a field we know that Rp vi∗ Λ = 0 for p ∈ / {0, 1}, R0 vi∗ Λ = ΛDi and R1 vi∗ Λ = ci∗ Λ(−1). This immediately gives that Rp Ψf Λ = 0 for p ∈ / {0, 1}, R0 Ψf Λ = ΛXs and R1 Ψf Λ = c∗ Λ(−1).
1.4 Constructing the vanishing cycles formalism The goal of this section is to construct in the motivic context a specialization system (in the sense of 1.3.4) that behaves as much as possible like the nearby cycles functors in ´etale cohomology. We begin by explaining why the definition of RΨf given in paragraph 1.2.1 does not give the right functors in
The Motivic Vanishing Cycles and the Conservation Conjecture
the motivic context. Let S be as in 1.2.1. For an S-scheme f : X consider the functor
27 /S
/ DM(Xs )
Φf : DM(Xη )
defined by the formula Φf (A) = ¯i∗ ¯j∗ A|Xη¯ . It is easy to check that Φ is indeed a specialization system over S. There is at least one problem with this definition: we have Φid (Z) 6= Z (this means for example that Φid cannot be monoidal). Indeed, let k be an algebraically closed field of characteristic zero and suppose that S is the henselization of the affine line over A1k = /S Spec(k[T ]) in its zero section. In this case, S¯ is the limit of Sn where Sn = S[T 1/n ]. To compute ΦidS Z, we consider the diagrams: ηn (en )η
η
jn
j
/ Sn o
in
s
en
/So
i
s.
By definition, ΦidB Z is the colimit over n ∈ N× of i∗n jn∗ Z. By an easy computation, we have that i∗n jn∗ I = Z ⊕ Z(−1)[−1] and for n dividing m / i∗ jm∗ Z is given by the matrix: the morphism i∗n jn∗ Z m
1 0
0 m n
: Z ⊕ Z(−1)[−1]
/ Z ⊕ Z(−1)[−1] .
Because we are working with integral coefficients, it follows that ΦidS Z is isomorphic to Z ⊕ Q(−1)[−1]. This problem disappears in ´etale cohomology, where the colimit of the diagram (−× m n : Λ 7−→ Λ)n divides m is zero because Λ is torsion.
1.4.1 The idea of the construction We will construct the specialization system Ψ out of the canonical specialization system χ of example 1.3.8. It is possible to make the definition over an arbitrary base of dimension one, but unfortunately our main results are known to hold only over an equi-characteristic zero base. This is because Theorems 1.3.10 and 1.3.11 are not true (as they are stated here) when the special point s is of positive characteristic † even if one assumes resolution of singularities. For some details on more general situations, the reader can † Part 2 of Theorems 1.3.10 and 1.3.11 is valid over an arbitrary base of dimension 1 if in the condition we replace bn by any quasi-finite extension of B (see [3]).
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J. Ayoub
consult [3], chapter III. In this paper we consider only the base η = GmL
j
/ B = A1 o k
i
s=0
LL ttt LL ttttt t LL t t ttt LL LL ttttttt % t
k
where k is a field of characteristic zero (not necessarily algebraically closed) and s is the zero section of the affine line. Note that whenever we have a smooth affine curve C, a closed point x ∈ C and a function πC ∈ Γ(C, OC ) invertible on C − x with πC (x) = 0, we get by restriction a specialization / A1 . So the real restriction system over C induced by the map πC : C k is not to work over A1k but to work over an equi-characteristic zero base. In the rest of the section, we will denote by π the variable so that A1k = Spec(k[π]). We shall also use the notations in section 1.3. Theorem 1.3.11 shows that a specialization system is to a large extent determined by its values at Iη ∈ SH(η). Thus our main objective will be to find a specialization system Υ over B such that: ΥidB I = I. We then modify Υ by a variant of the ”Grothendieck trick” to get the nearby cycles specialization system Ψ. In order to obtain Υ from the canonical specialization system χ = i∗ j∗ of Example 1.3.8 one has to kill in χidB I = I ⊕ I(−1)[−1] the component I(−1)[−1]. A way to do this is to look for an object P ∈ SH(η) such that χid P = I and then take the specialization system χ(− ⊗ P ) as in Example 1.3.9. In DMQ there is a natural candidate for such P given by the motivic ”logarithmic sheaf” over Gm (see § 1.4.6). With integral coefficients, we do not know of any natural object P ∈ SH(Gm) with this property, but there is a diagram A∨ of motives in SH(Gmk ) such that the homotopy colimit of χidB (A∨ ) is indeed I. In fact, A∨ can be taken to be the ”simplicial motive” obtained from the ”cosimplicial motive” A of the next paragraph by applying Hom(−, I) component-wise. The simplicial motive A∨ gives the motivic analogue of (the dual of) the Rapoport-Zink bicomplex Q•,• in ´etale cohomology (see remark 1.4.6). So we will define the unipotent part Υ of the nearby cycles to be the ”homotopy colimit” of the simplicial specialization system χ(− ⊗ A∨ ).
1.4.2 The cosimplicial motive A and the construction of Υ We mentioned that our construction is inspired by the Rapoport-Zink bicomplex Q•,• . We also pointed out that this bicomplex is built from i∗ Rj∗ Λ / Λ(1)[1] . The motivic anaand the fundamental class morphism θ : Λ
The Motivic Vanishing Cycles and the Conservation Conjecture
29
logue of i∗ Rj∗ Λ is of course i∗ j∗ I. We describe the motivic fundamental class in Definition 1.4.1 below. Recall that given an X-scheme U and a section / U , we denote by (U, s) the X-scheme pointed by s. The motive s: X M (s)
/ M (U ) . M (U, s) of a pointed X-scheme (U, s) is the cofiber of M (X) Moreover, we have a canonical decomposition M (U ) = IX ⊕ M (U, s). For example, the motive of (GmX , 1) is by definition IX (1)[1] and M (GmX ) = IX ⊕ IX (1)[1].
Definition 1.4.1. The motivic fundamental class θ : I morphism in SH(Gm) defined by the diagram
id
/ IGm (1)[1]
θ
IGm
M [Gm 7−→ Gm]
∆
/ I(1)[1] is the
pr1 / M [Gm × Gm − 7 → Gm]
pr1 / M [Gm × (Gm, 1) − 7 → Gm] f
where M is the ”associated motive” functor, [X 7−→ Gmk ] denotes a Gmscheme X, pr1 is the projection to the first factor and ∆ is the diagonal immersion. In DM(Gm), one can equivalently define θ as an element of the motivic cohomology group H1,1 (Gm) = Γ(Gm, O× ) because of the identification homDM(Gmk ) (Z, Z(1)[1]) = homDM(k) (Gm, Z(1)[1]). It corresponds then to the class of the variable π ∈ k[π, π −1 ]. It is an easy exercise to check that the ´etale realization of θ gives indeed the classical fundamental class. One difference with the classical situation is that θ ◦ θ is non zero even in DM. In fact θ ◦ θ corresponds in the Milnor K-theory group K2M (k[T, T −1 ]) to the symbol {T, T } = {T, −1} which is 2-torsion. Of course one can kill 2-torsion, and try to find a representative θ of θ such that θ2 is zero in the model category. We shall do something different. Note first the following lemma: Lemma 1.4.2. Let C be a category having direct products. Consider a diagram in C: A
f
/Bo
f0
A0 .
˜ B A0 )• in C such that for n ∈ N, we There exists a cosimplicial object (A× have: ˜ B A0 )n = A × B × · · · × B × A0 = A × B n × A0 , • (A×
30
• • • •
J. Ayoub
d0 (a, b1 , . . . , bn , a0 ) = (a, f (a), b1 , . . . , bn , a0 ), dn+1 (a, b1 , . . . , bn , a0 ) = (a, b1 , . . . , bn , f 0 (a0 ), a0 ), For 1 ≤ i ≤ n, di (a, b1 , . . . , bn , a0 ) = (a, b1 , . . . , bi , bi , . . . , bn , a0 ), For 1 ≤ i ≤ n − 1, si (a, b1 , . . . , bn , a0 ) = (a, b1 , . . . , bi , bi+2 , . . . , bn , a0 )
where a, a0 and the bi are respectively elements of hom(X, A), hom(X, A0 ) and hom(X, B) for a fixed object X of C. Moreover, if f is an isomorphism / A0 is a cosimplicial cohomotopy ˜ B A0 ) then the obvious morphism (A× equivalence†, where A0 is the constant cosimplicial object with value A0 . We apply Lemma 1.4.2 to the following diagram in the category Sm /Gm of smooth Gm-schemes: id
[Gm 7−→ Gm]
∆
(x,1) pr1 id / [Gm × Gm − 7 → Gm] o [Gm 7−→ Gm].
We denote by A• the cosimplicial Gm-scheme thus obtained. We will usually look at A• as a cosimplicial object in the model category of T -spectra over Gm: SpectTs (Gm) or SpectTtr (Gm). We claim (and show in remark 1.4.6) that for a semi-stable B-scheme f the simplicial object χf Hom(A• , I) is the motivic analogue of the double-complex Q•,• of the paragraph 1.2.2. This motivates the following definition: / A1 be a morphism of schemes. Let E Definition 1.4.3. Let f : X k T be an object of Spects (Xη ). We put 1
1
Υf (E) = Tot [LA i∗ RA j∗ Hom(fη∗ A• , (E)A1 −F ib )] 1
1
where (E)A1 −F ib is a functorial fibrant replacement and LA i∗ and RA j∗ are the left and right derived functors of i∗ and j∗ on the level of T -spectra. (Everything being with respect to the stable A1 -model structure.) This functor sends stable A1 -weak equivalences to stable A1 -weak equivalences / SH(Xs ) which we and induces a triangulated functor Υf : SH(Xη ) call the motivic unipotent nearby cycles functor. Remark 1.4.4. Recall that the functor Tot associates to a simplicial object in a model category M its homotopy colimit. It is simply the left derived / M that associates to a simplicial functor of the functor π0 : ∆op M // E . In our case object E• the equalizer of the two first cofaces: E1 0 the Tot functor has a simple description. Indeed, a simplicial object in the † By a cosimplicial cohomotopy equivalence we mean that if we view this cosimplicial morphism as a simplicial morphism between simplicial objects with values in the category C op , then it is a simplicial homotopy equivalence. Note that the notion of a simplicial homotopy equivalence is combinatorial and makes sense for any category.
The Motivic Vanishing Cycles and the Conservation Conjecture
31
category of simplicial sheaves is simply a bisimplicial sheaf and its homo/∆×∆. topy colimit is given by the restriction to the diagonal ∆ Similarly, the homotopy colimit of a simplicial object in the category of complexes of sheaves is given by the total complex associated to the double complex obtained by taking the alternating sum of the cofaces. Remark 1.4.5. A better way to define the functors Υf is to use the categories SH(−, ∆) which are obtained as the homotopy categories of the model categories ∆op SpectTs (−). One can take for example the Reedy model structure induced from the stable A1 -model structure on SpectTs (−) (or another one depending on the functor we want to derive). Our functor Υf is then the following composition of triangulated functors: SH(Xη )
Hom(A• ,−)
/ SH(Xη , ∆)
j∗
/ SH(X, ∆)
i∗
/ SH(Xs , ∆)
Tot
/ SH(Xs ).
Even better, one can use the notion of algebraic derivator to define Υf using only basic operators of the form a∗ , a∗ and a# . This is the point of view we use in [3]. Remark 1.4.6. Let us explain the relation between our definition and the Rapoport-Zink bicomplex Q•,• . We will work with Nisnevich sheaves with transfers over Sm /Gm. Let N(A) be the normalized complex of sheaves with transfers associated to the cosimplicial sheaf Ztr (A• ). The complex N(A) is concentrated in (homological) negative degrees and is given by i n N(A)−n = Ker(⊕n−1 i=1 s : Ztr (A )
/ ⊕n−1 Ztr (An−1 ) ) i=1
for n ≥ 0.
pr1
Recall that An = [(Gm)n+1 7−→ Gm] and si is given by the projection that forgets the (i + 1)–st coordinate. Because of the decomposition Ztr (GmX ) = Ztr (X) ⊕ Ztr (GmX , 1) it follows that N(A)−n is isomorphic to Ztr [Gm × (Gm, 1)∧n 7−→ Gm]. In particular, viewed as a complex of objects in DM(Gm), the complex N(A) looks like: ...
/0
/ IGm
/ IGm (1)[1]
/ ...
/ IGm (n)[n]
/ ....
/ IGm (1)[1] is It is easy to check that the first non-zero differential IGm given by the motivic fundamental class θ of Definition 1.4.1. One can prove / IGm (n)[n] is always given that the n-th differential IGm (n − 1)[n − 1] by θ + where, is zero in ´etale cohomology. It is now clear that when we apply Hom(−, I) component-wise and then the functor χf we get a motivic analogue of Q•,• .
Remark 1.4.7. Markus Spitzweck gave us a topological interpretation of the
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J. Ayoub
functor Υf which gives yet another motivation for our definition. His interpretation is as follows. One can look at the cosimplicial object A• as the space of paths in Gm with end-point equal to 1. This means that A• is in a sense the universal cover of Gm. When taking Hom(A• , E), we are looking at the sections over the universal cover of Gm with values in E. Finally, when applying i∗ j∗ , we are taking the restriction of these sections to the ”boundary” of the universal cover. This picture is of course very similar to the classical one we have in the analytic case. Proposition 1.4.8. The family (Υ? ) extends naturally to a specialization system over A1k . It is called the unipotent nearby cycles specialization system. Proof We have to define the natural transformations α? and prove that the axioms of Definition 1.3.4 hold. Suppose given a morphism of A1k -schemes g
/X AA AA f A
Y A A
A1k .
We define a natural transformation αg : gs∗ Υf composition
/ Υf ◦g gη∗ by taking the
gs∗ Tot i∗ j∗ Hom(fη∗ A• , −) O
∼
Tot gs∗ i∗ j∗ Hom(fη∗ A• , −)
/ Tot i∗ j∗ gη∗ Hom(fη∗ A• , −)
Tot i∗ j∗ Hom(gη∗ fη∗ A• , gη∗ (−)). It is easy to check that these α? are compatible with composition (see the third chapter of [3] for details). Furthermore, αg is an isomorphism when g is smooth by the ”base change theorem by a smooth morphism” and the formula gη∗ Hom(−, −) = Hom(gη∗ (−), gη∗ (−)). We still need to check that βg is an isomorphism for g projective. It is easy to see that βg is given by the
The Motivic Vanishing Cycles and the Conservation Conjecture
33
composition Tot i∗ j∗ Hom(fη∗ A• , gη∗ (−))
∼
/ Tot gs∗ i∗ j∗ Hom(gη∗ fη∗ A• , −)
Tot i∗ j∗ gη∗ Hom(gη∗ fη∗ A• , −)
gs∗ Tot i∗ j∗ Hom(gη∗ fη∗ A• , −). The first map is an adjunction formula and is always invertible. The second is an isomorphism when g is projective due to the ”base change theorem by a projective morphism” (proved in chapter I of [3]). The last morphism is also an isomorphism when g is projective because then gs∗ = gs! (see also the first chapter of [3]) and the operation gs! commutes with colimits. / I . This morphism induces Let us denote a the natural morphism A• / Υf which is a morphism of a natural transformation a : i∗ j∗ = χf specialization systems. We have the following normalization, which is the main reason for our definition: / i∗ j∗ IGm
Proposition 1.4.9. The composition: Is isomorphism.
/ Υid IGm is an
Proof Recall the commutative diagram j
GmB
/ A1 o k
BB BB q BBB !
i
s
p
k.
We define a natural transformation: q∗ sition: q∗
∼
/ p∗ j∗
/ p∗ i∗ i∗ j∗
/ i∗ j∗ by the following compo∼
/ i∗ j∗ .
Note that this natural transformation is an isomorphism when applied to q ∗ . / i∗ j∗ I(m) are isomorphisms for every In particular, the maps q∗ I(m) m ∈ Z. This implies that the natural map of simplicial objects q∗ Hom(A• , I) is an isomorphism.
/ i∗ j∗ Hom(A• , I)
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J. Ayoub
To prove the proposition, we only need to show that the composition I
/ Tot q∗ Hom(A• , I)
/ q∗ I
is invertible. By the adjunction formula we have an identification: Hom(q# A• , I) ' q∗ Hom(A• , I). It is then sufficient to check that the morphism of simplicial / Hom(q# A• , I) (where I is considered as a constant simpliobjects I cial motive) is a simplicial homotopy equivalence. The latter is induced / I which we check to be a from a map of cosimplicial objects q# A• cosimplicial cohomotopy equivalence. The cosimplicial motive q# A• is the one associated to the cosimplicial k-scheme obtained by forgetting in A• the structure of Gm-scheme. An easy computation shows that this cosimplicial scheme is obtained using Lemma 1.4.2 from the diagram of k-schemes id
Gm
/ Gm o
1
k.
/ I is induced via the projection to the Furthermore the map q# A• ˜ Gm k. By the last assertion of the Lemma 1.4.2, this is second factor of Gm× indeed a cosimplicial cohomotopy equivalence.
1.4.3 The construction of Ψ Now we come to the construction of the nearby cycles functors. For this / A1 which are given by elevawe introduce the morphisms en : A1k k tion to the n-th power. Note that these morphisms are isomorphic to the / B we introduce in paragraph 1.3.3 when B = A1 . Given a en : Bn k morphism f : X
/ A1 , we form the cartesian square k
Xn fn
A1k
en
en
/X
f
/ A1 . k
Lemma 1.4.10. For any non zero positive integer n there is a natural transformation µn : Υf
/ Υf n (en )∗η .
Moreover, if d is another non zero positive integer, we have: (f n )d = f nd , end = en ◦ ed and µnd is given by the composition Υf
µn
/ Υf n (en )∗η
µd
/ Υ(f n )d (ed )∗η (en )∗η ' Υf nd (end )∗η .
The Motivic Vanishing Cycles and the Conservation Conjecture
Υf
Proof There is an obvious transformation: given by the composition: Υf
αen
/ (en )s∗ (en )∗ Υf s
35
/ (en )s∗ Υen ◦f n (en )∗η
/ (en )s∗ Υf ◦en (en )∗η = (en )s∗ Υen ◦f n (en )∗η .
To define µn we need to specify a transformation νn : (en )s∗ Υen ◦f n
/ Υf n .
/ Xs induces an isomorphism on the associated reAs (en )s : (X n )s duced schemes, the functor (en )s∗ is an equivalence of categories. Moreover, the two functors (en )s∗ χen ◦f n and χf n are naturally isomorphic. To obtain our νn , it is then sufficient to define a map of cosimplicial objects: / (en ◦ f n )∗η A• . (f n )∗η A• First note that (en )∗η A• is the cosimplicial scheme obtained by Lemma 1.4.2 from the diagram (x,xn )
id
[Gm 7−→ Gm]
(x,1) pr1 id / [Gm × Gm − 7 → Gm] o [Gm 7−→ Gm].
The commutative diagram of Gm-schemes id
[Gm 7−→ Gm]
id
∆
(x,xn )
[Gm 7−→ Gm]
pr1 / [Gm × Gm − 7 → Gm] o
id
[Gm 7−→ Gm]
(x,y n )
(x,1) pr1 id / [Gm × Gm − 7 → Gm] o [Gm 7−→ Gm]
/ (en )∗η A• . This gives for any
induces a map of cosimplicial schemes A• A1k -scheme f a map (f n )∗η (A• )
∆
/ (f n )∗η (en )∗η (A• ) ' (en )∗η fη∗ (A• ) ' (f n ◦ en )∗η (A• ).
The last assertion is an easy verification which we leave to the reader. Definition 1.4.11. We define the (total) motivic nearby cycles functor Ψf : SH(Xη )
/ SH(Xs )
by the formula: Ψf = HoColim n∈N× Υf n (en )∗η . Remark 1.4.12. Because the homotopy colimit is not functorial in a triangulated category, one needs to work more to get a well–defined triangulated functor. A way to do this is to define categories SH(−, N× ) corresponding to N× -diagrams of spectra. Then extend the functor Υf to a more elaborate one that goes from SH(Xη ) to SH(Xs , N× ) and associates to A the full diagram (Υf n (en )∗η A)n . Finally apply the colimit functor
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J. Ayoub
/ SH(Xs ) . For more details, the reader can consult the SH(Xs , N× ) third chapter of [3].
Proposition 1.4.13. The family (Ψ? ) extends naturally to a specialization system over A1k . It is called the (total) nearby cycles specialization system. We have the following simple lemma: Lemma 1.4.14.
1) Suppose that the morphism f : X
/ A1 is k
/ Ψf I is an isomorsmooth. Then the canonical morphism: Υf I phism. ∼ / Ψf 2) For every n, there exist a natural isomorphism Ψf n (en )∗η making the triangle / Ψf n (en )∗η MMM MMM ∼ MMM MM&
Υf n (en )∗η
Ψf
commutative. Proof The first point is easy, and comes from the fact that the two objects are isomorphic to I. The second point is left as an exercise. Our next step is the computation of Ψen I and Ψe0n I (see the notations of paragraph 1.3.3): Proposition 1.4.15. For every n ∈ N× the canonical morphisms Υ(en )m I
/ Ψ(en ) I
and
Υ(e0n )m I
/ Ψe0 I n
are isomorphisms when m is divisible by n. Proof In both cases, the proofs are exactly the same in the two cases and are based on the fact that the normalizations of (en )m and (e0n )m are smooth A1k -schemes. Indeed let us denote B = A1 , Bn and Bn0 as in paragraph 1.3.3. Then we have: • (Bn )m = Bn ×B Bm = Spec(k[π][t1 ]/(tn1 −π)[t2 ]/(tm 2 −π)). When m is m is Spec(k[π][v]/(v n − divisible by n, the normalization Qm of (B ) n n m
m n etale over Bm . 1)[t2 ]/(tm 2 − π)) with v = t1 /t2 . In particular Qn is ´ 0 m 0 −1 n • (Bn ) = Bn ×B Bm = k[π][t1 , u, u ]/(t1 − u.π)[t2 ][tm 2 − π]. When −1 ][t ]/(tm − π) m is divisible by n its normalization Q0m is k[π][w, w 2 n 2 with wn = u. In particular Q0m n is smooth over Bm .
The Motivic Vanishing Cycles and the Conservation Conjecture
37
/ (Bn )m and t0 : Q0m / (B 0 )m . Consider now the morphisms tm : Qm n m n n They are both finite, and induce isomorphisms on the generic fibers. We have the following commutative diagram:
(tm )s∗ Υ(en )m ◦tm I βtm
Υ(en )m I
(a)
/ (tm )s∗ Ψ(e )m ◦t I n m
(1.1)
βtm
/ Ψ(en )m I
(b)
/ Ψe . n
The two vertical arrows are the transformations βtm of Definition 1.3.4 modulo the identification (tm )η = id. As tm is a finite map, these two arrows are invertible. By Lemma 1.4.14, we know that the horizontal arrows labeled (a) and (b) are also invertible. Thus we are done with the first case. The second case is settled using exactly the same argument. The proof of Proposition 1.4.15 gives a more precise statement. It computes exactly the motives Ψen I and Ψe0n I. Indeed in the diagram (1.1) we have Υ(en )m ◦tm I = I. It follows that: • Ψen I = (tn )s∗ I, • Ψe0n I = (t0n )s∗ I. These are Artin 0-motives and they are constructible. Theorem 1.3.10 then applies to give us the following: Theorem 1.4.16. For any quasi-projective f , the functor Ψf takes constructible motives of SH(Xη ) to constructible motives of SH(Xs ). Later on, we will need the following result: / A1 be a finite type morphism. For any Theorem 1.4.17. Let f : X k constructible object A ∈ SH(Xη ) there exists an integer N such that the / Ψf (A) is an isomorphism for non natural morphism: Υf m (em )∗η A zero m divisible by N .
Proof Note that for every non-zero n, the family of functors (Υf n (en )η )f is a / Ψf specialization system over A1 and the obvious morphisms Υf n (en )∗η give a morphism of specialization systems. The conclusion of the theorem follows from Proposition 1.4.15 and a refined version of Theorem 1.3.11. / sp0 I and Indeed, suppose that in 1.3.11 we only knew that spen I en spe0n I
/ sp0e0 I are invertible for n dividing a fixed number N . Then it is n
38
J. Ayoub
/ sp0 (A) is invertible for A ”comstill possible to conclude that spf (A) f ing” from varieties with semi-stable reduction over BN . For more details, see the third chapter of [3].
1.4.4 Pseudo-monoidal structure We continue our study of the functors Ψf by constructing a pseudo-tensor structure on them. We denote A• ⊗ A• the bicosimplicial Gm-scheme (or its associated motive) obtained by taking fiber products over Gmk . We will denote (A⊗A)• the cosimplicial object obtained from A• ⊗A• by restricting / ∆ × ∆ . We have the following lemma: to the diagonal ∆ Lemma 1.4.18. The cosimplicial scheme (A⊗A)• is the one obtained from Lemma 1.4.2 applied to the following diagram in Sm /Gm:
id
[Gm 7−→ Gm]
∆3
(x,1,1)
pr1 / [Gm × Gm × Gm − 7 → Gm] o
id
[Gm 7−→ Gm]
where ∆3 is the diagonal embedding. / (A ⊗ A)• be the morphism of cosimplicial objects inLet m : A• duced by the commutative diagram
id
[Gm 7−→ Gm]
id
[Gm 7−→ Gm]
∆
∆3
pr1 / [Gm × Gm − 7 → Gm] o
(x,1)
id
[Gm 7−→ Gm]
(x,y,y)
(x,1,1) pr1 id / [Gm × Gm × Gm − 7 → Gm] o [Gm 7−→ Gm].
Definition 1.4.19. Let f : X
/ A1 be a morphism. We define a bik
natural transformation m : Υf (−) ⊗ Υf (−0 )
/ Υf (− ⊗ −0 ) by the com-
The Motivic Vanishing Cycles and the Conservation Conjecture
39
position Tot i∗ j∗ Hom(A• , −) ⊗ Tot i∗ j∗ Hom(A• , −0 )
Tot [i∗ j∗ Hom(A• , −) ⊗ i∗ j∗ Hom(A• , −0 )]
Tot i∗ j∗ [Hom(A• , −) ⊗ Hom(A• , −0 )]
Tot i∗ j∗ Hom(A• ⊗ A• , − ⊗ −0 ) (a) ∼
Tot i∗ j∗ Hom((A ⊗ A)• , − ⊗ −0 )
Hom(m,−)
Tot i∗ j∗ Hom(A• , − ⊗ −0 ) where the arrow labelled (a) is the identification of the homotopy colimit of a bisimplicial object with the homotopy colimit of its diagonal. One checks (as in chapter III of [3]) that: Proposition 1.4.20. The bi-natural transformation m of the above definition makes Υf into a pseudo-monoidal functor. Moreover the natural / Υf is compatible with the pseudo-monoidal transformation χf = i∗ j∗ structures. Note that the above proposition defines a ”χ-module” structure on Υ in the sense that there exists a binatural transformation m0 : χf (−) ⊗ Υf (−0 )
/ Υf (− ⊗ −0 )
/Υ which is nothing but the composition of the canonical morphism χ 0 with the morphism of the definition. It is easy to check that m is given by
40
J. Ayoub
the following composition: i∗ j∗− ⊗Tot i∗ j∗ Hom(A• , −0 )
Tot i∗ j∗− ⊗ i∗ j∗ Hom(A• , −0 )
/ Tot i∗ j∗ (− ⊗ Hom(A• , −0 ))
Tot i∗ j∗ Hom(A• , − ⊗ −0 ). Corollary 1.4.21. For any object A in SH(Xη ) the composition Υf (A)
Υf (A) ⊗ I
/ Υf (A) ⊗ Υf I
/ Υf (A)
is the identity. Proof Consider the commutative diagram Υf (A)
O
/ Υf (A) ⊗ Υf I O
/ Υf (A)
/ Υf (A) ⊗ f ∗ χid I s
/ Υf (A) ⊗ χf I
/ Υf (A)
Υf (A) ⊗ I (a)
Υf (A)
where the arrow labelled (a) is the one induced from the canonical splitting: χid I → Υid I = I. So we need only to check that the composition of the bottom sequence is the identity. For this we can use the description of the ”χ-module” structure given above. Going back to the definition of Υ, we see that it suffices to check that the composition χf B
/ χf B ⊗ fs∗ i∗ j∗ I
/ χf B ⊗ χf I
/ χf B
is the identity for B ∈ SH(Xη ). This is an easy exercise. In order to extend the pseudo-monoidal structure from Υ to Ψ we use the following lemma: Lemma 1.4.22. With the notations of paragraph 1.4.3, we have a commutative diagram of binatural transformations / Υf n (en )∗η (−) ⊗ Υf n (en )∗η (−0 )
Υf (−) ⊗ Υf (−0 )
Υf (− ⊗ −0 )
/ Υf n (en )∗η (− ⊗ −0 )
∼
/ Υf n (en )∗η (−) ⊗ (en )∗η (−0 ).
The Motivic Vanishing Cycles and the Conservation Conjecture
41
Proof Going back to the definitions we see that we must check the commutativity of the corresponding diagram of cosimplicial objects (A ⊗ A)• O
/ diag[(en )∗η A• ⊗ (en )∗η A• ]
/ (en )∗η (A ⊗ A)• ) O / (en )∗η A•
A• This diagram is obviously commutative.
Lemma 1.4.22 allows us to define a bi-natural transformation Ψf (−) ⊗ Ψf (−0 )
/ Ψf (− ⊗ −0 )
by taking the colimit of the bi-natural transformations Υf n (en )∗η (−) ⊗ Υf n (en )∗η (−0 )
/ Υf n (en )∗η (− ⊗ −0 ).
We have: / A1 , the functor Ψf is naturally Theorem 1.4.23. For every f : X k a pseudo-monoidal functor. Furthermore, the morphisms
χf
/ Υf
/ Ψf
are natural transformations of pseudo-monoidal functors. We have the following important result: Theorem 1.4.24. Let F be an object of SH(η). Then for any f : X and any object A of SH(Xη ), the composition: Ψf (A) ⊗ fs∗ Ψid (F )
/ Ψf (A) ⊗ Ψf fη∗ F
/ A1
/ Ψf (A ⊗ fη∗ F )
is an isomorphism. In particular, Ψid is a monoidal functor. Proof We will apply Theorem 1.3.11 to a well chosen morphism between two specialization systems. These specialization systems are (see Example 1.3.9): (a)
(i) Ψ(a) , given by the formula: Ψf (A) = Ψf (A) ⊗ fs∗ Ψid F , (b)
(ii) Ψ(b) , given by the formula: Ψf (A) = Ψf (A ⊗ fη∗ F ). One sees immediately that the composition in the statement of the theorem / Ψ(b) . Note also defines a morphism of specialization systems: Ψ(a) (a)
(b)
that Ψf and Ψf both commute with infinite sums. So by Theorem 1.3.11, we only need to consider the two special cases:
k
42
J. Ayoub
• f = en and A = I, • f = e0n and A = I. The proofs in these two cases are similar to the proof of Proposition 1.4.15. We will concentrate on the first case and use the notations in the proof of / (Bn )m the normalization of 1.4.15. Recall that we denoted tm : Qm n m (Bn ) . We can suppose that F is of finite type. By Theorem 1.4.17 we can choose a sufficiently divisible m such that: • Ψen (I) ⊗ (en )∗s Ψid (F ) ' Υ(en )m (I) ⊗ ((en )m )∗s Υid (em )∗η (F ) ' (tm )s∗ Υ(en )m ◦tm (I) ⊗ ((en )m )∗s Υid (em )∗η (F ), • Ψen (I) ⊗ Ψen (en )∗η F ' Υ(en )m (I) ⊗ Υ(en )m (em )∗η ((en )m )∗η F ' (tm )s∗ Υ(en )m ◦tm (I) ⊗ (tm )s∗ Υ(en )m ◦tm (em )∗η ((en )m )∗η F • Ψen (I ⊗ (en )∗η F ) ' Υ(en )m (em )∗η (I ⊗ ((en )m )∗η F ) ' (tm )s∗ Υ(en )m ◦tm (em )∗η (I ⊗ ((en )m )∗η F ). Denoting by f the smooth morphism (en )m ◦tm , we end up with the following problem: is the composition Υf I ⊗ fs∗ Υid F
/ Υf I ⊗ Υf fη∗ F
/ Υf (I ⊗ fη∗ F )
invertible? This is indeed the case by Corollary 1.4.21.
1.4.5 Compatibility with duality It is a well-known fact that in ´etale cohomology the nearby cycles functors commute with duality (see for example [15]). We extend this result to the motivic context. We first specify our duality functors. / A1 . We define two duality functors Definition 1.4.25. Let f : X k Dη and Ds on SH(Xη ) and SH(Xs ) by:
(i) Dη (−) = Hom(−, fη! I), (ii) Ds (−) = Hom(−, fs! I). (The ”extraordinary inverse image” operation (−)! is constructed in the first chapter of [3].) Remark 1.4.26. Note that Dη differs by a Tate twist and a double suspension from the usual duality functor on SH(Xη ). Indeed we used fη! I instead of the dualising motive (q ◦ fη )! I (where q is the projection of Gm to k). We define for any f : X in the following way:
/ A1 a natural transformation δf : Ψf Dη k
/ Ds Ψf
The Motivic Vanishing Cycles and the Conservation Conjecture
43
(i) First note that for an object A ∈ SH(Xη ) there is a natural pairing / f ! I. η
A ⊗ Dη (A)
/ f ! I by the following s
(ii) We define a pairing Ψf (A) ⊗ Ψf Dη (A) composition: Ψf (A) ⊗ Ψf Dη (A)
/ Ψf (A ⊗ Dη A)
/ Ψf f ! I η
/ f ! Ψf I = f ! I. s s
(1.2) (iii) Using adjunction, we get from the above pairing the desired natural / Ds Ψf (A) . morphism δf : Ψf Dη (A) Theorem 1.4.27. When A is constructible in SH(Xη ) , the morphism δf : Ψf Dη (A)
/ Ds Ψf (A)
is an isomorphism. Proof Once again the proof is based on Theorem 1.3.11. First note that when A is constructible, Dη Dη (A) = A (by [3], chapter II). Thus we only / Ds Ψf Dη is need to prove that the natural transformation δf0 : Ψf an isomorphism when evaluated on constructible objects. Note that δf0 is nothing but the second adjoint deduced from the pairing (1.2). Now we have two specialization systems: Ψ and Ds ΨDη and a morphism δ?0 between them. Using Theorem 1.3.11 we only need to check the theorem when f = en or e0n and A = I. This is done using the same method as in the proof of Proposition 1.4.15. For more details, the reader can consult the third chapter of [3].
1.4.6 The monodromy operator In this section, we construct the monodromy operator on the unipotent part of the nearby cycles functors Υf . In order to do this we will work in DMQ (−). Note that one extends our definition of the specialization systems Υ and Ψ from SH(−) to DM(−) by using the same definitions. The main result of this paragraph is: / A1 be an A1 -scheme. There exists a Theorem 1.4.28. Let f : X k natural (in A) distinguished triangle in the triangulated category of motives with rational coefficients DMQ (Xs ) :
Υf (A)(−1)[−1]
/ χf (A)
/ Υf (A)
N
/ Υf (A)(−1).
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J. Ayoub
We call N the monodromy operator. Moreover, when A is of finite type, this operator is nilpotent. Our strategy is as follows: we introduce a new specialization system log on DMQ (−) for which we can easily construct a monodromy sequence. Then / Υ and prove we construct a morphism of specialization system log that it is an isomorphism. The specialization system log is defined using a pro-motive Log called the logarithmic pro-motive. To define Log we first need the motivic Kummer torsor: Definition 1.4.29. The motivic Kummer torsor is the object K of DM(Gm) defined up to a unique isomorphism by the distinguished triangle ZGm (1)
/K
/ ZGm (0)
θ
/ ZGm (1)[1]
where θ is the motivic fundamental class of Definition 1.4.1. Remark 1.4.30. The uniqueness of the motivic Kummer torsor follows from the vanishing of the group homDM(Gm) (Z(0) ⊕ Z(1)[1], Z(1)) = H1,0 (k) ⊕ H0,−1 (k) (see [40]). Indeed, let a be an automorphism of the above triangle which is the identity on Z(1) and Z(0). To prove that a is the identity, we look at id − a. This gives a morphism of distinguished triangles Z(1) 0
Z(1)
/K
~
/K
/ Z(0) u
e
0
/ Z(0)
e
/ Z(1)[1]
0
/ Z(1)[1].
It is easy to see that factors through some morphism u. To show our claim, it suffices to prove that the group homDM(Gm) (Z(0), K) is zero. For this we look at the exact sequence of hom groups in DM(Gm): hom(Z(0), Z(1)) −→ hom(Z(0), K) −→ hom(Z(0), Z(0)) −→ hom(Z(0), Z(1)[1]). Because hom(ZGm (0), ZGm (1)) is zero, we only need to show the injectivity of Z = hom(ZGm (0), ZGm (0))
/ hom(ZGm (0), ZGm (1)[1]) = Γ(Gm, O× ).
This morphism sends 1 ∈ Z to the class of the variable π ∈ k[π, π −1 ]. The desired injectivity follows from the fact that this element is non-torsion. Remark 1.4.31. When the base field is a number field, there is a way to think about K as an extension of Tate motives in some abelian sub-category of DM(Gm). Indeed, the Beilinson-Soul´e conjecture is known for Gmk and
The Motivic Vanishing Cycles and the Conservation Conjecture
45
all its points when k is a number field. It is then possible to define a motivic t-structure on the sub-category of Tate-motives over Gmk . We will use this (non-elementary) point of view to simplify the proofs of Lemmas 1.4.36 and 1.4.45. Note that these two lemmas admit elementary proofs that can be found in the third chapter of [3]. Definition 1.4.32. For n ∈ N, we define the object Log n of DMQ (Gm) by Log n = Symn (K) where Symn is the symmetric n-th power. This object is called the n-th logarithmic motive. Remark 1.4.33. The definition of the logarithmic motive Log n only makes sense after inverting some denominators. Indeed, the projector Symn is given by 1 X σ |Σn | σ∈Σn
where Σn is the n-th symmetric group. Remark 1.4.34. Logarithmic motives, or at least their realizations, are well– known objects in the study of Beilinson’s conjectures and polylogarithms. Lemmas 1.4.35, 1.4.36 and 1.4.45 are surely well-known. Lemma 1.4.35. Let n and m be integers. We have two canonical morphisms: • αn,n+m : Log n (m)
/ Log n+m
• βn+m,m : Log n+m
/ Log m
Moreover, if l is a third integer, we have: αn+m,n+m+l ◦ αn,n+m = αn,n+m+l and βm+l,l ◦ βn+m+l,m+l = βn+m+l,l . We also have a commutative square Log n+m (l)
/ Log n+m+l
/ Log m+l .
Log m (l)
Proof Consider the Q-algebras Q[Σn ] and Q[Σm ] as sub-algebras of Q[Σn+m ] corresponding to the partition n + m. In Q[Σn+m ] we have three projectors: Symn , Symm and Symn+m with the relations: Symn .Symm = Symm .Symn
Symn+m = Symn+m .Symn = Symn+m .Symm .
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J. Ayoub
We then see immediately that Symn+m K is canonically a direct factor of Symn K ⊗ Symm K. On the other hand, we have natural morphisms: / Symm K ,
• Q(m)
/ Q(0) .
• Symn K
We get the desired morphisms by taking the compositions: / Symm K ⊗ Symn K
• Q(m) ⊗ Symn K
/ Symm K ⊗ Symn K
• Symn+m K
/ Symn+m K , / Symm K ⊗ Q(0) .
We leave the verification of the two identities and the commutativity of the square to the reader (see [3], chapter III). Lemma 1.4.36. There is a canonical distinguished triangle Log n (m + 1)
α
/ Log n+m+1
β
/ Log n (m + 1)[+1].
/ Log m
Proof We have chosen to give a short and simple proof of Lemma 1.4.36 based on a non-elementary result, rather than a complicated and self-contained one (see [3] for an elementary proof). The non-elementary result we shall use is the existence of an abelian category MTM(Gm) of mixed Tate motives over Gm, which is the heart of a motivic t-structure on the sub-category of DM(Gm) generated by Q(i) for i ∈ Z. Of course, MTM(Gm) is known to exist only when the base field is a number field. So we first construct our distinguished triangle when our base field is Q and then extend it to arbitrary field of characteristic zero by taking its pull-back. Let us first prove that β ◦ α is zero. It clearly suffices to prove that for any two subsets I and J of E = {1, . . . , n + m + 1} of respective cardinality n and m, the composition / K⊗E
K⊗I ⊗ Q(1)⊗E−I
/ K⊗J ⊗ Q(0)⊗E−J
is zero. But this is indeed the case, because (E − I) ∩ (E − J) is always nonempty. The next step will be to prove that the sequence: 0
/ Log n (m + 1)
α
/ Log n+m+1
β
/ Log m
/0
is a short exact sequence in MTM(Gm). This will imply our statement. One can easily see that α is injective and β surjective. So we have to prove exactness at the middle term. For this, we use the fact that MTM(Gm) is a neutral Tannakian Q-linear category and all its non-zero objects have a
The Motivic Vanishing Cycles and the Conservation Conjecture
47
strictly positive dimension (given by the trace of the identity). So to prove the exactness at the middle term we only need to show that dim(Log n+m+1 ) = dim(Log n ) + dim(Log m ). But this is true because dim(Log l ) = l + 1, which is an easy consequence of dim(K) = 2. By Lemma 1.4.35, the logarithmic motives define a pro-object in the category of mixed Tate motives over Gm. This pro-object (Log n+1 → Log n )n will be denoted by Log. We will use this particular case of 1.4.36 to get our monodromy sequence: Corollary 1.4.37. There is canonical pro-distinguished triangle Log n−1 (1)
/ Log n
/ Q(0)
/ Log n−1 (1)[1]
in DMQ (Gm). We make the following definition: / A1 and an object k
Definition 1.4.38. Given an A1k -scheme f : X A ∈ SH(Xη ), we define
logf (A) = Colimn χf Hom(fη∗ Log n , A). The arguments used in the proof of Proposition 1.4.8 show that this formula extends to a specialization system log. Proposition 1.4.39. For any f , there is a natural distinguished triangle logf (A)(−1)[−1]
/ χf (A)
/ logf (A)
N
/ logf (A)(−1).
Proof This is clear from Corollary 1.4.37. To obtain the first part of Theorem 1.4.28 from Proposition 1.4.39 we need to compare the two specialization systems Υ and log. We do this in three steps: Step 1. If E • is a cosimplicial object in an additive category we will denote by cE • the usual complex associated to it (by taking the alternating sum of the faces). Given a complex K = K • in some additive category, we denote K ≤n the complex obtained by replacing the objects K i by a zero object for all i ≥ n. Given a smooth X-scheme U let us simply denote by X (and not Qtr (X)) the Nisnevich sheaf of Q-vector spaces with transfers
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J. Ayoub
represented by X. Consider the complexes of sheaves with transfers cA≤n . We have canonical morphisms / cA≤n
cA≤n+1
that give a pro-object (cA≤n )n∈N . First remark that (cA≤1 ) maps naturally to the motivic Kummer torsor. Indeed, we take the morphism induced by the morphisms of complexes cA≤1 '
[id : Gm → Gm]
K'
[id : Gm → Gm]
∆−(x,1)
θ
/ [pr1 : Gm × Gm → Gm] / [pr1 : Gm × (Gm, 1) → Gm]
where the two horizontal arrows are the first and only non-zero differentials of cA≤1 and K. This canonical morphism will be denoted by γ1 . By com/ (cA≤1 ) we get for n ≥ 1 posing with the obvious morphisms (cA≤n ) canonical morphisms γ1 : (cA≤n )
/ K.
Passing to the symmetric powers, we get morphisms γr : Symr (cA≤n )
/ Log r .
/ Q(0) we can define in the same Using the obvious morphism: (cA≤n ) way as for Log a pro-object structure on (Symr (cA≤n ))r∈N , and the family (γr ) becomes a morphism of pro-objects for n ≥ 1. Given a morphism of / A1 , we get from (γr ) a natural transformation k-schemes f : X
logf
/ Colimn,r i∗ j∗ Hom(f ∗ Symr (cA≤n ), −). η
(1.3)
Step 2. Let us denote by (A⊗r )• the cosimplicial object obtained by taking self products of A in the category of cosimplicial Gm-schemes. There is an action of Σr on (A⊗r )• , so that the symmetric part (Symr A)• can be defined in the category of cosimplicial sheaves of Q-vector spaces. Us/ Gm , we get a pro-object of cosimplicial sheaves ing the projection A ( (Symr+1 A)•
/ (Symr A)• )r . As in the first step, we consider the com-
plexes c(Symr A)≤n and c(A⊗r )≤n . We have an obvious Σr -equivariant mor/ c(A⊗r )≤n . Passing to the symmetric part, we get phism: (cA≤n )⊗r morphisms Symr (cA≤n )
/ c(Symr A)≤n
The Motivic Vanishing Cycles and the Conservation Conjecture
of N × N-pro-objects. This induces for any A1 -scheme f : X natural transformation Colimn,r i∗ j∗ Hom(fη∗ c(Symr A)≤n , −)
49 / A1 a
(1.4)
Colimn,r i∗ j∗ Hom(fη∗ Symr (cA≤n ), −). Lemma 1.4.40. The natural transformation (1.4) is an isomorphism. Proof It suffices to fix r and to prove that the natural transformation: Colimn i∗ j∗ Hom(fη∗ c(Symr A)≤n , −)
Colimn i∗ j∗ Hom(fη∗ Symr (cA≤n ), −) is invertible. This natural transformation is a direct factor of Colimn i∗ j∗ Hom(fη∗ c(A⊗r )≤n , −)
/ Colimn i∗ j∗ Hom(f ∗ (cA≤n )⊗r , −). η
Let us show that the latter is invertible. The left hand side is nothing but the total space of the simplicial space i∗ j∗ Hom(fη∗ (A⊗r )• , −). The right hand side is the total space of the r-simplicial space i∗ j∗ Hom(fη∗ (A• ⊗ · · · ⊗ A• ), −) and the morphism we are looking at is the one induced from the identification of i∗ j∗ Hom(fη∗ (A⊗r )• , −) with the restriction of i∗ j∗ Hom(fη∗ (A• )⊗r , −) to / ∆ × · · · × ∆ . But it is a well– the diagonal inclusion of categories ∆ known fact that the total space of an r-simplicial object is quasi-isomorphic to the total space of its diagonal. Step 3. Using Lemma 1.4.40 and the two natural transformations (1.3) and / A1 a natural transformation (1.4), we get for any k-morphism f : X k logf
/ Colimn,r i∗ j∗ Hom(f ∗ c(Symr A)≤n , −). η
(1.5)
/ (A⊗r )• . Now consider the diagonal embedding of cosimplicial schemes A• One easily sees that it is Σr -equivariant. So it factors uniquely through
A•
/ (Symr A)• .
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This gives us a morphism of pro-objects / c(Symr A)≤n
cA≤n
and a natural transformation of functors Colimn,r i∗ j∗ Hom(fη∗ c(Symr A)≤n , −)
Colimn i∗ j∗ Hom(fη∗ cA≤n , −) = Υf . Composing with (1.5), we finally get the natural transformation γf : logf
/ Υf .
We leave the verification of the following lemma to the reader: Lemma 1.4.41. The family of natural transformations (γf ) is a morphism of specialization systems. Moreover, we have a commutative triangle / log @@ @@ @@ γ @
χ@
Υ.
The rest of this section is mainly devoted to the proof of the following result: Proposition 1.4.42. The morphism γ : log
/ Υ is an isomorphism.
We break up the proof into several lemmas, which are of independent interest: Lemma 1.4.43. For every non-zero natural number n, the composition Q
/ χe Q n
/ Υe Q n
is an isomorphism. Proof This is a generalization of Proposition 1.4.9 that holds when working in DMQ (−). We repeat exactly the same proof of 1.4.9, replacing everywhere A• with (en )∗η A• . We end up with the following problem: is the morphism Q
/ Tot(Gm× ˜ Gm,(en )η k)
The Motivic Vanishing Cycles and the Conservation Conjecture
51
invertible in DMQ (k) ? The difference with 1.4.9 is that the cobar cosimplicial object on the right hand side is the one obtained by applying Lemma 1.4.2 to (en )η
Gm
/ Gm o
1
k.
To answer this question, we look at the obvious morphism of cosimplicial objects / Gm× ˜ Gm k
˜ Gm,(en )η k Gm×
and check that it is level-wise an A1 -weak equivalence (up to torsion). On the level i this morphism is given by / Gmi+1 .
(en )η × id×i : Gmi+1
It is well–known that elevation to the n-th power on Gm induces the identity on Q and multiplication by n on Q(1)[1] modulo the decomposition M (Gm) ' Q ⊕ Q(1)[1] in DMQ (k). Lemma 1.4.44. The motives K and (en )∗η K are isomorphic. Proof Indeed, the motive (en )∗η K corresponds to the extension Q(1)
/ Q(0)
/ (en )∗η K
n.θ
/ Q(1)[1].
We have a commutative square θ
Q(0)
n.θ
Q(0)
/ Q(1)[1]
×n
/ Q(1)[1]
which we extend into a morphism of distinguished triangles /K
Q(1)
×n
Q(1)
/ Q(0)
θ
/ Q(1)[1]
a
/ (en )∗η K
/ Q(0)
n.θ
×n
/ Q(1)[1].
The morphism a is clearly invertible. / k . For every n ∈ N, Lemma 1.4.45. Denote by q the projection Gm there is a canonical distinguished triangle which splits:
Q(n + 1)[1]
/ q# Log n
/ Q(0)
/
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J. Ayoub
Moreover, the diagram Q(n + 2)[1] 0
Q(n + 1)[1]
/ q# Log n+1
/ Q(0)
/
/ q# Log n
/ Q(0)
/
is a morphism of distinguished triangles. Proof This is a well-known fact to people working on Polylogarithms. The simplest way to prove it is to work over a number field and in the abelian category of mixed Tate motives MTM(Gm). We gave an elementary proof in the third chapter of [3]. We use the finite filtration: Log n−i (i) ⊂ Log n to produce a spectral sequence of mixed Tate motives E1i,j = hi+j M q# Q(i)
+3 hi+j q Log n M #
where h•M is the truncation with respect to the motivic t-structure. We have: • hrM q# Q(i) = 0 except for r = −1, 0, 0 • h−1 M q# Q(i) = Q(i + 1) and hM q# Q(i) = Q(i). So our spectral sequence on MTM(k) looks like: O j
Q(0) Q(1)
/ i
• •
Q(1) /• Q(2) •
Q(2) /• Q(3) • /
Q(n − 1) • Q(n) •
Q(n) /• Q(n + 1) •
The Motivic Vanishing Cycles and the Conservation Conjecture
53
It is easy to show that the non-zero differentials of the E1 -page are the identity. Indeed they are given by / h−1 q# Q(i + 1)[+1] M
h−1 M q# Q(i)
/ Q(1)[1] / Q(i+1)[1] is the motivic fundamental class Q(0) where Q(i) twisted by Q(i) (due to Lemma 1.4.36). So it suffices to compute the re/ q# Q(1)[1] to Q(1)[1]. By definition q# θ is striction of q# θ : q# Q(0) the diagonal embedding
Gm
/ Gm × (Gm, 1) .
This shows that q# induces the identity on Q(1)[1]. In particular, our spectral sequence degenerates at E2 and the only nonzero terms that we get are Q(0) and Q(n + 1). This proves the lemma. Corollary 1.4.46. For every nonzero natural number n, the composition Q
/ χe Q n
/ loge Q n
is an isomorphism. Proof Due to Lemma 1.4.44, it suffices to consider the case n = 1. Once again we apply the argument in the proof of Proposition 1.4.9. We end up with the following question: Is the morphism Q
/ Colimn Hom(p# Log n , Q)
an isomorphism? The answer is yes by Lemma 1.4.45. Lemma 1.4.43 and Corollary 1.4.46 together imply that for any n, the morphism logen Q
/ Υe Q n
is an isomorphism. This proves Proposition 1.4.42 by applying Theorem 1.3.11, part 2. We have proved the first part of Theorem 1.4.28. For the nilpotency of N , first remark that due to Lemma 1.4.43 and Theorem 1.3.10 we know that Υf sends constructible objects to constructible objects. So we can apply the following general result: Proposition 1.4.47. Let S be a scheme of finite type over a field k of characteristic zero. Let A and B be constructible objects in DM(S). Then for N large enough, the groups homDM(S) (A, B(−N )[∗]) are zero.
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Proof One may assume that A is the motive of a smooth S-scheme f : / S . Then we see that U hom(A, B(−N )[∗]) = hom(ZU , f ∗ B(−N )[∗]) = hom(Z, (πU )∗ f ∗ B(−N )[∗]) with πU the projection of U to k. Thus it suffices to consider the case where A = Z and B ∈ DMct (k). Denoting D = Hom(−, Z) the duality operator, we have: hom(Z, B(−N )[∗]) = hom(D(B), Z(−N )[∗]). We finally see that it suffices to prove that for a smooth variety V over k we have hom(V, Z(−N )[∗]) = 0 for N large enough. But by the cancellation theorem of Voevodsky [38], we know that we can take any N ≥ 1.
1.5 Conservation conjecture. Application to Schur finiteness of motives 1.5.1 The statement of the conjecture Recall that a category C is pointed if it has an initial and a terminal object that are isomorphic (via the unique map between them). The initial and ter/ C2 minal objects will be called zero objects. Usually a functor F : C1 between two abstract categories is said to be conservative if it detects isomorphisms, that is, an arrow f is an isomorphism if and only if F (f ) is an isomorphism. For our purposes, it will be more convenient to say that a / C2 between two pointed categories is conservative if functor F : C1 it detects the zero objects. That is, an object A ∈ C1 is zero if and only if F (A) is zero. When F is a triangulated functor between two triangulated categories, then the two notions coincide. In general, they are quite different. Conjecture 1.5.1. Let S be the spectrum of a geometric DVR of equi-characteristic zero. We denote as usual η and s the points of S, and we fix a uniformizer / DMct (s) is conserπ ∈ Γ(S, OS ). Then the functor ΨidS : DMct Q (η) Q vative. Remark 1.5.2. We do not know if it is reasonable to expect that the functor / DMct (s) is conservative without killing torsion. ΨidS : DMct (η) / SHct (s) is conRemark 1.5.3. One can also ask if ΨidS : SHct Q (η) Q servative. When (−1) is a sum of squares in OS this is equivalent to 1.5.1. Indeed, by Morel [27] the category SHQ (k) decomposes into a direct product of triangulated categories DMQ (k)⊕?(k) where ?(k) is zero if and only if (−1) is a sum of squares in k. When (−1) is not a sum of
The Motivic Vanishing Cycles and the Conservation Conjecture
55
squares in OS , the functor ΨidS may fail to be conservative for obvious reasons. Indeed, one can prove that ΨidS is compatible with the decomposition: SHQ (−) = DMQ (−)⊕?(−). In particular, for a base S such that (−1) as a sum of squares over s but not over η, the functor ΨidS takes the non-zero subcategory ?(η) to zero. The main reason why one believes in the conservation conjecture is because it is a consequence of the conservation of the realization functors. In/ D+ (−, Q` ) deed, assuming that the `-adic realization functor R` : DMct Q (−) (see [19]) is conservative for fields it is easy to deduce conjecture 1.5.1 using the commutative diagram (up to a natural isomorphism) DMct Q (η) R`
D+ (η, Q` )
Ψ
/ DMct (s) Q
Ψ
R`
/ D+ (s, Q` )
that expresses the compatibility of our motivic nearby cycles functor with the classical `-adic one. Indeed, the functor Ψ on the level of continuous Galois modules is nothing but a forgetful functor which associates to a Gal(¯ η /η)-module the Gal(¯ s/s)module with the same underlying Q` -vector space obtained by restricting the action using an inclusion Gal(¯ s/s) ⊂ Gal(¯ η /η). The latter inclusion is obtained using the choice of a uniformizer (in equi-characteristic zero). Maybe it is worth pointing that our conservation conjecture is weaker than the conservation of the realizations, which seems out of reach for the moment. Furthermore, the statement of 1.5.1 is completely motivic. So we hope it is easier to prove.
1.5.2 About the Schur finiteness of motives Let us first recall the notion of Schur finiteness due to Deligne (see [7]). Let (C, ⊗) be a Q-linear tensor category. For an object A of C, the n-th symmetric group Σn acts on A⊗n = A ⊗ · · · ⊗ A. By linearity, we get an action of the group algebra Q[Σn ] on A⊗n . If C is pseudo-abelian, then for any idempotent p of Q[Σn ] we can take its image in A⊗n obtaining in this way an object Sp (A) ∈ C. Definition 1.5.4. An object A of C is said to be Schur finite if there exists an integer n and a non-zero idempotent p of the algebra Q[Σn ] such that Sp (A) = 0.
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This notion is a natural generalization of the notion of finite dimensionality of vector spaces. Indeed a vector space E is of finite dimension if and n
only if for some n ≥ 0, the n-th exterior product ΛE is zero. The notion of Schur finiteness makes sense in many contexts. One can speak about Schur finiteness of mixed motives in DMQ (k). For more about this notion the reader can consult [22]. Another finiteness notion of the same spirit is the Kimura finiteness (see [21]). One of the reasons why Schur finiteness is more flexible than Kimura finiteness is the following striking result, proved in [22]: Lemma 1.5.5. Suppose given a distinguished triangle in DMQ (k): A0
/A
/ A00
/ A[+1] .
If two of the three objects A, A0 and A00 are Schur finite, then so is the third†. It is conjectured that any constructible object of DMQ (k) is Schur finite. For Kimura finiteness one can at most hope that this property holds for pure objects of DMQ (k), that is, objects coming from the fully faithful embedding (see [40]) Chow(k)Q
/ DMQ (k)
where Chow(k) is the category of Chow motives. The problem is that Lemma 1.5.5 fails for Kimura finiteness. Lots of unsolved problem would follow if one could prove that some motives are Schur or Kimura finite. For an overview, the reader may consult [1], [12] and [21]. Let us only mention the Bloch conjecture‡ for surfaces with pg = 0. Unfortunately, the only way we know to construct Schur finite motives is by the following proposition (see [21]): Proposition 1.5.6. If C is a smooth k-curve then its motive M (C) is Schur (even Kimura) finite. It formally follows from Lemma 1.5.5 and the above proposition that all objects of the triangulated tensor subcategory DMAbelian (k) of DMct Q Q (k) generated by motives of curves are Schur finite. It is remarkable that there (k) for which Schur is not a single motive that does not belong to DMAbelian Q finiteness has been established. † This property is not specific to DMQ (k). It holds for any triangulated Q-linear tensor category T coming from a monoidal Quillen model category (see [11]). ‡ Actually it is not clear that the Bloch conjecture follows from the Schur finiteness of the motives of surfaces, but it does follow from their Kimura finiteness. For more information the readers can consult [23].
The Motivic Vanishing Cycles and the Conservation Conjecture
57
One of the applications of the theory of vanishing cycles is the following reduction: Proposition 1.5.7. Suppose that k is of infinite transcendence degree over Q. To show that every constructible object of DMQ (k) is Schur finite, it suffices to check that for any n ∈ N, and any general† smooth hypersurface H of Pn+1 , the motive M (H) is Schur finite. k In the rest of the paragraph we give a proof of 1.5.7. We work under the assumption that the motive M (H) is Schur finite whenever H is a general smooth hypersurface of some Prk . Remark that if k 0 is another field and H 0 is a general hypersurface of Prk0 then the motive M (H 0 ) is Schur finite in DMQ (k 0 ). Indeed, one can replace k 0 by an extension so that H 0 is isomor/ Spec(k) . phic to the pull-back of H along some morphism Spec(k 0 ) ct The subcategory DMQ (k) of constructible motives is generated (up to Tate twist and direct factors) by motives M (X) with X a smooth projective variety (see [40]). By Lemma 1.5.5, we need only to check that these motives are Schur finite. We argue by induction on the dimension of X. Let n be the dimension of X. Denote DMct Q (k)≤n−1 the triangulated ct subcategory of DMQ (k) generated by motives of smooth projective varieties of dimension ≤ n − 1 and their Tate twists. By induction, the objects of DMct Q (k)≤n−1 are Schur finite. Let X 0 be a smooth (possibly open) k-variety birational to X. Using: • Resolution of singularities (see [14]) and the weak factorization theorem (see [41]), • The blow up with smooth center formula for motives (see [40]), • The Gysin distinguished triangle for the complement of a smooth closed subscheme (see [40]), one sees that the motive of X is obtained from the motive of X 0 and some objects of DMct Q (k)≤n−1 by successive fibers and cofibers. It follows that the Schur finiteness of X is equivalent to the Schur finiteness of X 0 . We would like to deform X to a smooth hypersurface. This is impossible in general but we have: Lemma 1.5.8. There exists a projective flat morphism f : E Spec(k[π]) such that:
/ A1 = k
(i) E is smooth, † Here ”general” means that the coefficients of the equation of H are algebraically independent in k.
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J. Ayoub
(ii) The generic fiber of f is a general smooth hypersurface in Pn+1 k(π) , (iii) E0 = f −1 (0) is a reduced normal crossing divisor, (iv) The fiber E0 contains a branch D which is birational to X. Proof The variety X is birational to a possibly singular hypersurface X0 ⊂ Pn+1 of degree d. By taking a general pencil of degree d hypersurfaces k passing through X0 we get a flat morphism f 0 : E 0
/ A1 , such that: k
• The generic fiber of f 0 is a general smooth hypersurface in Pn+1 k(π) , • The fiber E00 is the reduced scheme X0 . By pulling back the family f 0 along the elevation to the m-th power em : A1k
(−)m
/ A1 (for some sufficiently divisible m) and resolving singularities k
/ A1 with semi-stable reduction. This we get by [29] a morphism f : E k f has the properties (1)-(3). Property (4) for f follows immediately from the fact that f 0 is smooth in the neighborhood of the generic point of X0 .
Let η = Spec(k(π)) be the generic point of A1k , and denote by s its zero section. The motive M (Eη ) ∈ DMQ (η) is Schur finite. Let us denote by / DMQ (s) our nearby cycles functor. By Theorem 1.4.24 Ψ : DMQ (η) this functor is monoidal. It follows that the motive Ψ(M (Eη )) ∈ DMQ (s) is Schur finite. Proposition 1.5.7 is a consequence of the following result (which we state with integral coefficients): / A1 be a flat projective morphism of relative Lemma 1.5.9. Let f : E dimension n. Suppose that E is smooth, and that Es = f −1 (s) is a reduced normal crossing divisor. Let us write Es = D1 ∪ · · · ∪ Dr , where Di are the smooth branches. We let Di0 be the open scheme of Di defined by Di − ∪j6=i Dj . There is a distinguished triangle in DM(s):
⊕i M (Di0 )
/ Ψ(M (Eη ))
/N
/
with N in the triangulated subcategory DMct (s)≤n−1 ⊂ DMct (s) generated by Tate twists of motives of smooth projective varieties with dimension less than n − 1. Proof The main ingredient in the proof of this lemma is Theorem 1.3.13. We first work on Es and then push everything down using fs! . Let w : / Es the / Es be the obvious inclusion, and denote c : C ∪i Di0
The Motivic Vanishing Cycles and the Conservation Conjecture
59
complement. We have an exact triangle in DM(Es ): / c∗ c∗ Ψf Z .
/ Ψf Z
w! w∗ Ψf Z
Because f is projective, we have fs! Ψf = Ψfη! . To show what we want, it suffices to prove that: • fη! Z is up to a twist M (Eη ), • fs! w! w∗ Ψf Z equals ⊕i M (Di0 ) up to a twist, • fs! c∗ c∗ Ψf Z is in the subcategory DMct (s)≤n−1 . The first point is easy. Indeed we have fη! Z = fη# Z(−n)[−2n] = M (Eη )(−n)[−2n] because f is smooth (see [3], chapter I). The second point follows in the same way, using the equality w∗ Ψf Z = Z∪i D0 and smoothness of ∪i Di0 . For the i last point, it suffices to prove that c∗ Ψf I is in the triangulated subcategory /C of DMct (C) generated by objects of the form t∗ I(m) where t : Z is a closed immersion, Z smooth and m an integer. To do this, we need some notation. For non-empty I ⊂ [1, r] we de/ Es its note CI = ∩i∈I Di the closed subscheme of Es and cI : CI / CJ be the obvious inclusions. inclusion. For J ⊂ I we let cI,J : CI When card(I) ≥ 2, the subscheme CI is inside C. In this case, we call / C the inclusion. Note also the following commutative diadI : CI grams: Di0
vi
/ Di O cI,i
DI
ui
/ Es > } }} } }} c }} I
for i ∈ I. / C for card(I) = 2 form a cover by closed subsets of C. The dI : CI By a variant of the Mayer-Vietoris distinguished triangle for covers by closed subschemes (see [3], chapter II), one proves that any object A ∈ DM(C) is in the triangulated subcategory generated by the set of objects {dI∗ d∗I A | I ⊂ [1, r] and card(I) ≥ 2}. To finish the proof, we will show that for ∅ 6= I ⊂ [1, r] the object c∗I Ψf I is in the triangulated subcategory generated by the set of objects {(cK,I )∗ Z(m) | I ⊂ K ⊂ [1, r] and m ∈ Z}. Using Theorem 1.3.13 one has: c∗I Ψf I ' c∗I,i u∗i Ψf I ' c∗I,i vi∗ I. It is wellknown that vi∗ I is in the triangulated subcategory generated by (cK,i )∗ I(−m) for K ⊂ [1, r] containing i and m an integer. This implies our claim.
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Remark 1.5.10. The proof of Proposition 1.5.7 gives the following more precise statement: the category DMct (k) is generated by the motives Ψid (M (H)), H a general hypersurface of Pn+1 k(π) . This fact is interesting for its own sake. For instance, using the Chow-Kunneth decomposition for smooth hyper˜ M (H)[n] we conclude that surfaces M (H) = Q(0)[0] ⊕ · · · ⊕ Q(n)[2n] ⊕ h n ct ˜ M (H)). Note that these DMQ (k) can be generated by the motives Ψid (h n generators have the following nice properties: • They are in the heart of the conjectural motivic t-structure i.e. the real˜ M (H)) are concentrated in degree zero. izations of Ψid (h n • They are equipped with a non-degenerate pairing ˜ M (H)) ⊗ Ψid (h ˜ M (H)) Ψid (h n n
/ Q(2n)
˜ M (H)) ' D(Ψid (h ˜ M (H)))(2n) where D = inducing an isomorphism Ψid (h n n Hom(−, Q) is the duality functor. • They are conjecturally Kimura finite (and not simply Schur finite).
1.5.3 The conservation conjecture implies the Schur finiteness of motives. A way to prove the Schur finiteness of objects in DMct Q (k) is to prove the conservation conjecture. Indeed: Proposition 1.5.11. Assume conjecture 1.5.1. Then every constructible motive of DMQ (k) is Schur finite. Proof We have seen in 1.5.7 that to check the Schur finiteness of constructible motives, one only needs to consider the motive of a general smooth hypersurface of some projective space. Let H ⊂ Pn+1 be a general smooth hypersurface of degree d. One can find a projective flat morphism f : / A1 such that E0 = f −1 (0) is a Fermat hypersurface and E1 = E k f −1 (1) is H. It is well known that the motive of a Fermat hypersurface is a direct factor of the motive of a product of projective smooth curves. It follows from Proposition 1.5.6 that M (E0 ) is Schur finite. Fix Sp , a non-zero projector of Q[Σm ] such that Sp M (E0 ) = 0. Let us consider for ? ∈ {0, 1} the vanishing cycles functors Ψ? : DMQ (η)
/ DMQ (?).
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We know that Ψ? (M (Eη )) = M (E? ) and that Ψ? is monoidal. We have Ψ0 (Sp M (Eη )) = Sp Ψ0 (M (Eη )) = Sp M (E0 ) = 0. The conservation of Ψ0 tell us that Sp M (Eη ) = 0. Applying Ψ1 , we get: 0 = Ψ1 (Sp M (Eη )) = Sp Ψ1 (M (Eη )) = Sp M (E1 ) = Sp (M (H)). This proves that the motive of H is Schur finite. Remark 1.5.12. The proof of the above proposition was suggested to us by Kimura. Our original proof was more complicated and very similar to the proof of Proposition 1.5.7. It was by induction on the degree d. The idea was to degenerate a hypersurface of degree d to the union of two hypersurface of degree d − 1 and 1. This original proof was more elementary as it did not use Proposition 1.5.6.
1.5.4 Some steps toward the Conservation conjecture In this final paragraph, we shall explain some reductions of the conservation conjecture. With our definition of Ψ, it seems too difficult to study the conservation conjecture. Our first result says that the conservation of Ψ is equivalent to the conservation of a simpler functor Φ already introduced in the beginning of section 1.4. Let us recall the definition of the functor Φ. As in paragraph 1.4.3, we call / A1 the elevation to the n-th power. We let η be the generic en : A1k k point of A1k and s its zero section. We consider the commutative diagrams ηn
j
(en )η
η
j
/ A1 o k
i
s
i
s.
en
/ A1 o k
We then define Φ(A) = Colimn∈N× i∗ j∗ (en )∗η A for every object A of DMQ (η). Proposition 1.5.13. The following two statements are equivalent: • The functor Ψ : DMct Q (η)
/ DMct (s) is conservative, Q
• The functor Φ : DMct Q (η)
/ DMct (s) is conservative. Q
Proof Indeed, let A be a finite type object of DMQ (η). Replacing A by a (en )∗η A with n sufficiently divisible ((en )∗η is a conservative functor), we may assume by Theorem 1.4.17 (and its variant for Φ and χ) that:
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• Ψ(A) = Υ(A), • Φ(A) = χ(A) = i∗ j∗ (A). By the monodromy Theorem 1.4.28 we have a distinguished triangle Υ(A)(−1)[−1]
/ χ(A)
/ Υ(A)
N
/ Υ(A)(−1).
Now, suppose that χ(A) = 0. Then N is an isomorphism. But we know by the same theorem that N is nilpotent (because A is of finite type). This / Υ(A)(−m) is an isomorphism for means that the zero map of Υ(A) sufficiently divisible m. This of course implies that Υ(A) is zero. On the other hand, if χ(A) = 0 one sees that Υ(A) = 0 by looking at the definition of Υ(A). Thus we have proved the equivalence Ψ(A) = 0 ks
+3 Φ(A) = 0 .
This clearly implies the statement of the proposition. One can go further and prove that the conservation of Φ is a consequence of the conservation of a very concrete functor φ defined on the level of homotopy sheaves with transfers. Before doing this, we need to introduce a t-structure on DMeff (k) and DM(k). Definition 1.5.14. 1- The category DMeff (k) is equipped with a natural t-structure called the homotopy t-structure. The heart of this t-structure is denoted by HI(k). The objects of HI(k) are the homotopy invariant Nisnevich sheaves with transfers on Sm /k (see [40]). 2- The category DM(k) is equipped with a natural t-structure also called the homotopy t-structure. The heart of this t-structure is denoted by HIM(k). The objects of HIM(k) are modules on the the Milnor K-theory spectrum K∗M ; we shall call them A1 -homotopy modules. The category HIM(k) is equivalent to the category of Rost modules by a result of Deglise [4]. Let us briefly explain what an A1 -homotopy module is. An A1 -homotopy module is a collection (Fi )i∈Z of homotopy invariant sheaves with transfers ∼ / on Sm /k together with assembly isomorphisms Fi Hom(K1M , Fi+1 ) . They are in some sense analogous to topological spectra, where the topological spheres are replaced by the Milnor K-theory sheaves. Let us return to our specialization functors. The reason why the homotopy t-structure is interesting for the conservation conjecture is the following result, obtained in the second and third chapters of [3]: Lemma 1.5.15. The two functors Φ, Ψ : DM(η) t-exact with respect to the homotopy t-structures.
/ DM(s) are right
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This is a little bit surprising, because in ´etale cohomology or in Betti cohomology, these two functors turn out to be left exact with respect to the canonical t-structures. Of course, the point is that the homotopy tstructure is specific to motives and does not correspond via realization to any reasonable t-structure in the ´etale or the Betti context. Another way to say this is that the homotopy t-structure is not the dreamt of motivic t-structure. / HIM(s) be the functor defined Corollary 1.5.16. Let φ : HIM(η) by φ(−) = τ≤0 (Φ(−)) with τ≤0 being the truncation with respect to the homotopy t-structure. Then φ is a right exact functor between abelian categories.
There is a natural notion of finite type and finitely presented objects in HIM(k). The subcategory of finite type objects† in HIM(k) is denoted by HIMtf (k). We conjecture that: Conjecture 1.5.17. Suppose that k is of characteristic zero. The functor / HIMQ (s) is conservative. φ : HIMtf Q (η) The conservation of φ implies the conservation of Φ. Indeed, if A is a constructible object then hi (A) = 0 for i small enough (where hi means the homology object of A with respect to the homotopy t-structure). So if A is non zero, we can assume that h0 (A) 6= 0 and hi (A) = 0 for i < 0. The constructibility of A implies that h0 (A) is of finite type (and even finitely presented). But then we would have φ(h0 (A)) = h0 (Φ(A)). Thus if Φ(A) = 0, then h0 (A) would be zero, contradicting our assumption that A is non zero. It is also possible to consider the effective version Φeff : / DMeff (s) of Φ. We can still prove that Φeff is right exact. DMeff (η) / HI(s) be the induced functor on the hearts. We We let φeff : HI(η) think that it is easy to show that the functor φ is conservative (on objects of finite type and rational coefficients) if and only if its effective version φeff is conservative (on objects of finite type and rational coefficients). Such a reduction could be interesting. Indeed, the functor φeff is rather explicit and defined on sheaves. Unfortunately, we do not know how to prove that
φeff : HItf Q (η)
/ HIQ (s) is conservative. We should also say that Srinivas
gave us a counterexample to the conservation of φeff for fields of positive characteristic. We end by recalling his example. / B be the universal family of elliptic Example 1.5.18. Let e : E curves over a field of positive characteristic k. Fix s ∈ B such that the fiber † Warning: this category is not abelian. Indeed, kernels are not necessarily of finite type.
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Es is super-singular. Then define the relative surface S/B by a desingularization of (E ×B E)/(Z/2Z) where the group Z/2Z is acting by: (x, y) 7−→ (−x, −y). Finally let η be the generic point of B. Then it is known that: • CH0 (Sη ) is infinite dimensional (in the sense of Mumford), • Ss is a uniruled surface. ˜ 0 (Sη ) is non-zero, but In particular, the reduced Suslin homology sheaf h ˜ 0 (Ss ) = 0. Now, it is expected that φeff (h0 (Sη )) = h0 (Ss ). This means h ˜ 0 (Sη ) (which is of finite type). that φeff kills the non-zero object h Consequently, any proof of 1.5.1 via the functor φeff should use in a nontrivial way the assumption that the base field is of characteristic zero.
References ´: Motifs de dimension finie. S´eminaire Boubaki, 2004. [1] Y. Andre [2] M. Artin and A. Grothendieck: Th´eorie des topos et cohomologie ´etale des sch´emas. In S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie (SGA 4 I-IIIII). Dirig´e par M. Artin et A. Grothendieck. Lecture Notes in Mathematics, Vol. 269, 270 and 305. Springer-Verlag, Berlin-New York, 1972-73. [3] J. Ayoub: Th`ese de Doctorat de l’Universit´e Paris 7: Les six op´erations de Grothendieck et le formalisme des cycles ´evanescents dans le monde motivique. Preprint, December 12, 2005, K-theory Preprint Archives. http://www.math.uiuc.edu/K-theory/0761/. ´glise: Th`ese de Doctorat de l’Universit´e Paris 7: Module homotopiques [4] F. De avec transfers et motifs g´en´eriques. Preprint, January 16, 2006, K-theory Preprint Archives. http://www.math.uiuc.edu/K-theory/0766/. ´glise: [5] F. De Finite correspondances and transfers over a regular base. Preprint, January 16, 2006, K-theory Preprit Archives. http://www.math.uiuc.edu/K-theory/0765/. ´glise and C. Denis-Charles: Pr´emotifs. n preparation. [6] F. De [7] P. Deligne: Cat´egories tensorielles. Dedicated to Yuri I. Manin on the occasion of his 65th birthday, Mosc. Math. J. 2 (2002), no. 2, 227–248 . [8] P. Deligne: Voevodsky’s lectures on cross functors, Fall 2001, Preprint. http://www.math.ias.edu/~vladimir/seminar.html. [9] P. Deligne and N. Katz: Groupes de monodromie en g´eom´etrie alg´ebrique. II in S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1967-1969 (SGA 7 II), Dirig´e par P. Deligne et N. Katz, Lecture Notes in Mathematics, 340. Springer-Verlag, Berlin-New York, 1973. [10] A. Grothendieck: Groupes de monodromie en g´eom´etrie alg´ebrique, I. in S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie 1967-1969 (SGA 7 I), Dirig´e par A. Grothendieck. Lecture Notes in Mathematics, Vol. 288. SpringerVerlag, Berlin-New York, 1972. [11] V. Guletskii: Finite dimensional objects in distinguished triangles, K-theory Preprint Archives. http://www.math.uiuc.edu/K-theory/0637/. [12] V. Guletskii and C. Pedrini: Finite dimensional motives and the conjectures of Beilinson and Murre, K-theory Preprint Archives. http://www.math.uiuc.edu/K-theory/0617/.
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[13] P. Hirschhorn: Model categories and their localizations. Mathematical Serveys and Monographs, Vol. 99 (2003). American Mathematical Society. [14] H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Annals of Mathematics (2) 79 (1964) 109-203: ibid. (2) 79 (1964) 205-326. [15] L. Illusie: Autour du th´eor`eme de monodromie locale. P´eriodes p-adiques (Bures-sur-Yvette, 1988), Ast´erisque No. 223 (1994), 9–57. [16] L. Illusie: On semistable reduction and the calculation of nearby cycles, in Geometric aspects of Dwork theory. Vol. I, II, 785–803, Walter de Gruyter GmbH & Co. KG, Berlin, 2004. [17] L. Illusie: Perversit´e et variation, Manuscripta Math. 112 (2003), no. 3, 271– 295. [18] L. Illusie: Sur la formule de Picard-Lefschetz, in Algebraic geometry 2000, Azumino (Hotaka), 249-268, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002. [19] F. Ivorra: Th`ese de Doctorat de l’Universit´e Paris 7: R´ealisation l-adique des motifs triangul´es g´eom´etriques. Preprint, January 2, 2006, K-theory Preprint Archives http://www.math.uiuc.edu/K-theory/0762/. [20] J. F. Jardine: A1 -local symmetric spectra, Preprint. http://www.math.ias.edu/ vladimir/seminar.html. [21] S. Kimura: Chow groups are finite dimensional, in some sense, Math. Ann. 331 (2005), no. 1, 173-201. [22] C. Mazza: Schur functors and motives, to be published in K-Theory Journal (an old version can be found on the K-theory preprint archive). [23] C. Mazza and A. D. Padrone: Schur-Finiteness and Nilpotency. Work in progress. http://math.ias.edu/~carlo/preprints.html. [24] F. Morel: Th´eorie homotopique des sch´emas, Ast´erisque 256 (1999). [25] F. Morel: An introduction to A1 -homotopy theory, in Contemporary Developments in Algebraic K-theory, I.C.T.P Lecture notes, 15 (2003), pp. 357–441, [26] F. Morel: On the motivic stable π0 of the sphere spectrum, in Axiomatic, Enriched and Motivic Homotopy Theory, 219–260, J.P.C. Greenlees (ed.), 2004 Kluwer Academic Publishers. [27] F. Morel: Rationalized motivic sphere spectrum and rational motivic cohomology. Work in progress. http://www.mathematik.uni-muenchen.de/~ morel/listepublications.html. [28] F. Morel and V. Voevodsky: A1 -homotopy theory of schemes, Publications Mathematiques de l’H.I.E.S 90 (1999), p. 45-143. [29] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat: Toroidal Embeddings I. Lecture Notes in Mathematics, 339, Springer-Verlag. [30] A. Neeman: Triangulated categories. Annals of Mathematics Studies 148 (2001). Princeton University Press. [31] D. Quillen: Homotopical algebra. Lecture Notes in Mathematics, 43 (1967). Springer-Verlag. ¨ [32] M. Rapoport and T. Zink: Uber die lokale Zetafunktion von Shimuravariet¨ aten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. (German) [On the local zeta function of Shimura varieties. Monodromy filtration and vanishing cycles in unequal characteristic], Invent. Math. 68 (1982), no. 1, 21–101. [33] J. Riou: Th´eorie homotopique des S-sch´emas, M´emoire de DEA, 2002. http://www.math.jussieu.fr/~riou/dea/.
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¨ ndigs: Functoriality in motivic homotopy theory. Preprint. [34] O. Ro http://www.math.uni-bielefeld.de/∼oroendigs/ [35] M. Spitzweck: Operads, algebras and modules in model Categories and Motives, PhD Dissertation. http://www.uni-math.gwdg.de/spitz/. [36] J. Steenbrink: Limits of Hodge structures, Invent. Math. 31 (1975/76), no. 3, 229–257. [37] V. Voevodsky: A1 -homotopy theory. Documenta Mathematica, Extra Volume ICM I (1998), 579-604. http://www.math.uni-bielefeld.de/documenta/xvol-icm/00/. [38] V. Voevodsky: Cancellation theorem, Preprint, January 28, 2002, K-theory Preprint Archives. http://www.math.uiuc.edu/K-theory/0541/. [39] V. Voevodsky: Cohomological theory of presheaves with transfers, in Cycles, transfers, and motivic homology theories, 87–137, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000. [40] V. Voevodsky: Triangulated categories of motives over a field, in Cycles, transfers, and motivic homology theories, 188–238, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000. [41] J. Wlodarczyk: Toroidal varieties and the weak factorization theorem. Preprint, June 2001, math. AG/9904076.
2 On the Theory of 1-Motives Luca Barbieri-Viale Dipartimento di Matematica Pura e Applicata, Universit` a degli Studi di Padova Via G. Belzoni, 7, Padova – I-35131, Italy
[email protected]
Dedicated to Jacob Murre
Abstract This is an overview and a preview of the theory of mixed motives of level ≤ 1 explaining some results, projects, ideas and indicating a bunch of problems.
Let k be an algebraically closed field of characteristic zero to start with and let S = Spec(k) denote our base scheme. Recall that Murre [47] associates to a smooth n-dimensional projective variety X over S a Chow cohomological Picard motive M 1 (X) along with the Albanese motive M 2n−1 (X). The projector π1 ∈ CHn (X × X)Q defining M 1 (X) is obtained via the isogeny Pic0 (X) → Alb(X) between the Picard and Albanese variety, given by the restriction to a smooth curve C on X since Alb(C) = Pic0 (C) (such a curve is obtained by successive hyperplane sections). For a survey of classical Chow motives see [55] (cf. also [4]). In the case of curves M 1 (X) is the Chow motive of X minus the lower and higher trivial components, i.e., M 0 (X) and M 2 (X), such that, for smooth projective curves X and Y Hom(M 1 (X), M 1 (Y )) ∼ = Hom(Pic0 (X), Pic0 (Y ))Q
(2.1)
by Weil (see [59, Thm. 22 on p. 161] and also a remark of Grothendieck and Manin [42]). Furthermore, the semi-simple abelian category of abelian varieties up to isogeny is the pseudo-abelian envelope of the category of Jacobians and Q-linear maps. Thus, such a theory of pure motives of 67
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smooth projective curves is known to be equivalent to the theory of abelian varieties up to isogeny, as pointed out by Grothendieck: one-dimensional (pure) motives are abelian varieties. Formula (2.1) suggests that we may take objects represented by Picfunctors as models for larger categories of mixed motives of any kind of curves over arbitrary base schemes S. However, non representability of Pic for open schemes, forces to refine our models. Let X be a closure of X with divisor at infinity X∞ , i.e., X = X − X∞ . For X smooth we have that Pic(X) is the cokernel of the canonical map Div∞ (X) → Pic(X) which for a divisor D on X supported at infinity, is the map D 7→ O(D). Thus, when X is smooth over S = Spec(k), following Deligne [23] and Serre [56], mapping divisors at infinity which are algebraically equivalent to zero to line bundles, we may take for our models [Div0∞ (X) → Pic0 (X)].
(2.2)
Therefore, a vague definition of our categories of 1-motives M can be visioned as two term - complexes (up to quasi-isomorphisms) of the following kind M := [L → G] where L is discrete-infinitesimal and G is continuous-connected. Moreover, we expect that a corresponding formula (2.1) would be available in the larger category of mixed motives.
2.1 On Picard Functors Let π : X → S and consider the Picard functor T PicX/S (T ) on the category of schemes over S obtained by sheafifying the functor T Pic(X ×S T ) with respect to the fppf-topology (= flat topology). This means that if π : X ×S T → T then 0 PicX/S (T ) := Hfppf (T, R1 π∗ (Gm |X×S T )).
If π∗ (OX ) = OS or by reducing to this assumption, e.g., if π is proper, the Leray spectral sequence along π and descent yields an exact sequence 2 2 0 → Pic(S) → Pic(X) → PicX/S (S) → Hfppf (S, Gm ) → Hfppf (X, Gm ).
Here the ´etale topology suffices since by a theorem of Grothendieck (see [30, i (−, G ) for all i ≥ VI.5 p. 126 & VI.11 p. 171]) we have H´eit (−, Gm ) ∼ = Hfppf m 0. If there is a section of π we then have that PicX/S (S) ∼ = Pic(X)/ Pic(S). If π : X → S is proper and flat over a base, the Picard fppf-sheaf PicX/S
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would be possibly representable by an algebraic space only. For a general theory we should stick to algebraic spaces not schemes (see [19, 8.3]). However, as long as S = Spec(k) is a field, we may just consider group schemes: by Grothendieck and Murre (see [46] and [19, 8.2]) we have that PicX/k is representable by a scheme locally of finite type over k. As a group scheme Pic0 usually stands for the connected component of the identity of Pic, and Pic0X/k is an abelian variety (known classically as the Picard variety, cf. [19, 8.4]) as soon as X is also smooth and k has zero characteristic. Here NSX := π0 (PicX/k ) is finitely generated. In positive characteristic, for X smooth and proper over k perfect, the connected component of the identity endowed with its reduced structure Pic0,red X/k is an abelian variety. More information, e.g., on the universal line bundle P on X ×S PicX/S , can be obtained from [19, §8]. For example, if X is a singular projective curve, over an algebraically ˜ is the normalization of X we then closed field of zero characteristic, and X have an extension ˜ →0 0 → V ⊕ T → Pic0 (X) → Pic0 (X) where V = Gra is a vector group and T = Gsm is a torus; in the geometric case, i.e., when k is algebraically closed, PicX/k (k) ∼ = Pic(X) and the torus splits. The additive part here is non homotopical invariant, that is, the semi-abelian quotient is homotopical invariant, e.g., consider the well known example of X = projective rational cusp: its first singular cohomology group is zero but Pic0X/k = Ga . For proper schemes in zero characteristic, we can describe the semi-abelian quotient of Pic-functors as follows.
2.1.1 Simplicial Picard Functors Let π : X· → X be a smooth proper hypercovering of X over S = Spec(k). Recall that X· is a simplicial scheme with smooth components obtained roughly as follows: X0 is a resolution of singularities of X, X1 is obtained by a resolution of singularities of X0 ×X X0 , etc. Such hypercoverings were introduced by Deligne [23] in characteristic zero (after Hironaka’s resolution of singularities) but are also available over a perfect field of positive characteristic (after de Jong’s theory [34]) by taking X0 an alteration of X (in this case X0 → X is only generically ´etale). Actually, in characteristic zero, it is possible to refine such a construction, obtaining a (semi)simplicial scheme X· such that dim(Xi ) = dim(X) − i so that the corresponding complex of algebraic varieties (in the sense of [9]) is bounded.
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2.1.1. Denote Pic(X· ) ∼ = H1fppf (X· , O∗X ) ∼ = H´e1t (X· , O∗X ) the group of iso-
·
·
morphism classes of simplicial line bundles on X· , i.e., of invertible OX · modules. Let PicX /S be the associated fppf-sheaf on S. Over S = Spec(k) · such PicX /S is also representable (see [11, A]). The canonical spectral se· quence for the components of X· yields the following long exact sequence of fppf-sheaves: ker((π1 )∗ Gm,X1 → (π2 )∗ Gm,X2 ) ,→ PicX /S → ker(PicX0 /S → PicX1 /S ) · Im((π0 )∗ Gm,X0 → (π1 )∗ Gm,X1 ) ker((π2 )∗ Gm,X2 → (π3 )∗ Gm,X3 ) (2.3) → Im((π1 )∗ Gm,X1 → (π2 )∗ Gm,X2 ) where πi : Xi → S are the structure morphisms. By pulling back along π : X· → X we have the following natural maps π∗
PicX/S → PicX
·/S → ker(PicX /S → PicX /S ). 0
1
The most wonderful property of hypercoverings is cohomological descent: H´e∗t (X, F) ∼ = H´e∗t (X· , π ∗ (F)) for any sheaf F on S´et (as well as for other usual topologies). In particular, for the ´etale sheaf µm ∼ = Z/m of m-rooths of unity on S = Spec(k), k = k and (m, char(k)) = 1, by (simplicial) Kummer theory (see [11, 5.1.2]) and cohomological descent we get the following commutative square of isomorphisms H1 (X , µm ) ∼ = Pic(X )m-tor ´ et
·
k 1 H´et (X, µm )
·
k ∼ = Pic(X)m-tor .
The simplicial N´eron-Severi group NS(X· ) := Pic(X· )/Pic0 (X· ) is finitely generated. Therefore the Tate module of Pic(X· ) is isomorphic to that of Pic0 (X· ) and, by cohomological descent, to that of Pic0 (X). Moreover, Pic0 (X· ) is the group of k-points of a semi-abelian variety, in which torsion points are Zariski dense. Scholium 2.1.2 ([11, 5.1.2]). If X is proper over S = Spec(k), k = k of characteristic 0, and π : X· → X is any smooth proper hypercovering, then π ∗ : Pic0 (X)→ →Pic0 (X· ) is a surjection with torsion free kernel.
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As a consequence, we see that the simplicial Picard variety Pic0 (X· ) is the semi-abelian quotient of the connected commutative algebraic group Pic0 (X). Moreover, if X is semi-normal, then π∗ (O∗X ) = O∗X , and so π ∗ :
·
Pic(X) ,→ Pic(X· ) is injective, by the Leray spectral sequence for the sheaf O∗X along π; therefore, from Scholium 2.1.2 we get
·
Pic0 (X) ∼ = Pic0 (X· ) ∼ = ker0 (Pic0 (X0 ) → Pic0 (X1 )) whenever Pic(X) ,→ Pic(X0 ) is also injective (here ker0 denotes the connected component of the identity of the kernel). Thus, if X is normal Pic0 (X) is an abelian variety which can be represented in terms of X0 and X1 only. If X is only semi-normal a similar argument applies and Pic0 (X) ∼ = Pic0 (X· ) is semi-abelian. 2.1.3. Homotopical invariance of units and Pic, i.e., H i (X, Gm ) ∼ = H i (A1S ×S 1 X, Gm ) for i = 0, 1 induced by the projection AS ×S X → X, is easily deduced for X smooth. Let A1S ×S X· → X· be the canonical projection; considering sequence (2.3) we see that Pic(X· ) ∼ = Pic(A1S ×S X· ) since X· has smooth components. Therefore, the semi-abelian quotient of Pic0 (X) is always homotopical invariant. By dealing with homotopical invariant theories we just need to avoid the additive factors, and Pic0 is the ‘motivic’ object corresponding to M 1 of proper (arbitrarily singular) S-schemes, i.e., Pic+ in the notation adopted in [11] (cf. [52] and also the commentaries below Conjecture 2.3.1). 2.1.4. In positive characteritic p > 0 the picture is more involved and a corresponding Scholium 2.1.2 is valid up to p-power torsion only. However, the semi-abelian scheme Pic0,red (X· ) is independent of the choices of the hyper1 covering X· (see [1, A.2]) furnishing a motivic definition of Hcrys (described in [1], cf. Conjecture 2.3.4 below).
2.1.2 Relative Picard Functors For a pair (X, Y ) consisting of a proper k-scheme X and a closed sub-scheme Y we have a natural long exact sequence H 0 (X, O∗X ) → H 0 (Y, O∗Y ) → Pic(X, Y ) → Pic(X) → Pic(Y )
(2.4)
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induced by the surjection of Zariski (or fppf) sheaves Gm,X → i∗ Gm,Y where i : Y ,→ X is the inclusion; here Pic(X, Y ) = H1 (X, Gm,X → i∗ Gm,Y ) is the group of isomorphism classes of pairs (L, ϕ) such that L is a line bundle on X and ϕ : L |Y ∼ = OY is a trivialization on Y (see [11, §2]). For (X, Y ) as above the fppf-sheaf associated to the relative Picard functor T
Pic(X ×k T, Y ×k T )
is representable by a k-group scheme which is locally of finite type over k (cf. [11, A] where the smoothness assumption can be removed). If Pic0 (X) is abelian, e.g., X is normal, the sequence (2.4) yields a semi-abelian group scheme Pic0 (X, Y ) (cf. [11, 2.1.2]) which can be represented as an extension (say of k-points over k = k of characteristic zero) H 0 (Y, O∗Y ) ,→ Pic0 (X, Y )→ → ker0 (Pic0 (X) → Pic0 (Y )) Im H 0 (X, O∗X )
(2.5)
where Pic0 (X, Y ) is the connected component of the identity of Pic(X, Y ), the k-torus is coker ((πX )∗ Gm,X → (πY )∗ Gm,Y ) where πX : X → Spec k, πY : Y → Spec k are the structure morphisms and where ker0 denotes the connected component of the identity of the kernel (the abelian quotient is further described below). 2.1.5. For example, assume X proper normal and Y = ∪Yi , where Yi are the (smooth) irreducible components of a reduced normal crossing divisor Y . ` Consider the normalization π : Yi → Y and observe that π ∗ : Pic(Y ) → ⊕ Pic(Yi ) is representable by an affine morphism (see [11, 2.1.2]). Therefore ker0 (Pic0 (X) → Pic0 (Y )) = ker0 (Pic0 (X) → ⊕ Pic0 (Yi )). Moreover, for any such pair (X, Y ), we have that (cf. [11, 2.2]) any relative Cartier divisor D ∈ Div(X, Y ), i.e., a divisor on X such that the support |D| ∩ Y = ∅, provides (OX (D), 1) which defines an element [D] ∈ Pic(X, Y ) where 1 denotes the tautological section of OX (D), trivializing it on X −|D|. Here a Cartier divisor D ∈ Div(X, Y ) is algebraically equivalent to zero relative to Y if [D] ∈ Pic0 (X, Y ). Denote Div0Z (X, Y ) ⊂ DivZ (X, Y ) the subgroup of relative Cartier divisors supported on a closed sub-scheme Z ⊂ X which are algebraically equivalent to zero relative to Y . We also have a ‘motivic’ object [Div0Z (X, Y ) → Pic0 (X, Y )] which morally corresponds to M 1 (X − Z, Y ).
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2.1.6. Starting from an open scheme X let X be a closure of X with boundary X∞ , i.e., X = X − X∞ . For Z = X∞ and Y = ∅ from the pair (X, ∅) we get (2.2) and for Y = X∞ we have [Div0Z (X, X∞ ) → Pic0 (X, X∞ )] (cf. [11, 2.2.1]). 2.1.3 Higher Picard Functors Let X be an equidimensional k-scheme. Let CHp (X) := Zp (X)/ ≡rat be the Chow group of codimension p-cycles modulo rational equivalence. Recall that CH1 (X) = Pic(X) if X is smooth but the Chow functor T CHp (X ×k T ) for 1 < p ≤ dim(X) doesn’t provide a representable functor even in the case when X is smooth and proper over k = k. 2.1.7. To supply this defect several proposed generalizations have been investigated (see [48], [41] and [32]). Consider the sub-group CHp (X)alg of those cycles in CHp (X) which are algebraically equivalent to zero and let NSp (X) := CHp (X)/ CHp (X)alg denote the N´eron-Severi group. Denote CHp (X)ab the sub-group of CHp (X)alg of those cycles which are abelian equivalent to zero†, i.e., CHp (X)ab is the intersection of all kernels of regular homomorphisms from CHp (X)alg to abelian varieties (see [48] for definitions and references). The main question here is about the existence of an ‘algebraic representative’, i.e., a universal regular homomorphism from CHp (X)alg to an abelian variety. In modern terms, one can rephrase it (equivalently or not) by asking if the homotopy invariant sheaf with transfers (see [58] for this notion) CHpX/k associated to X smooth is provided with a universal map to a 1-motivic sheaf (see [8] and [2], also Scholium 2.2.36 below). The abelian category Shv1 (k) of 1-motivic ´etale sheaves is given by those homotopy invariant sheaves with transfers F such that there is map f
G→F where G is continuous-connected (e.g., semi-abelian); ker f and coker f are discrete-infinitesimal (e.g., finitely generated). The paradigmatic example is F = PicX/k for X a smooth k-variety (see [8]). Starting from CHpX/k we may seek for cp : CHpX/k → (CHpX/k )(1) with (CHpX/k )(1) ∈ Shv1 (k) universally. Remark that the key point is to † Note that CHp (X)alg and CHp (X)ab are divisible groups.
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provide a finite type object as such a universal Ind-object always exists (see (1) [2]). Namely, CHp (X)alg will be related to the ‘algebraic representative’. 2.1.8. Assume the existence of a universal regular homomorphism ρp : CHp (X)alg → ApX/k (k) to (the group of k-points) of an abelian variety ApX/k defined over the base field k. This is given by Murre’s theorem for p = 2 (see [48]) and it is clear for p = 1, dim(X) by the theory of the Picard and Albanese varieties. We then quote the following functorial algebraic filtration Fa∗ on CHp (X) (cf. [5]): • Fa0 CHp (X) = CHp (X), • Fa1 CHp (X) = CHp (X)alg • Fa2 CHp (X) = CHp (X)ab , i.e., is the kernel of the universal regular homomorphism ρp above, • and the corresponding extension 0 → ApX/k (k) → CHp (X)/Fa2 → NSp (X) → 0.
(2.6)
Remark that Bloch, Beilinson and Murre (see [35]) conjectured the exis∗ on CHp (X) (with rational coefficients) such tence of a finite filtration Fm Q 1 CHp (X) is given by CHp (X) that Fm hom , i.e., by the sub-group of those codimension p cycles which are homologically equivalent to zero for some ∗ CHp (X) should be functorial and compatible Weil cohomology theory, Fm ∗ will be inducing the with the intersection pairing. The motivic filtration Fm ∗ ∩ CHp (X) algebraic (or 1-motivic) filtration Fa∗ somehow, e.g., Fa∗ = Fm alg for ∗ > 0. Remark that we may even push further this picture by seeking for the 1-motivic algebraically defined extension of codimension p cycles modulo numerical equivalence by ApX/k (k) (which pulls back to the extension (2.6), see also point 2.3.10 below).
2.2 On 1-Motives A free 1-motive over S (here S is any base scheme) in Deligne’s definition u (a 1-motif lisse cf. [23, §10]) is a complex M := [L → G] of S-group schemes where G is semi-abelian, i.e., it is an extension of an abelian scheme A by a torus T over S, the group scheme L is, locally for the ´etale topology on S, isomorphic to a finitely-generated free abelian constant group, and u : L → G is an S-homomorphism. A 1-motive M can be represented by a
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diagram
0→ T
L ↓ → G → A →0
An effective morphism of 1-motives is a morphism of the corresponding complexes of group schemes (and actually of the corresponding diagrams). Any such a complex can be regarded as a complex of fppf-sheaves. Following the existing literature, L is placed in degree −1 and G in degree 0 (however, for some purposes, e.g., in order to match the conventions in Voevodsky triangulated categories, it is convenient to shift L in degree 0 and G in degree 1, cf. [8]). We let Mfr 1 denote the category of Deligne 1-motives (cf. § 2.2.9 below).
2.2.1 Generalities Mfr 1
has kernels and cokernels but images and coimages, It is easy to see that u in general, don’t coincide. For kernels, if kerc (φ) = [ker(f ) → ker(g)] is the kernel of φ = (f, g) : M → M 0 as a map of complexes then ker(φ) = u [ker0 (f ) → ker0 (g)] is the pull-back of ker0 (g) along u, where ker0 (g) is the connected component of the identity of the kernel of g : G → G0 and ker0 (f ) ⊆ ker(f ). u0
Similarly, for cokernels, if cokerc (φ) = [coker(f ) → coker(g)] is the cokernel as complexes and T is the torsion subgroup of coker(f ), as group schemes, then coker(φ) = [coker(f )/T → coker(g)/u0 (T)] is a Deligne’s 1-motive which is clearly a cokernel of φ. Associated to any 1-motive M there is a canonical extension (as two terms complexes) 0 → [0 → G] → M → [L → 0] → 0
(2.7)
2.2.1. Actually, a 1-motive M is canonically equipped with an increasing weight filtration by sub-1-motives as follows: M i≥0 [0 → G] i = −1 Wi (M ) = [0 → T ] i = −2 0 i ≤ −3
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W In particular we have grW −1 (M ) = [0 → A] and gr0 (M ) = [L → 0].
u
2.2.2. For S = Spec(k) a 1-motive M = [L → G] over k (a perfect field) is equivalent to the given semi-abelian k-scheme G, a finitely generated free u abelian group L which underlies a Gal(k/k)-module, a 1-motive [L → Gk ] over k, such that u is Gal(k/k)-equivariant, for the given module structure on ¯ In fact, the morphism L, and the natural semi-linear action on Gk = G×k k. u is determined uniquely by base change to k, i.e., by the morphism uk : Lk → Gk , which is Gal(k/k)-equivariant. 2.2.3. It is easy to see that there are no non-trivial quasi-isomorphisms between Deligne 1-motives. Actually, there is a canonical functor ι : Mfr 1 → Db (Sfppf ) which is a full embedding into the derived category of bounded complexes of sheaves for the fppf-topology on S. Scholium 2.2.4 ([53, Prop.2.3.1]). Let M and M 0 be free 1-motives. Then HomMfr (M, M 0 ) ∼ = HomDb (Sfppf ) (ι(M ), ι(M 0 )). 1
Proof The naive filtration of M = [L → G] and M 0 = [L → G0 ] yields a spectral sequence M E1p,q = Extq (i M, j M 0 )⇒Extp+q (M, M 0 ) −i+j=p
yielding complexes E1·,q Extq (G, L0 ) → Extq (G, G0 ) ⊕ Extq (L, L0 ) → Extq (L, G0 ), where the left-most non-zero term is in degree −1. We see that Ext0 (G, L0 ) = Hom(G, L0 ) = 0 since G is connected and Ext(G, L0 ) = 0 since L0 is free. Thus E20,0 = HomMfr (M, M 0 ) is Ext0 (M, M 0 ) = HomDb (Sfppf ) (ι(M ), ι(M 0 )). 1
2.2.2 Hodge Realization The Hodge realization THodge (M ) of a 1-motive M over S = Spec(C) (see [23, 10.1.3]) is (TZ (M ), W∗ , F ∗ ) where TZ (M ) is the lattice given by the pull-back of u : L → G along exp : Lie (G) → G, W∗ is the integrally defined
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weight filtration† TZ (M ) i≥0 H1 (G) i = −1 Wi T (M ) := H (T ) i = −2 1 0 i ≤ −3 and F ∗ is the Hodge filtration defined by F 0 (TZ (M ) ⊗ C) := ker(TZ (M ) ⊗ C → Lie (G)). Then we see that THodge (M ) is a mixed Hodge structure and ∼ we have grW −1 THodge (M ) = H1 (A, Z) as pure polarizable Hodge structures of weight −1. 2.2.5. The functor '
fr THodge : Mfr 1 (C) −→ MHS1
is an equivalence between the category of 1-motives over C and the category of torsion free Z-mixed Hodge structures of type {(0, 0), (0, −1), (−1, 0), (−1, −1)} such that grW −1 is polarizable. Deligne (cf. [23, §10.1.3]) observed that such a H ∈ MHSfr 1 is equivalent to a 1-motive over the complex numbers. In fact, for H ∈ MHSfr 1 the canonical extension of mixed Hodge structures 0 → W−1 (H) → H → grW 0 (H) → 0
(2.8)
yields an extension class map (cf. [21]) eH : HomMHS (Z, grW 0 (H)) → ExtMHS (Z, W−1 (H)) which provides a 1-motive with lattice L := grW 0 (HZ ) mapping to the semiabelian variety with complex points G(C) := ExtMHS (Z, W−1 (H)). Summarizing, any 1-motive M over C has a covariant Hodge realization M
THodge (M )
and the exact sequence (2.7) gives rise to the exact sequence (2.8) of Hodge realizations. 2.2.6. We have that THodge ([0 → Gm ]) = Z(1) is the Hodge structure (pure of weight −2 and purely of type (−1, −1)) provided by the complex expo∗ nential √ exp : C → C , i.e., here TfrZ ([0 → Gm∨]) is the free Z-module on 2π −1. Recall that for H ∈ MHS1 we get H := Hom(H, Z(1)) ∈ MHSfr 1 † Note that H1 (G) is the kernel of exp : Lie (G) → G.
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where Hom is the internal Hom in MHS (see [23]). We have that Z∨ = Z(1). Moreover '
op fr ( )∨ := Hom( , Z(1)) : (MHSfr 1 ) → MHS1
is an anti-equivalence providing MHSfr 1 of a natural involution. We may introduce a contravariant Hodge realization given by M
T Hodge (M ) := THodge (M )∨
and an induced involution on Mfr 1 (C) defined by the formula −1 M ∨ := THodge ◦ T Hodge (M ).
Actually, such an involution can be made algebraic (see § 2.2.7 below) and is known as Cartier duality for 1-motives. 2.2.7. Remark that MHSfr 1 ⊂ MHS1 where we just drop the assumption that the underlying Z-module is torsion free and we have that the category MHS1 is a thick abelian sub-category of (graded polarizable) mixed Hodge structures. In [9, §1] an algebraic description of MHS1 is given (see § 2.2.9 below). For H ∈ MHS let H(1) denote the maximal sub-structure of the considered type (= largest 1-motivic sub-structure, for short) and let H (1) be the largest 1-motivic quotient. For H 0 ∈ MHS1 we clearly have HomMHS (H 0 , H) = HomMHS1 (H 0 , H(1) ) and HomMHS (H, H 0 ) = HomMHS1 (H (1) , H 0 ). In other words the embedding MHS1 ⊂ MHS has right and left adjoints given by the functors H 7→ H(1) and H 7→ H (1) respectively. Moreover, it is quite well known to the experts that Ext1MHS1 is right exact and the higher extension groups ExtiMHS1 (i > 1) vanish since similar assertions hold in MHS (by Carlson [21]) and the objects of MHS1 are stable by extensions in MHS. As a consequence, the derived category Db (MHS1 ) is a full subcategory of Db (MHS). ´ 2.2.3 Flat, `-adic and Etale Realizations u
Let M = [L → G] be a 1-motive over S which we consider as a complex of fppf-sheaves over S with L in degree −1 and G in degree 0. Consider the cone M/m of the multiplication by m on M . The exact sequence (2.7) of 1-motives yields a short exact sequence of cohomology sheaves 0 → H −1 (G/m) → H −1 (M/m) → H −1 (L[1]/m) → 0
(2.9)
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as soon as L is torsion-free, i.e., H −2 (L[1]/m) = ker(L → L) vanishes, since multiplication by m on G connected is an epimorphism of fppf-sheaves, i.e., m H 0 (G/m) = coker(G → G) vanishes. Here H −1 (G/m) = m-torsion of G and H −1 (L[1]/m) = L/m whence the sequence above is given by finite group schemes. The flat realization TZ/m (M ) := H −1 (M/m) 1 is a finite group scheme, flat over S, which is ´etale if S is defined over Z[ m ]. By taking the Cartier dual we also obtain a contravariant flat realization T Z/m (M ) := Hom fppf (H −1 (M/m), Gm ).
2.2.8. If ` is a prime number then the `-adic realization T` (M ) is the inverse limit over ν of TZ/`ν (M ). We have T` ([0 → Gm ]) = Z` (1) by the Kummer sequence. The `-adic realization of an abelian scheme A is the `-adic Tate module of A. In characteristic zero then Y T (M ) = T` (M ) Tb(M ) := lim Z/m ←− m
`
is called the ´etale realization of M . For S = Spec(k), Tb(Mk ), along with a natural action of Gal(k/k), is a (filtered) Galois module which is a free b Z-module of finite rank. Over S = Spec(k) and k = k we just have TZ/m (M )(k) =
{(x, g) ∈ L × G(k) | u(x) = −mg} . {(mx, −u(x)) | x ∈ L}
b 2.2.9. If k = C we then have a comparison isomorphism Tb(M ) ∼ = TZ (M )⊗ Z where TZ (M ) is the Z-module underlying to THodge (M ) (cf. [11, §1.3]). 2.2.4 Crystalline Realization Let S0 be a scheme and p a prime number such that p is locally nilpotent on S0 . Now let S0 ,→ Sn be a thickening defined by an ideal with nilpotent divided powers. Actually, over S0 = Spec(k) a perfect field of characteristic p > 0 and W(k) the Witt vectors of k (with the standard divided power structure† on its maximal ideal) a thickening Sn = Spec(Wn+1 (k)) is given by the affine scheme defined by the truncated Witt vectors of length n+1 (or equivalently by W(k)/pn+1 ). Suppose that M0 := [L0 → G0 ] is a 1-motive defined over S0 . Consider M0 [p∞ ] := lim TZ/pν (M0 ) −→ ν
† Note that for p = 2 the standard divided power structure of Wn (k) is not nilpotent.
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the direct limit being taken, in terms of the explicit formula above, for µ ≥ ν, by sending the class of a point (x, g) in L0 × G0 (k) to the class of (pµ−ν x, g). Such M0 [p∞ ] is a p-divisible (or Barsotti-Tate) group and the sequence (2.9) yields the exact sequence 0 → G0 [p∞ ] → M0 [p∞ ] → L0 [p∞ ] → 0
(2.10)
where L0 [p∞ ] := L0 ⊗ Qp /Zp . For M0 := [0 → A0 ] an abelian scheme we get back the Barsotti-Tate group of A0 . 2.2.10. Let D be the contravariant Dieudonn´e functor from the category of p-divisible groups over S0 = Spec(k) to the category of Dk -modules, for the Dieudonn´e ring Dk := W(k)[F, V ]/(F V = V F = p). This D is defined as the module of homomorphisms from the p-divisible group to the group of Witt covectors over k and provides an anti-equivalence from the category of p-divisible groups over k to the category of Dk -modules which are finitely generated and free as W(k)-modules (see [26]). For any such a thickening S0 ,→ Sn the functor D can be further extended to define a crystal on the nilpotent crystalline site on S0 that is (equivalently given by) the Lie algebra of the associated universal Ga -extension of the dual p-divisible group, by lifting it to Sn (cf. [43], [1]). Therefore, by taking D(M0 [p∞ ]) we further obtain a filtered F -crystal on the crystalline site of S0 , associated to the Barsotti-Tate group M0 [p∞ ]. Recall that (see [1]) the category of filtered F -W(k)-modules consists of finitely generated W(k)-modules endowed with an increasing filtration and a σ-linear† operator, the Frobenius F , respecting the filtration. Filtered F -crystals are the objects whose underlying W(k)-modules are free and there exists a σ −1 linear operator, the Verschiebung V , such that V ◦ F = F ◦ V = p. 2.2.11. The crystalline realizations of M0 over S0 = Spec(k) are the following filtered F -crystals (see [1, §1.3] where are also called Barsotti-Tate crystals of the 1-motive M0 and cf. [27] and [37, 4.7]). The contravariant one is T crys (M0 ) := lim D(M0 [p∞ ])(S0 ,→ Sn ) ←− n
and the covariant is ∞ ∨ Tcrys (M0 ) := lim D M [p ] (S0 ,→ Sn ) 0 ←− n
where M0 [p∞ ]∨ is the Cartier dual. It follows from the sequence (2.10) † Here σ is the Frobenius on W(k).
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that Tcrys (M0 ) admits Frobenius and Verschiebung operators and a filtration (respected by Frobenius and Verschiebung). 2.2.12. We get Tcrys ([0 → Gm ]) = W(k)(1) which is the filtered F -crystal W(k), with filtration Wi = W(k) if i ≥ −2 and Wi = 0 for i < −2 and with the σ-linear operator F given by 1 7→ 1 and the σ −1 -linear operator V defined by 1 7→ p.
2.2.5 De Rham Realization u
The De Rham realization of a 1-motive M = [L → G] over a suitable base scheme S is obtained via Grothendieck’s idea of universal Ga -extensions (cf. [43, §4], [23, 10.1.7] and [1]). Consider Ga as a complex of S-group schemes concentrated in degree 0. If G is any S-group scheme such that Hom (G, Ga ) = 0 and Ext (G, Ga ) is a locally free OS -module of finite rank, the universal Ga -extension is an extension of G by the (additive dual) vector group Ext (G, Ga )∨ (see [43]). u
2.2.13. Now for any 1-motive M = [L → G] over S, we have Hom (M, Ga ) = 0, and by the extension (2.7) Ext (M, Ga ) is of finite rank. Thus we obtain a universal Ga -extension M \ , in Deligne’s notation [23, 10.1.7], where u\
M \ = [L → G\ ] is a complex of S-group schemes† which is an extension of M by Ext (M, Ga )∨ considered as a complex in degree zero. Here we have an extension of S-group schemes 0 → Ext (M, Ga )∨ → G\ → G → 0 such that G\ is the push-out of the universal Ga -extension of the semiabelian scheme G along the inclusion of Ext (G, Ga )∨ into Ext (M, Ga )∨ . The canonical map u\ : L → G\ such that the composition u\
L → G\ → Hom (L, Ga )∨ is the natural evaluation map. The De Rham realization of M is then defined as TDR (M ) := Lie G\ , with the Hodge-De Rham filtration given by F 0 TDR (M ) := ker(Lie G\ → Lie G) ∼ = Ext (M, Ga )∨ † Note that G\ is not the universal Ga -extension of G unless L = 0.
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2.2.14. Over a base scheme on which p is locally nilpotent there is a canonical and functorial isomorphism (see [1, Prop. 1.2.8]) '
(M [p∞ ])\ ×M [p∞ ] G[p∞ ] −→ G\ ×G G[p∞ ] where (M [p∞ ])\ also denotes the universal Ga -extension of a Barsotti-Tate group. In particular, we have a natural isomorphism of Lie algebras '
Lie (M [p∞ ])\ −→ Lie G\
(2.11)
2.2.15. For S0 a scheme such that p is locally nilpotent and M0 = [L0 → G0 ] a 1-motive over S0 , let S0 ,→ S be a locally nilpotent pd thickening of S0 . Let M and M 0 be two 1-motives over S lifting M0 . We have proven (see [1, §3]) that there is a canonical isomorphism M \ ∼ = (M 0 )\ showing that the universal Ga -extension is crystalline. Define the crystal of (2-terms complexes of) group schemes M0\ on the nilpotent crystalline site of S0 as follows M0\ (S0 ,→ S) := M \ which we called the universal extension crystal of a 1-motive (see [1, §3]). Applying it to M0 defined over S0 = Spec(k) a perfect field and Sn = Spec(Wn+1 (k)) we see that the De Rham realization is a crystal indeed. Actually (see [1, §4] for details) the isomorphism (2.11) yields: Scholium 2.2.16 ([1, Thm. A0 ]). There is a comparison isomorphism of F -crystals Tcrys (M0 ) = TDR (M ) for any (formal) lifting M over W(k) of M0 over k. 2.2.17. If k = C then the De Rham realization is also compatible with the Hodge realization; we have TDR (M ) = THodge (M ) ⊗ C as bifiltered C-vector spaces, i.e., we have that H1 (G\ , Z) = H1 (G, Z) thus TC (M ) := TZ (M ) ⊗ C ∼ = Lie G\ and M \ = [L → TC (M )/H1 (G, Z)], see [23, §10.1.8].
2.2.6 Paradigma Let X be a (smooth) projective variety over k = k. Let PicX/k be the Picard scheme and Pic0,red X/k the connected component of the identity endowed with
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its reduced structure. Recall that NSX := π0 (PicX/k ) is finitely generated and Pic0,red X/k is divisible. Recall that we always have 1 Hfppf (X, µn ) = Pic(X)n-tor .
Therefore 1 T` ([0 → Pic0,red X/k ]) = Hfppf (X, Z` (1))
If ` 6= char(k) then the ´etale topology will be enough. 2.2.18. Let Pic\ (X) be the group of isomorphism classes of pairs (L, ∇) where L is a line bundle on X and ∇ is an integrable connection , on L. In characteristic zero then there is the following extension 0 → H 0 (X, Ω1X ) → Pic\,0 (X) → Pic0 (X) → 0 where Pic\,0 is the the subgroup of those pairs (L, ∇) such that L ∈ Pic0 . The above extension is the group of k-points of the universal Ga -extension of the abelian variety Pic0X/k , Lie Pic0 (X) = H 1 (X, OX ) and 1 Lie Pic\,0 (X) = HDR (X/k)
as k-vector spaces (as soon as the De Rham spectral sequence degenerates). Moreover, for k = C, the exponential sequence gives THodge ([0 → Pic0X/C ]) = H 1 (X, Z(1)). 2.2.19. In general, for an abelian S-scheme A (in any characteristics cf. [43, §4]) we have (A∨ )\ = Pic\,0 A/S , so that the dual of A has De Rham realization 1 TDR ([0 → A∨ ]) = HDR (A/S)(1)
where the twist (1) indicates that the indexing of the Hodge-De Rham filtration is shifted by 1 (cf. [11, §2.6.3]). However, for X (smooth and proper) over a perfect field k of characteristic 1 (X/k) cannot be recovered from the Picard p > 0, the k-vector space HDR scheme (as remarked by Oda [49]). The subspace obtained via the Picard scheme is closely related to crystalline cohomology (see [49, §5]). 2.2.20. Let X be smooth and proper over a perfect field k of characteristic p > 0. Let Piccrys,0 X/Sn be the sheaf on the fppf site on Sn = Spec(Wn+1 (k)) given by the functor associating to T the group of isomorphism classes of crystals of invertible Ocrys X×Sn T /T -modules (which are algebraically equivalent to 0 when restricted to the Zariski site). Such Piccrys is the natural substitute 1 for the previous functor Pic\ and we can view Hcrys as Lie Piccrys (see [1] for
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details, cf. [16]). In fact, the Sn [ε]-points of Piccrys reducing to the identity modulo ε are the infinitesimal deformations of Ocrys X/Sn . For A0 , an abelian variety over S0 = Spec(k), and an abelian scheme An over Sn lifting A0 , the category of crystals of invertible Ocrys A0 ×Sn Sn [ε]/Sn [ε] -modules over the nilpotent crystalline site of A0 ×S0 S0 [ε] relative to Sn [ε] is equivalent to the category of line bundles over An [ε] with integrable connection. Hence, we have an isomorphism of sheaves over the fppf site of Sn \,0 crys,0 \ ∼ ∼ (A∨ n ) = PicAn /Sn = PicA0 /Sn
and passing to Lie we get a natural isomorphism of OSn -modules crys,0 ∼ \ ∼ 1 T crys (A0 ) ⊗ OSn ∼ = Lie (A∨ n ) = Lie PicA0 /Sn = Hcrys (A0 /Sn ).
2.2.21. By applying the previous arguments to the Albanese variety Alb(X) = crys,0 ∨ (Pic0,red (Alb(X)) can be identified to the X/k ) = A0 we see that Lie Pic Lie algebra of the universal extension of a (formal) lifting of Pic0,red X/k to the Witt vectors. The Albanese mapping is further inducing a canonical isomorphism† (cf. [38, II.3.11.2], [16] and [1]) '
Lie Piccrys,0 (Alb(X)) −→ Lie Piccrys,0 (X). Concluding, we have ∼ 1 Tcrys ([0 → Pic0,red X/k ]) = Hcrys (X/W(k)) for X a smooth proper k-scheme.
2.2.7 Cartier Duality For H = THodge (M ), H ∨ = Hom(H, Z(1)) is an implicit definition (see § 2.2.2) of the dual M ∨ of a 1-motive M over C. In general, Deligne [23, §10.2.11–13] provided an extension of Cartier duality to (free) 1-motives showing that is compatible with such Hodge theoretic involution. The main deal here is the yoga of Grothendieck biextensions (see [45], [31, VII 2.1)] and [23, §10.2.1]). 2.2.22. A Grothendieck (commutative) biextension P of G1 and G2 by H is an H-torsor on G1 × G2 along with a structure of compatible isomorphisms of torsors Pg1 ,g2 Pg10 ,g2 ∼ = Pg1 g10 ,g2 and Pg1 ,g2 Pg1 ,g20 ∼ = Pg1 ,g2 g20 (including associativity and commutativity) for all points g1 , g10 of G1 and g2 , g20 of G2 . † Note that H 0 (Alb(X), Ω1X ) 6= H 0 (X, Ω1X ) in general, in positive characteristics.
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Recall that an isomorphism class of a Grothendieck biextension (as commutative groups in a Grothendieck topos) can be essentially translated by the formula (see [31, VII 3.6.5]) L
Biext(G1 , G2 ; H) = Ext(G1 ⊗ G2 , H). L
Here we further have Ext(G1 ⊗ G2 , H) = Ext(G1 , RHom(G2 , H)) and the canonical spectral sequence E2p,q = Extp (G1 , Ext q (G2 , H)) ⇒ Extp+q (G1 , RHom(G2 , H)) yields an exact sequence of low degree terms 0 → Ext(G1 , Hom (G2 , H)) → Biext(G1 , G2 ; H) → Hom(G1 , Ext (G2 , H)) → Ext2 (G1 , Hom (G2 , H)) ∼ Hom(G1 , Ext (G2 , H)). If Hom (G2 , H) = 0 then ∂ : Biext(G1 , G2 ; H) = In particular, for H = Gm and G2 = A an abelian scheme, since A∨ = Ext (A, Gm ) for abelian schemes, this isomorphism ∂ reduces to the more classical isomorphism (cf. [8, 4.1.3] and § 2.2.8 below). '
Hom(−, A∨ ) −→ Biext(−, A; Gm ) given by f 7→ bixetension PA sentable by the If G1 and G2
(2.12)
(f × 1)∗ PtA pulling back the (transposed) Poincar´e Gm of A and A∨ , i.e., the functor Biext(−, A; Gm ) is repredual abelian scheme. are semi-abelian schemes we further have Biext(A1 , A2 ; Gm ) ∼ = Biext(G1 , G2 ; Gm )
by pullback from the abelian quotients A1 and A2 (see [31, VIII 3.5-6]). Actually, we can regard biextensions of smooth connected group schemes (over a perfect base field) G1 and G2 by Gm as invertible sheaves on G1 ×G2 birigified with respect to the identity sections (see [31, VIII 4.3]). u
2.2.23. Now let Mi = [Li →i Gi ] for i = 1, 2 be two 2-terms complexes of sheaves. A biextension (P, τ, σ) of M1 and M2 by an abelian sheaf H is given by i) a Grothendieck biextension P of G1 and G2 by H and a pair of compatible trivializations, i.e., ii) a biadditive section τ of the biextension (1 × u2 )∗ (P ) over G1 × L2 , and iii) a biadditive section σ of the biextension (u1 × 1)∗ (P ) over L1 × G2 , such that
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iv) the two induced sections τ|L1 ×L2 = σ|L1 ×L2 coincide. Let Biext(M1 , M2 ; H) denote the group of isomorphism classes of biextensions. We still have the following fundamental formula (see [23, §10.2.1]) L
Biext(M1 , M2 ; H) = Ext(M1 ⊗ M2 , H) L
here Ext(M1 ⊗ M2 , H) = Ext(M1 , RHom(M2 , H)) where Mi is considered a complex of sheaves concentrated in degree −1 and 0. u
2.2.24. Let M = [L → G] be a 1-motive where G is an extension of an abelian scheme A by a torus T . The main point is that the functor on 1-motives N 7→ Biext(N, M ; Gm ) u∨
is representable, i.e., there is a Cartier dual M ∨ = [T ∨ → Gu ] such that '
Hom(N, M ∨ ) −→ Biext(N, M ; Gm )
(2.13)
is given by pulling back the Poincar´e biextension generalizing the isomorphism (2.12). More precisely, it is given by ϕ 7→ (ϕ × 1)∗ PtM where the Poincar´e Gm -biextension PM is simply obtained from that of A and A∨ by further pullback to G and Gu according to the above (and below) description. See [23, 10.2.11] and [11, 1.5] for the construction of M ∨ and [8, 4.1.1] for the representability given by formula (2.13). The Cartier dual can be described in the following way: u∨
• For M = [0 → G] we have M ∨ = [T ∨ → A∨ ] where T ∨ = Hom(T, Gm ) is the character group of T and u∨ is the canonical homomorphism pushing out characters T → Gm along the given extension G of A by T . u • For M = [L → A] we have M ∨ = [0 → Gu ] where Gu denote the group scheme which represents the functor associated to Ext(M, Gm ). Here Ext(M, Gm ) consists of extensions of A by Gm together with a trivialization of the pull-back on L. In particular [L → 0]∨ = Hom(L, Gm ). u • In general, the standard extension M = [L → G] of M/W−2 M = [L → A] by W−2 M = [0 → T ] provides via Ext(M/W−2 M, Gm ) the corresponding extension Gu of A∨ by Hom(L, Gm ) and a boundary map u∨ : Hom(W−2 M, Gm ) → Ext(M/W−2 M, Gm ) lifting T ∨ → A∨ as above.
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2.2.25. A biextension is also providing natural pairings in realizations (see [31, VIII 2] and [23, 10.2]). In fact, for Grothendieck biextensions we also have an exact sequence 0 → Ext(G1 ⊗ G2 , H) → Biext(G1 , G2 ; H) → Hom(Tor (G1 , G2 ), H) → Ext2 (G1 ⊗ G2 , H) and a natural map T` (G1 )⊗T` (G2 ) → T` (Tor (G1 , G2 )) (see [31, VIII 2.1.13]) yielding a map Hom(Tor (G1 , G2 ), H) → Hom(T` (G1 ) ⊗ T` (G2 ), T` (H)) which in turns, by composition, provides a map (see [31, VIII 2.2.3]) Biext(G1 , G2 ; H) → Hom(T` (G1 ) ⊗ T` (G2 ), T` (H)). Similarly (non trivially! cf. [23, §10.2.3-9] and [13]) a biextension P of 1-motives M1 and M2 by H = Gm provides the following pairings: T` (M1 ) ⊗ T` (M2 ) → T` (Gm ) and TDR (M1 ) ⊗ TDR (M2 ) → TDR (Gm ). This latter pairing on De Rham realizations is obtained by pulling back P to a \-biextension P \ of M1\ and M2\ by Gm . The Poincar´e biextension PM of M and M ∨ by Gm is then providing compatibilities between the Cartier dual of a 1-motive and the Cartier dual of its realizations. Moreover, over a base such that p is locally nilpotent, the Poincar´e biextension is crystalline (see [1, 3.4]) providing the Poincar´e crystal of biextensions P\0 of M0\ and (M0∨ )\ thus a pairing of F -crystals Tcrys (M0 ) ⊗ Tcrys (M0∨ ) → Tcrys (Gm ). We also have: Scholium 2.2.26 ([23, 10.2.3]). If M1 and M2 are defined over C then there is a natural isomorphism Biext(M1 , M2 ; Gm ) ∼ = HomMHS (THodge (M1 ) ⊗ THodge (M2 ), Z(1)) Over C, all these pairings on the realizations are deduced from Hodge theory.
2.2.8 Symmetric Avatar u
u∨
For a Deligne 1-motive M = [L → G] and its Cartier dual M ∨ = [T ∨ → Gu ] the Poincar´e biextension PM = (PA , τ, σ) of M and M ∨ by Gm is canonically trivialized on L × T ∨ by ψ := τ|L×T ∨ = σ|L×T ∨ given by the push-out map
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ψχ : G → χ∗ G along the character χ : T → Gm , i.e., we have 0→
T χ↓ 0 → Gm
→
G ψχ ↓ → χ∗ G
→ A →0 k → A →0
→A and ψ(x, χ) = ψχ (u(x)) ∈ χ∗ Gu(x) = (PA )(u(x),u∨ (χ)) where u : L → G→ and χ∗ G = u∨ (χ). Actually, the data of u : L → A, u∨ : T ∨ → A∨ and ψ determine both M and M ∨ under the slogan trivializations
⇐⇒
liftings
For example, for χ1 , . . . , χr a basis of T ∨ we can regard G as the the pullback of A diagonally embedded in Ar as follows 0→
T → G → A →0 k ↓ ↓ r 1 r 0 → Gm → χ∗ G × · · · × χ∗ G → Ar → 0
and (ψ(x, χ1 ), . . . , ψ(x, χr )) provides a point of G lifting u(x). u
u0
2.2.27. The symmetric avatar can be abstractly defined as (L → A, L0 → A0 , ψ) where L, L0 are lattices, A0 is dual to A and ψ : L×L0 → (u×u0 )∗ (PA ) is a trivialization of the Poincar´e biextension when restricted to L × L0 (cf. [23, 10.2.12]). In order to make up a category we define morphisms between symmetric avatars by pairs of commutative squares such that the trivializations are compatible, i.e., a map u
u0
u0
u
(L1 →1 A1 , L01 →1 A01 , ψ1 ) → (L2 →2 A2 , L02 →2 A02 , ψ2 ) is a map f : A1 → A2 along with its dual f 0 : A02 → A01 and a pair of liftings g : L1 → L2 of f u1 and g 0 : L02 → L01 of f 0 u02 such that ψ1 |L1 ×L02 = ψ2 |L1 ×L02 . Here we have used the property that (f × 1)∗ (PA2 ) = (1 × f 0 )∗ (PA1 ) for Poincar´e biextensions. Denote Msym this category. 1 Scholium 2.2.28 ([23, 10.2.14]). There is an equivalence of categories u
u∨
'
sym M 7→ (L → A, T ∨ → A∨ , ψ) : Mfr 1 −→ M1
Under this equivalence Cartier duality is u
u∨
u∨
u
(L → A, T ∨ → A∨ , ψ) 7→ (T ∨ → A∨ , L → A, ψ t ).
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2.2.29. For the sake of exposition we sketch how to construct a map of symmetric avatars out of any biextension (almost proving formula (2.13), see [8, 4.1] for more details). Let (P, τ, σ) be a Gm -biextension of Deligne 1motives M1 and M2 . Translating via extensions, P corresponds to a map f : ∨ A1 → A∨ ¯1 . 2 and the trivialization τ corresponds to a lifting g : L1 → T2 of f u Here we have P fu ¯1 0 ∈ Hom(L1 , Ext(G2 , Gm )) = Biext(L1 , G2 ; Gm ) where u∨
2 0 → Hom(G2 , Gm ) → T2∨ → A∨ 2 → Ext(G2 , Gm ) → 0
granting the existence of g such that u∨ ¯1 . Moreover f u ¯1 [E] = 2 g = fu 0 ∈ Ext(L1 ⊗ G2 , Gm ) = Biext(L1 , G2 ; Gm ) and any section (= trivialization) of such trivial Gm -extension E is exactly given by an element g ∈ Hom(L1 , T2∨ ) = Hom(L1 ⊗ T2 , Gm ) as above. Since P t corresponds to the dual f ∨ : A2 → A∨ 1 we have that σ also corresponds to a lift0 ∨ ing g 0 : L2 → T1∨ of f ∨ u2 yielding u∨ 1 g = f u2 . Moreover, since P is a pull-back of (f × 1A2 )∗ (PtA2 ) then the trivialization τ is the pull-back along g × 1 : L1 × G2 → T2∨ × G2 of the canonical trivialization ψ2t on T2∨ × G2 given by the identity.† Since P is also a pull-back of (1A1 × f ∨ )∗ (PA1 ) the trivialization σ on G1 × L2 is the pull-back of the canonical trivialization ψ1 on G1 × T1∨ along 1 × g 0 : G1 × L2 → G1 × T1∨ . Thus, if we further pull-back to L1 × L2 we get ψ2t |L1 ×L2 = τ |L1 ×L2 = σ |L1 ×L2 = ψ1 |L1 ×L2 by assumption. We therefore get a map u
u∨
u∨
u
1 2 ∨ 2 ∨ t A∨ (L1 →1 A1 , T1∨ → 1 , ψ1 ) → (T2 → A2 , L2 → A2 , ψ2 )
which provides a map M1 → M2∨ . 2.2.9 1-Motives with Torsion u
An effective 1-motive which admits torsion (see [9, §1] and [8]) is M = [L → G] where L is a locally constant (for the ´etale topology) finitely generated abelian group and G is a semi-abelian scheme. Here L can be represented by an extension 0 → Ltor → L → Lfr → 0 u
where Ltor is finite and Lfr is free. An effective map from M = [L → G] u0
to M 0 = [L0 → G0 ] is a commutative square and Homeff (M, M 0 ) denote † Note that for the Poincar´ e biextension we have that the resulting f , f ∨ , g and g 0 are all identities.
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the abelian group of effective morphisms. The corresponding category is u fr eff denoted Meff 1 . We clearly have that M1 ⊂ M1 . For M = [L → G] we set (see [9, §1] and [8]) u
Mfr := [Lfr → G/u(Ltor )] Mtor := [ker(u) ∩ Ltor → 0]
(2.14)
u
Mtf := [L/ ker(u) ∩ Ltor → G] considered as effective 1-motives. We say that M is torsion if L is torsion and G = 0, M is torsion-free if ker(u) ∩ Ltor = 0 and free if L is free. There are canonical effective maps M → Mtf , Mtor → M and Mtf → Mfr . 2.2.30. A quasi-isomorphism (q.i. for short) of 1-motives M → M 0 is a q.i. of complexes of group schemes. Actually, an effective map of 1-motives u
u0
M = [L → G] → M 0 = [L0 → G0 ] is a q.i. of complexes if and only if we have that ker(u) = ker(u0 ) and coker(u) = coker(u0 ) and thus ker and coker of L → L0 and G → G0 are equal. Then coker(G → G0 ) = 0, since it is connected and discrete, and ker(G → G0 ) is a finite group. Therefore a q.i. of 1-motives is given by an isogeny G → G0 such that L is the pull-back of L0 , i.e., 0→ E → k 0→ E →
G u ↑ L
G0 u0 ↑ → L0 →
→0 →0
where E is a finite group. We then define morphisms of 1-motives by localizing Meff 1 at the class of q.i. and thus set ˜ , M 0) Hom(M, M 0 ) := lim Homeff (M −→ q.i.
˜ → M as above. We then have a wellwhere the limit is taken over q.i. M defined composition of morphisms of 1-motives (see [9, 1.2]) Hom(M, M 0 ) × Hom(M 0 , M 00 ) → Hom(M, M 00 ). ˜ 0 → M 0 , there ˜ → M 0 and any q.i. M In fact, for any effective morphism M c→M ˜ together with an effective morphism M c→M ˜ 0 making exists a q.i. M 0 c→M ˜ is uniquely determined). up a commutative diagram (such that M 2.2.31. Denote the resulting category by M1 , i.e., objects are effective 1˜ →M motives and morphisms from M to M 0 can be represented by a q.i. M ˜ → M 0 . This category has been introduced and an effective morphism M
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in [9] and it is further investigated in [8]. The main basic facts are the following: • M1 is an abelian category where exact sequences can be represented by effective exact sequences of two terms complexes; • Mfr 1 ⊂ M1 is a Quillen exact sub-category such that M 7→ Mfr is leftadjoint to the embedding, i.e., we have Homeff (Mfr , M 0 ) = Hom(M, M 0 ) for M ∈ M1 and M 0 ∈ Mfr 1. Actually, we have Homeff (M, M 0 ) = Hom(M, M 0 ) for M ∈ M1 and M 0 ∈ Mfr 1 . Clearly, this is according with a corresponding Scholium 2.2.4 for the functor ι : M1 → Db (kfppf ) which is still faithful but, in general, not full for effective morphisms. A key point in order to show that M1 is abelian is the following. Scholium 2.2.32 ([9, Prop. 1.3]). Any effective morphism M → M 0 can be factored as follows M −→ M 0 & % ˜ M ˜ is an effective morphism such that the kernel of the morphism where M → M ˜ → M0 of semi-abelian varieties is connected, i.e., a strict morphism, and M is a q.i. For example, in the following canonical factorisation induced by the equations (2.14) M −→ Mfr & % Mtf the effective map M → Mtf is a strict epimorphism with kernel Mtor and Mtf → Mfr is a q.i. We then always have a canonical exact sequence in M1 0 → Mtor → M → Mfr → 0 We further have that the Hodge realization (see § 2.2.2) naturally extends to M1 (C) (see [9, Prop. 1.5]) and the functor '
THodge : M1 (C) −→ MHS1 is an equivalence between the category of 1-motives with torsion over C
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and the category of Z-mixed Hodge structures introduced in § 2.2.2 above. Similarly, the other realizations extend to M1 , e.g., (cf. [9], [8] and § 2.2.3) let M/`ν be the torsion 1-motive (= finite group) given by the cokernel of `ν : M → M the effective multiplication by `ν which is fitting in an exact sequence (of finite groups) 0 → `ν M → `ν L → `ν G → M/`ν → L/`ν → 0 and set M/`ν T` (M ) := lim ←− ν
Remark that Cartier duality does not extends to M1 : such category M1 is just an algebraic version of MHS1 . 2.2.33. In [8] (cf. §2.2.10 below) we also consider larger categories of nonconnected 1-motives, e.g., [L → G] where G is a reduced group scheme locally of finite type over k such that G0 is semi-abelian† and π0 (G) is finitely generated. If M = [L → G] is non-connected we get an effective 1-motive M 0 := [L0 → G0 ] where L0 ⊆ L is the subgroup of those elements mapping to G0 and π0 (M ) := [L/L0 ,→ π0 (G)] is a discrete object.
2.2.10 1-Motives up to Isogenies For any additive category C denote CQ the Q-linear category obtained from C by tensoring morphisms by Q. Let C1 := C [−1,0] (Shv(k´et )) be the category of complexes of ´etale sheaves of length 1 over Spec k. Then C1 and CQ 1 are abelian categories. We may eff as full subcategories of C , hence Mfr,Q and Meff,Q as a view Mfr and M 1 1 1 1 1 full subcategory of CQ . The abelian category of 1-motives up to isogenies 1 can be regarded via the following equivalences eff,Q ∼ Q ∼ Mfr,Q = M1 = M1 1
since torsion 1-motives vanish and q.i. of 1-motives are isomorphism in Meff,Q . 1 nc Furthermore, let M1 be the full subcategory of C1 consisting of non-connected 1-motives, i.e., complexes of the form [L → G] where L is finitely generated † Note that this condition can also be achieved by Murre’s axiomatic [11, Appendix A.1].
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and G is a commutative algebraic group whose connected component of the identity G0 is semi-abelian (see [8]). We have that Mnc 1 ⊂ C1 is an nc abelian (thick) subcategory of C1 . For M ∈ M1 we have that M 0 ,→ M ' and M 0 → →Mfr0 are isomorphisms in Mnc,Q . Thus Mfr,Q −→ Mnc,Q is an 1 1 1 equivalence of abelian categories. Scholium 2.2.34 ([8, 1.1.3]). The category of Deligne 1-motives up to isogeny is equivalent to the abelian Q-linear category given by complexes of ´etale sheaves [L → G] where L is (locally constant) finitely generated and G is a commutative algebraic group whose connected component of the identity G0 is semi-abelian. Finally, this category MQ 1 is of cohomological dimension ≤ 1, i.e., if i 0 Ext (M, M ) 6= 0, for M, M 0 ∈ MQ 1 , then i = 0 or 1 ([50, Prop. 3.13]) and, clearly, Scholium 2.2.4 holds for MQ 1 as well.
2.2.11 Universal Realization and Triangulated 1-Motives I briefly mention some results from [58], [50] and [8]. Considering the derived category of Deligne 1-motives up to isogeny we have a ‘universal realization’ in Voevodsky’s triangulated category of motives. Notably, this realization has a left adjoint: the ‘motivic Albanese complex’. 2.2.35. Recall that any abelian group scheme may be regarded as an ´etale sheaf with transfers (see [58] for this notion and cf. [44]). Moreover, a 1motive M = [L → G] is a complex of ´etale sheaves where L and the extension G of A by T are clearly homotopy invariants. Thus a 1-motive M gives rise to an effective complex of homotopy invariant ´etale sheaves with transfers, hence to an object of DMeff −,´ et (k) (see [58, Sect. 3] for motivic complexes over a field k). Regarding 1-motives up to isogeny Nisnevich sheaves will be enough as eff ∼ DMeff − (k; Q) = DM−,´ et (k; Q) is an equivalence of triangulated categories (see [58, Prop. 3.3.2] and [44, Th. 14.22]). The triangulated category of effective geometrical motives DMeff gm (k; Q) eff is the full triangulated sub-category of DM− (k; Q) generated by motives of smooth varieties: here the motive of X denoted M (X) ∈ DMeff − (k) is defined in [58] by the Suslin complex C∗ of the representable presheaf with transfers L(X) associated to X smooth over k. The motivic complexes provided by 1-motives up to isogeny actually belong to DMeff gm (k; Q) (cf. [50] and [8]). Scholium 2.2.36 ([58, Sect. 3.4, on page 218] [50]). There is a fully
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faithful functor '
eff eff Tot : Db (MQ 1 ) → d≤1 DMgm (k) ⊆ DMgm (k; Q)
whose essential image is the thick triangulated subcategory d≤1 DMeff gm (k) ⊆ eff DM− (k) generated by motives of smooth varieties of dimension ≤ 1. b Actually, in [8] we show that Db (Mfr 1 ) = D (M1 ) and we also refine this embedding to an integrally defined embedding of Db (M1 )[1/p] (where p is the exponential characteristic) into the ´etale version DMeff gm,´ et (k) of Voevodsky’s eff category. The homotopy t-structure on DM−,´et (k) induces a t-structure on Db (M1 ) ∼ = d≤1 DMeff gm,´ et (k) with heart the category Shv1 (k) of 1-motivic sheaves. Here we also have that Tot([0 → Gm ]) = Gm [−1] ∼ = Z(1) (see [44, Th. 4.1] and [8]).
2.2.37. For M ∈ DMeff gm there is an internal (effective) Hom(M, −) ∈ eff DM− (see [58, 3.2.8]). Set D≤1 (M ) := Hom(M, Z(1))
(2.15)
eff for any object M ∈ DMeff gm . Actually, D ≤1 (M ) ∈ d≤1 DMgm (see [8, 3.1.1]) and restricted to d≤1 DMeff gm is an involution (see [8, 3.1.2]). On the other hand, Cartier duality for 1-motives M 7→ M ∨ is an exact functor and extends to Db (M1 ). A key ingredient of [8] is that, under Tot, Cartier duality is transformed into the involution M 7→ Hom(M, Z(1)) on d≤1 DMeff gm (k; Q) given by the internal (effective) Hom above.
Scholium 2.2.38 ([8, 4.2]). We have a natural equivalence of functors '
η : ( )∨ −→ Tot−1 D≤1 Tot i.e., under the equivalence Tot we have '
eff Db (MQ 1 ) −→ d≤1 DMgm (k; Q) ∨ ↓ ( ) ↓ D≤1 ' eff b D (MQ ) −→ d DM ≤1 gm (k; Q) 1
Regarding Tot as the universal realization functor we expect that any Q other realization of Db (MQ 1 ) (hence of M1 ) will be obtained from a realizaeff tion of DMgm (k; Q) by composition with Tot and Cartier duality will be interchanging homological into cohomological theories. 2.2.39. We show in [8] that Tot has a left adjoint LAlb. Dually, composing with Cartier duality, we obtain RPic. In order to construct LAlb, let 2 eff d≤1 := D≤1 : DMeff gm (k; Q) → d≤1 DMgm (k; Q)
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denote the functor d≤1 (M ) = Hom(Hom(M, Z(1)), Z(1)) ∈ d≤1 DMeff gm (k; Q).
(2.16)
The evaluation map yields a canonical map aM : M → d≤1 (M ) that induces an isomorphism ∼
Hom(d≤1 M, M 0 ) −→ Hom(M, M 0 ) eff 0 0 for M ∈ DMeff gm (k; Q) and M ∈ d≤1 DMgm (k; Q). In fact, M = D ≤1 (N ) for some N ∈ d≤1 DMeff gm (k; Q) and if C is the cone of aM then
Hom(C, M 0 ) = Hom(C, D≤1 (N )) = Hom(C, Hom(N, Z(1))) = Hom(C ⊗ N, Z(1)) = Hom(N ⊗ C, Z(1)) = Hom(N, D≤1 (C)) = 0 3 =D . since D≤1 ≤1
Scholium 2.2.40 ([8, Sect. 2.2]). Define Q b LAlb : DMeff gm (k; Q) → D (M1 )
as the composition of d≤1 := D2≤1 in formula (2.16) and Tot−1 . It is left adjoint to the embedding eff Tot : Db (MQ 1 ) ,→ DMgm (k; Q)
and M 7→ aM is the unit of this adjunction. The Cartier dual of LAlb is RPic = Tot−1 D≤1 . 2.2.41. These functors provide natural complexes of 1-motives (up to isogeny) of any algebraic variety X over a field k if char(k) = 0 (for X smooth and k perfect even if char(k) > 0). Their basic properties are investigated in [8]. We have: • LAlb(X) := LAlb(M (X)) the homological Albanese complex which is covariant on X and, e.g., it is homotopy invariant and satisfies MayerVietoris; • LAlbc (X) := LAlb(M c (X)) the Borel-Moore Albanese complex which is covariant for proper morphisms and LAlb(X) = LAlbc (X) if X is proper;
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• LAlb∗ (X) := LAlb(M (X)∗ (n)[2n]) the cohomological Albanese complex of X purely n-dimensional, which is contravariant for maps between varieties of the same dimension and LAlbc (X) = LAlb∗ (X) if X is smooth (by motivic Poincar´e duality M c (X) = M (X)∗ (n)[2n], see [58, Th. 4.3.2]); and the Cartier duals: • RPic(X) := RPic(M (X)) the cohomological Picard complex which is contravariant in X; • RPicc (X) := RPic(M c (X)) the compactly supported Picard complex such that RPic(X) = RPicc (X) if X is proper; • RPic∗ (X) := RPic(M (X)∗ (n)[2n]) the homological Picard complex of X purely n-dimensional, which is covariant for maps between varieties of the same dimension and RPicc (X) = RPic∗ (X) if X is smooth. Remark that the unit aX : M (X) → Tot LAlb(X)
(2.17)
provide a universal map in DMeff gm (k; Q), the motivic Albanese map, which is an isomorphism if dim(X) ≤ 1 and it refines the classical Albanese map and the less classical map in [57]. 2.2.12 1-Motives with Additive Factors In order to keep care of non homotopical invariant theories we do have to include additive factors. This is also suitable in order to include, in the 1motivic world, the universal Ga -extension M \ of a Deligne 1-motive M . In order to make Cartier duality working we cannot simply take [L → G] where L is (free) finitely generated and G is a (connected) algebraic group: the b a , i.e., the connected formal additive Cartier dual of Ga is the formal group G k-group (see [26]and [28] for formal groups). Laumon [39] introduced a generalization of Deligne’s 1-motives in the following sense.
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2.2.42. Laumon’s 1-motives over a field k of characteristic zero are given by u
M := [F → G] where F is a torsion free formal group and G is a connected algebraic group, i.e., F has a presentation by a splitting extension 0 → F 0 → F → F´et → 0 where F´et ´etale over k is further assumed torsion free (which means F´et (k) = b a) Zr ) and F 0 is infinitesimal (that is given by a finite number of copies of G and G has a presentation 0→T +V →G→A→0 where T is a k-torus, V is a k-vector group and A is an abelian variety. The map u : F → G is any map of abelian fppf-sheaves so that an effective map M → M 0 is given by a map of complexes concentrated in degrees −1 and 0. denote this category. Let Ma,fr 1 2.2.43. Recall [28, 2.2.2] that we have an antiequivalence between (affine) algebraic groups and (commutative) formal groups, and, moreover, the following formula (see [39, 5.2.1]) holds: if such a formal group F has Cartier dual F ∨ and A has dual Pic0 (A) = A∨ then Hom(F, A) = Ext(A∨ , F ∨ ). Note that if F = F 0 is infinitesimal then F ∨ := Lie (F )∨ (= dual k-vector space of the Lie algebra) and the extension associated to F → A is here obtained from the universal Ga -extension Pic\ of A∨ by push-out along H 0 (A, Ω1A ) = Lie (A)∨ → Lie (F )∨ . The Cartier dual (cf. § 2.2.7) of M = u [F → G] is given by an extension Gu of A∨ by F ∨ associated to the composite F → G→ →A and a lifting of u∨ : (T + V )∨ → A∨ to Gu yielding u∨
M ∨ := [(T + V )∨ −→ Gu ] Moreover the Poincar´e biextension of M and M ∨ by Gm is obtained by pull-back from that of A and A∨ as usual (see [39, 5.2] for details). 2.2.44. We have the following paradigmatic examples (cf. § 2.2.6 and [39, 5.2.5]). If X is a proper k-scheme then [0 → Pic0X/k ] is a 1-motive defined by the Picard functor whose Cartier dual (= the homological Albanese 1motive) is [F → Alb(X)] where Alb(X) = coker(Alb(X1 ) → Alb(X0 )) is dual to the abelian quotient of Pic0X/k , F´et = Zr is the character group of b d corresponds to d-copies of Ga the torus, see sequence (2.3), and F 0 = G a
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in Pic0X/k . Let A be an abelian variety and let [0 → Pic\,0 A/k ] the 1-motive ∨ determined by the universal Ga -extension of the dual A . The Cartier dual b → A] where A b is the the completion at the origin of A. For A = Alb(X) is [A and X smooth proper over k (of zero characteristic) we have so described the Cartier dual of [0 → Pic\,0 X/k ]. 2.2.45. It seems possible to modify such a category, as we did (see § 2.2.9) for Deligne 1-motives, in order to include torsion, obtaining an abelian category. Just consider effective 1-motives M = [F → G] where F is any formal group, so that F´et may have torsion. However Cartier duality doesn’t extend (here F ∨ would be any, also non connected, algebraic group). Let Ma1 denote this category. Similarly (cf. Scholium 2.2.34) the category of Laumon 1-motives up to isogeny is equivalent to the abelian Q-linear category given by complexes of sheaves [F → G] where F is a formal group and G is a commutative algebraic group. 2.2.46. A related matter is the Hodge theoretic counterpart of Laumon’s 1-motives over C providing a generalized Hodge structure catching such additive factors (see [3]). Provisionally define a formal Hodge structure (of level ≤ 1) as follows. A formal group H and a two steps filtration on a C-vector space V , i.e., b d and V 0 ⊆ V 1 ⊆ V = Cn , along H = H 0 × H´et , H´et = Zr + torsion, H 0 = C with a mixed Hodge structure on the ´etale part, i.e., say H´et ∈ MHS1 for short, and a map v : H → V . Regarding the induced map v´et : H´et → V we require the following conditions: for HC := H´et ⊗ C with Hodge filtration 0 0 the canonical map, the following and c : H´et → HC /FHodge FHodge v
´ et H´et −→ V c ↓ ↓ pr ' 0 HC /FHodge −→ V /V 0
(2.18)
0 ∼ commutes in such a way that v´et yields an isomorphism HC /FHodge = V /V 0 restricting to an isomorphism W−2 HC ∼ = V 1 /V 0 . Denote (H, V ) for short such a structure and let FHS1 denote the category whose objects are (H, V ) and the (obvious) morphisms given by commutative squares compatibly with the data and preserving the conditions given by diagram (2.18), e.g., inducing a map of mixed Hodge structures on the ´etale parts. Here we then get a forgetful functor (H, V ) 7→ H´et from FHS1 0 to MHS1 , left inverse of the embedding H 7→ (H, HC /FHodge ). Actually we 0 can define (H, V )´et := (H´et , V /V ) and say that a formal Hodge structure is ´etale, if (H, V ) = (H, V )´et , i.e., if H 0 = V 0 = 0. The full subcate-
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gory FHS´1et of ´etale structures is then equivalent to MHS1 via the forgetful functor and the functor (H, V ) 7→ (H, V )´et is a left inverse of the inclusion FHS´1et ⊂ FHS1 . Remark that (H, V ) with H´et pure of weight zero exists if and only if V = V 1 = V 0 . Thus if v restricts to a map v 0 : H 0 → V 0 then (H 0 , V 0 ) is a formal substructure of (H, V ) and we have a ‘non canonical’ extension 0 → (H 0 , V 0 ) → (H, V ) → (H, V )´et → 0
(2.19)
Say that (H, V ) is connected if (H, V )´et = 0 and that it is special if (H 0 , V 0 ) := (H, V )0 is a substructure of (H, V ) or, equivalently, (H, V )´et is a quotient of (H, V ): the above extension (2.19) is then characterizing special structures. 2.2.47. Extending Deligne’s Hodge realization (cf. § 2.2.2) for a given 1motive M = [F → G] consider the pull-back TH (F ) of F → G along Lie (G) → G. Here TH (F ) is a formal group and the canonical map TH (F ) → Lie (G) provides the ‘formal Hodge realization’ of M TH (M ) := (TH (F ), Lie (G)) as follows. u For M = [F → G] over k let V (G) := Gna ⊆ G be the additive factor and display G as follows 0 → V (G) → G → G× → 0
(2.20)
where G× is the semi-abelian quotient. We have that Lie (G) is the pull-back of Lie (G× ) and H1 (G) = H1 (G× ). Moreover F = F 0 ×k F´et (canonically) and we can set M´et := [F´et → G× ]. The functor M 7→ M´et is a left inverse of the inclusion of Deligne’s 1-motives. We have that TH (F )´et is an extension of F´et by H1 (G× ) so that, by construction, the formal group TH (F ) has canonical extension 0 → F 0 → TH (F ) → TZ (M´et ) → 0 where TZ (M´et ) is the Z-module of the usual Hodge realization (see § 2.2.2) providing the formula TH (F )´et = TZ (M´et ) = the pullback of F´et ,→ F along TH (F ) → F . Thus (TH (F ), Lie (G)) ∈ FHS1 where TH (F )´et is the underlying group of the Hodge structure THodge (M´et ), the filtration V (G) ⊆ V (G) + Lie (T ) ⊆ Lie (G) is the two steps filtration and the condition (2.18) is provided by construction (see § 2.2.2) since TC (M´et ) := TZ (M´et ) ⊗ C ∼ = Lie (G\ ) (see § 2.2.5,
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here M´e\t = [F´et → G\ ] is the universal Ga -extension of M´et ), i.e., v
´ et TZ (M´et ) −→ Lie (G) c ↓ ↓ pr ' 0 TC (M´et )/FHodge −→ Lie (G× )
commutes and W−2 TC (M´et ) ∼ = Lie (T ). Moreover, if u restricted to F 0 is mapped to V (G) we can further set V (M ) := [F 0 → V (G)] fitting in an extension 0 → V (M ) → M → M´et → 0
(2.21)
providing a ‘non canonical’ extension of the 1-motive (cf. formula (2.19), here the extension (2.21) becomes fthe extension (2.19) by applying TH ). Note that M´et is pure of weight zero if and only if TH (M ) = M . For example, if u u c→ W → V is a linear map between C-vector spaces, and M = [W V ] is the c is the formal completion at the origin, cf. [39, induced 1-motive (here W 5.2.5]) then TH (M ) = M . Scholium 2.2.48 ([3]). There is an equivalence of categories M
'
fr TH (M ) : Ma,fr 1 (C) −→ FHS1
between Laumon’s 1-motives and torsion free formal Hodge structures (of level ≤ 1) providing a diagram '
Mfr → MHSfr 1 1 (C) ↑↓ ↑↓ ' fr Ma,fr 1 (C) → FHS1 As regards to duality for (H, V ) = TH (M ) such that H´et is free, we can argue cheaply defining it as follows TH (M )∨ := TH (M ∨ ) After Cartier duality (cf. [39, 5.2]) it is easy to check that this is a self duality ∨ ∨ extending the one on MHSfr et , Z(1)) = (H, V )´ 1 . For example H´ et := Hom(H´ et as usual and the Cartier dual of (H, V ) with H´et of weight zero such that the sequence (2.21) splits M = V (M ) ⊕ M´et , is obtained as follows: V (M )∨ ∨ \ is given by bv : V → Lie (H 0 )∨ obtained from the induced map Lie (H 0 ) → V , taking the dual vector space map V ∨ → Lie (H 0 )∨ and its completion ∨ \ bv : V → V ∨ at the origin† thus ∨
\ (H 0 × Z(0)⊕r , V )∨ = (bv : V × Z(1)⊕r , Lie (H 0 )∨ × G⊕r a ) † Note that in characteristic zero there is a canonical equivalence of categories between Lie algebras and infinitesimal formal groups.
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where the map Z(1)⊕r → Ga⊕r is canonically induced from the exponential map, e.g., in particular (Z(0), 0)∨ = (Z(1), Ga ). 2.2.49. Formal Hodge structures† would have to fit in the following diagram Deligne’s 1-motives ↑↓ Laumon’s 1-motives
THodge
−→ TH
−→
MHS ↑↓ FHS
where • FHS would be a rigid tensor abelian category which is an enlargement of MHS and H 7→ H´et would yield a functor from FHS to MHS, left inverse of the embedding; • TH would be fully faithful so that under the realizations Cartier duality corresponds to a canonical Hom(−, Za (1)) involution.‡ 2.2.50. Similarly define other realizations, e.g., see [6] where we obtain the sharp De Rham realization T] . For example, if F 0 = 0 we can describe T] out of the universal Ga -extension M´e\t (see § 2.2.5), defining the algebraic group G] by pull-back via the extension (2.20) as an extension 0 → Ext(M´et , Ga )∨ → G] → G → 0 taking Lie (G] ). In this case we thus obtain a canonical extension 0 → V (G) → M ] → M´e\t → 0 and we can relate to (H, V ) = TH (M ), where H 0 = F 0 = 0, passing to Lie algebras, by the following pull-back diagram
0 0 → FHodge k 0 → Ext(M´et , Ga )∨
0 ↑ 0 →0 → HC → HC /FHodge ↑ ↑ → Lie (G] ) → Lie (G) →0 ↑ V (G) ↑ 0
† We here mean to deal with arbitrary Hodge numbers. However, for the sake of brevity, no more details on FHS are provided: generalizing our definition above it’s not that difficult but it’s more appropriate to treat such a matter separately. ‡ Here we clearly have the candidate TH ([0 → Gm ]) := Za (1) and a formal version of Scholium 2.2.26 should be conceivable.
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0 where H´et = TZ (M´et ), HC = Lie (G\ ), HC /FHodge = Lie (G× ), V 0 = V (G) ⊆
V = Lie (G). Set HC] := Lie (G] ) and by the universal property we get an induced map c] : HZ → HC] providing a splitting of the projection HC] → →HC and the diagram above can be translated in the following diagram 0 HC → HC /FHodge ↑ ↑ HC] → V ↑ HC
(2.22)
2.2.51. Remark that, from a different point of view Bloch and Srinivas [18] proposed a category of enriched Hodge structures EHS whose objects are pairs E := (H, V· ) where H is a mixed Hodge structure and V· is a diagram (not a complex) · · · = Va+1 = Va → Va−1 → · · · → Vb → 0 → 0 · · · of C-vector spaces such that Vi → HC /F i (compatibly with the diagram) and there is a map HC → Va such that HC → Va → HC /F a is the identity, thus F a = 0. There is a canonical functor E 7→ H to MHS (with a right adjoint). It is not difficult to see that EHS1 is equivalent to FHSs1 ⊂ FHS1 the subcategory of special formal Hodge structures given by formula (2.19). Sharp De Rham realization also clearly provides an enriched Hodge structure, e.g., via diagram (2.22). In fact, we can refine the construction (2.22) obtaining a functor T]s : FHSs1 → EHS1 by sending (H, V ) 7→ T]s (H, V ) := (H´et , HC] → V ) where HC] (along with the splitting) is just obtained by pull-back when H 0 = 0 (see [6] for the precise statements and further properties).
2.3 On 1-Motivic (Co)homology In the previous section 2.2 we have provided realizations as covariant and contravariant functors from categories of 1-motives to categories of various kind of structures. Here we draft a picture (which goes back to the algebraic geometry constructions of section 1) providing 1-motives, i.e., 1-motivic cohomology, whose realizations are the ‘1-motivic part’ of various existing (or forthcoming) homology and cohomology theories.
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2.3.1 Albanese and Picard 1-Motives Let X be a complex algebraic variety and let H ∗ (X, Z) be the mixed Hodge structure on the singular cohomology of the associated analytic space. De∗ (X, Z(·)) ⊆ H ∗ (X, Z(·)) the largest substructure and H ∗ (X, Z(·))(1) note H(1) the largest quotient in MHS1 (cf. § 2.2.2). Conjecture 2.3.1 (Deligne’s Conjecture, ([23, 10.4])). Let X be a complex algebraic variety of dimension ≤ n. There exist algebraically defined 1i (X, Z(1)) , H i (X, Z(i))(1) motives whose Hodge realizations over C are H(1) fr fr (1)
for i ≤ n and H i (X, Z(n))fr for i ≥ n and similarly for `-adic and De Rham realizations. The results contained in [22], [11], [51], [52], [9] and [1] show some cases of this conjecture. Over a field k, char(k) = 0, with the notation of [11]: • Pic+ (X) which reduces to (2.2) if X is smooth (or to the simplicial 1 (X, Z(1)) = Pic0 if X is proper) provides an algebraic definition of H(1) fr H 1 (X, Z(1)); • Alb+ (X) is an algebraic definition of H 2n−1 (X, Z(n))fr for n = dim(X); • Pic− (X) = Alb+ (X)∨ is an algebraic definition of H2n−1 (X, Z(1 − n)); • Alb− (X) = Pic+ (X)∨ which reduces to Serre Albanese if X is smooth, is an algebraic definition of H1 (X, Z)fr . Moreover, in [9] we have constructed effective 1-motives with torsion: i+1 • Pic+ (X, i) for i ≥ 0 providing an algebraic definition of H(1) (X, Z(1))fr up to isogeny.
Actually, for Y a closed subvariety of X we have Pic+ (X, Y ; i) (= Mi+1 (X, Y ) in the notation of [9]) such that Pic+ (X, ∅; 0) = Pic+ (X). These Pic+ (X, Y ; i) are obtained using appropriate ‘bounded resolutions’ which also provide a canonical integral weight filtration W on the relative cohomology H ∗ (X, Y ; Z) (see [9, 2.3]). Scholium 2.3.2 ([9, 0.1]). There exists a canonical isomorphism of mixed Hodge structures '
i+1 φfr : THodge (Pic+ (X, Y ; i))fr −→ W0 H(1) (X, Y ; Z(1))fr
and similarly for the l-adic and de Rham realizations. ∗ (X, Z(1)) up to isogeny and This implies Deligne’s conjecture on H(1) fr
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in cohomological degrees ≤ 2 even without isogenies by dealing with such 1-motives with torsion. The conjecture without isogeny is reduced to ∗ ∗ (X, Y ; Z)fr . (X, Y ; Z)fr = W2 H(1) H(1) j Here the semiabelian part of Pic+ (X, Y ; i) yields W−1 H(1) (X, Y ; Z(1))fr and j the torus corresponds to W−2 H(1) (X, Y ; Z(1))fr .
2.3.3. In [1] we have also formulated a statement like Conjecture 2.3.1 for the crystalline realization. Recall that de Jong [34, p. 51-52] proposed a definition of crystalline cohomology† forcing cohomological descent. Let X be an algebraic variety, over a perfect field k, de Jong’s theory [34] provides a pair (X· , Y· ) where X· is a smooth proper simplicial scheme, Y· is a normal crossing divisor in X· and X· − Y· is a smooth proper hypercovering of X. ∗ (X/W(k)) := H∗ Set Hcrys logcrys (X· , Log Y· ) where (X· , Log Y· ) here denotes the simplicial logarithmic structure on X· determined by Y· (see [1, §6]). ∗ (X/W(k)) is not a priori wellThe question here (cf. [34]) is that Hcrys defined. Similarly to [9, 2.3] we may also expect a weight filtration W∗ on ∗ ((X, Y )/W(k)) of a pair (X, Y ). the crystalline cohomology Hcrys Over a perfect field, using de Jong’s resolutions, it is easy to obtain an appropriate construction of Pic+ (X, Y ; i) such that Pic+ (X, ∅; 0) = Pic+ (X) as above. In [1, Appendix A] we have shown that Pic+ (X) is really welldefined and independent of the choices of resolutions or compactifications. However, it is not clear, for i > 0, if Pic+ (X, Y ; i) is integrally well-defined: the 1-motive is well-defined up to p-power isogenies in characteristic p by [1, A.1.1] and a variant of [9, Thm. 3.4] for `-adic realizations with ` 6= p. Conjecture 2.3.4 (Crystalline Conjecture, ([1, Conj. C])). Let ∗ ∗ ((X, Y )/W(k)) whose Hcrys,(1) ((X, Y )/W(k)) denote the submodule of W2 Hcrys image in gr2W is generated by the image of the discrete part of Pic+ (X, Y ; i) under a suitable cycle map. Then there is a canonical isomorphism (eventually up to p-power isogenies) '
i+1 Tcrys (Pic+ (X, Y ; i)) −→ Hcrys,(1) ((X, Y )/W(k))(1)
of filtered F -W(k)-modules (i.e., we expect a crystalline analogue of Conjecture 2.3.1). We can show this statement for i = 0 and Y = ∅ (see [1, Thm. B0 ]). The corresponding general statement for De Rham cohomology over a field of characteristic zero is [9, Thm. 3.5]. † Note that we can also deal with rigid cohomology and everything here can be rephrased switching crystalline to rigid.
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2.3.5. According with the program in [8] these Pic+ (X, Y ; i) would get linked to Voevodsky’s theory of triangulated motives as follows. The covariant functor M : Sm/k → DMeff gm (k) from the category of smooth schemes of finite type over k (a field admitting resolution of singularities) extends to all schemes of finite type (see [58, §4.1]). Thus the motivic Albanese complex LAlb(X) and the motivic Picard complex RPic(X) are well-defined for any such scheme X (and similarly for the other complexes, see § 2.2.41). Consider the (co)homology 1-motives (up to isogenies) Hi (LAlb(X)) := Li Alb(X) and H i (RPic(X)) := Ri Pic(X) for i ∈ Z. We have that Ri Pic(X) = Li Alb(X)∨ by motivic Cartier duality (see Scholium 2.2.38). Hypothesis 2.3.6 (The LAlb - RPic Hypothesis (cf. [8]). We assume that, up to isogeny) we are in the following set-up (1)
• THodge (Li Alb(X)) = Hi (X, Z)fr = 1-motivic singular homology mixed Hodge structure; (1) • THodge (Li Albc (X)) = HiBM (X, Z)fr = 1-motivic Borel-Moore homology mixed Hodge structure; (1) • THodge (Li Alb∗ (X)) = H 2n−i (X, Z(n))fr = 1-motivic Tate twisted singular cohomology mixed Hodge structure of X n-dimensional; and dually: i (X, Z(1)) = 1-motivic singular cohomology mixed • THodge (Ri Pic(X)) = H(1) fr Hodge structure; c i • THodge (Ri Pic (X)) = Hc,(1) (X, Z(1))fr = 1-motivic compactly supported cohomology mixed Hodge structure; ∗ • THodge (Ri Pic (X)) = H2n−i,(1) (X, Z(1 − n))fr = 1-motivic Tate twisted singular homology of X n-dimensional.
Similar statements for `-adic, De Rham and crystalline realizations are also workable (providing a positive answer to Conjectures 2.3.1 and 2.3.4). It is not difficult (see [8]) to compute these 1-motivic (co)homology groups for X smooth or a singular curve. We recover in this way the DeligneLichtenbaum motivic (co)homology of curves (cf. [23] and [40]). The set-up above also recovers the previously mentioned Picard and Albanese 1-motives as follows L1 Alb(X) = Alb− (X)
L1 Alb∗ (X) = Alb+ (X)
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and R1 Pic(X) = Pic+ (X)
∗
R1 Pic (X) = Pic− (X).
Finally, for i ≥ 1, we should get a formula like that Ri Pic(X, Y ) = Pic+ (X, Y ; i−1)
Li Alb(X, Y ) = Alb− (X, Y ; i−1)
with the obvious meaningful notation adopted above.
2.3.2 Hodge 1-Motives We briefly explain the point of view developed in [5] extending Deligne’s philosophy from Conjecture 2.3.1 to algebraic cycles in higher codimension (cf. § 2.1.7). The starting point is by looking at the side of 2.3.2 which provides a Lefschetz theorem on (1, 1)-classes, i.e., in degrees > 1. See also [10]. 2.3.7. Define NS+ (X, Y ; i) for i ≥ 0 as the quotient of Pic+ (X, Y ; i + 1) by its toric part and consider the extension 0 → W−2 → Pic+ (X, Y ; i + 1) → NS+ (X, Y ; i) → 0. It follows from 2.3.2 up to isogeny 2+i THodge (NS+ (X, Y ; i)) = W0 H(1) (X, Y ; Z(1))/W−2
given by the extension 2+i 0 → grW (X, Y ; Z)/W0 → grW 1 → W2 H 2 →0
pulling back (1, 1)-classes in grW 2 (and twisting by Z(1)). We may call + NS (X, Y ; i) the Hodge-Lefschetz 1-motive since, e.g., if X is smooth proper and Y = ∅ we obtain NS+ (X; 0) = NS(X) and NS+ (X; i) = 0 for i 6= 0. 2.3.8. Set H := H 2p+i (X, Y ; Z) for a fixed p ≥ 1 and i ≥ −1 and consider W 0 → grW 2p−1 H → W2p H/W2p−2 H → gr2p H → 0
given by the integral weight filtration (see [9]). Consider the integral (p, p)classes HZp,p := HomMHS (Z(−p), grW 2p H) and the associated intermediate jacobian J p (H) := Ext(Z(−p), grW 2p−1 H) which is just a complex torus if p > 1 (see [21]). Consider the largest abelian subvariety Ap (H) of the torus J p (H) W H purely which corresponds to the maximal polarizable substructure of gr2p−1 of types {(p−1, p), (p, p−1)}. Define the group of Hodge cycles H p (H) as the
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preimage in HZp,p of Ap (H) under the extension class map ep : HZp,p → J p (H). Define the Hodge 1-motive by ep : H p (H) → Ap (H) and the corresponding mixed Hodge structure H h ∈ MHS1 . Conjecture 2.3.9 (The Anodyne Hodge Conjecture (cf. [5, 2.3.4])). Let X be an algebraic variety and Y a closed subvariety defined over a perfect field k. There exist algebraically defined 1-motives with torsion Ξi,p (X, Y ) ∈ M1 (k) whose Hodge realization over k = C are H 2p+i (X, Y ; Z)h ∈ MHS1 , i.e., here H 2p+i (X, Y ; Z) is the associated mixed Hodge structure (for p ≥ 1 and i ≥ −1) so that THodge (Ξi,p (X, Y )) ∼ = H 2p+i (X, Y ; Z)h and similarly for `-adic, De Rham and crystalline realizations. For p = 1 this is ‘almost’ true (≈ Deligne’s Conjecture 2.3.1 and 2.3.2, but see Conjecture 2.3.4) and it follows from 2.3.7 and Ξi,1 (X, Y ) = NS+ (X, Y ; i). One can also easily formulate a homological version of 2.3.9. Recall that for X smooth proper purely n-dimensional and Y + Z normal crossing divisors on X (in particular when X = X − Z and Y ∩ Z = ∅) we have H 2p+i (X − Z, Y ; Z(p)) ∼ = H2r−i (X − Y, Z; Z(−r))
(p = n − r)
as mixed Hodge structures (see [11, 2.4.2]). For X smooth and proper (here we assume that Y = Z = ∅ and X = X) we get H 2p+i (X, Z)h 6= 0 if and only if i = −1, 0 and 2.3.9 reduces to the quest of an algebraic definition of Ap ⊆ J p or HZp,p respectively. Classical Grothendieck-Hodge conjecture then provides candidates up to isogeny. 2.3.10. For X a smooth proper C-scheme we can consider Jap (X) ⊆ J p (X) the image of CHp (X)alg (cf. § 2.1.7) under the Abel-Jacobi map: the usual Grothendieck-Hodge conjecture claims that Jap (X) is the largest abelian variety in J p (X), i.e., that Ap = Jap (up to isogeny) and H 2p−1 (X, Z)h is algebraically defined via the coniveau filtration. Similarly, the image of NSp (X) generates HZp,p (with Q-coefficients). In the most wonderful world (mathematics!?) the 1-motivic sheaf (CHpX )(1) in § 2.1.7 could make the job providing an algebraically defined extension of HZp,p by Jap compatibly with the extension (2.6) (here Jap would also coincide with the universal regular quotient of CHp (X)alg when X is smooth and proper). If X is only proper then let π : X· → X be a resolution and consider the Chow groups of each component Xi of X· (which are proper and smooth).
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Let (N S p )• and (Jap )• denote the complexes induced by the simplicial structure and similarly to the extension given by (2.6) we obtain an extension of (N S p )• by (Jap )• . By taking homology groups we then get boundary maps λia : H i ((N S p )• ) → H i+1 ((Jap )• ). Conjecture 2.3.11 (The Hodge Conjecture ([5, 2.3.4])). The boundary map λia behave well with respect to the extension class map ep yielding a motivic cycle class map, i.e., the following diagram λi
H i ((N S p )• ) →a ↓
H i+1 ((Jap )• ) ↓
ep
H 2p+i (X)p,p → J p (H 2p+i (X)) commutes.† The image 1-motive (up to isogeny) is the Hodge 1-motive Ξi,p (X) corresponding to H 2p+i (X, Z)h . Moreover, one might then guess that the complex of 1-motivic sheaves (CHpX )(1) would provide such Hodge 1-motive directly.
·
2.3.3 Non-homotopical Invariant Theories A typical problem occurring with homotopical invariant theories attached to singular varieties is that they do not catch all of the information coming from the singularities. In general, the cohomological Picard 1-motive Pic+ (X) of a proper scheme X is given by the semi-abelian quotient of Pic0 (X) (see Scholium 2.1.2). Forgetting its additive components we loose informations, e.g., we don’t see cusps. In order to reach the full picture here we have to enlarge our target to Laumon’s 1-motives at least. A natural guess is that our 1-motives are only the ´etale part of Laumon’s 1-motives, i.e., there a exists Pic+ a (X, Y ; i) ∈ M1 such that + Pic+ et = Pic (X, Y ; i) a (X, Y ; i)´ a and similarly RPica (X) ∈ Db (Ma1 ) (cf. Hypothesis 2.3.6), Ξi,p a (X) ∈ M1 i,p i,p such that Ξa (X)´et = Ξ (X) (cf. Conjecture 2.3.9), etc.‡ Their geometrical sources are additive Chow groups and their universal regular quotients, cf. [17] and, by the way, see [24] for a construction of an additive version of the cohomological Albanese Alb+ a (X) of a projective † Note that all maps in the square are canonically defined. ‡ Note that such Laumon 1-motives should rather be visible from a triangulated viewpoint! There should be a “sharp” cohomological motive M] (X) in a triangulated category DM] , related to Voevodsky category of motivic complexes, with a realisation in Db (FHS). The conjectural formalism for motivic complexes should be translated for ]-motivic complexes.
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+ ∗ variety X, i.e., here Alb+ et = Alb (X) = L1 Alb (X), etc. as above. a (X)´ Similarly, for X quasi-projective, we expect a formal part defining Alb+ a (X) as a Laumon 1-motive.
2.3.12. The forthcoming theories are eqn:sharp cohomology theories, e.g., ]-singular cohomology X H]∗ (X) ∈ FHS for X over C, ]-De Rham co∗ homology H]−DR (X) of X k-algebraic over a field k of zero characteristic and ]-crystalline cohomology in positive characteristics, which are nonhomotopical invariant theories. Sample H]1 (X) := TH (Pic+ a (X)) 0 Here Pic+ a (X) = [0 → Pic (X)] if X is proper: in this case, define the ] group scheme Pic by the following pull-back square (cf. diagram (2.22), the assertions in 2.2.18, Scholium 2.1.2 and [11, 4.5])
Pic\ (X· ) → Pic(X· ) ↑ ↑ ] Pic (X) → Pic(X)
(2.23)
such that • ker(Pic],0 (X)→ → Pic0 (X)) = H 0 (X· , Ω1X ) and
·
• ker(Pic],0 (X)→ →Pic\,0 (X· )) is the additive subgroup ⊆ Pic0 (X); then 1 (X) := Lie Pic],0 (X) H]−DR 1 (X) by the additive part of 1 (X) is an extension of HDR so that H]−DR Pic0 (X).
2.3.13. Similarly, remark that (see [18]) for E = (H, V· ) ∈ EHS there is a surjection ExtEHS (Z(0), E)→ → ExtMHS (Z(0), H) and the kernel of this map is a vector space if H = H 2r−1 (X, Z(r)) and i−1 Vi = H2r−1 (X, OX → · · · → ΩX )(r), where X is a proper C-scheme; in particular, if X is the cuspidal curve then ExtEHS (Z(0), E) is the additive group Ga = Pic0 (X). 2.3.4 Final Remarks Hoping to have puzzled the reader enough to procede on these matters I would finally remark that this exposition is far from being exhaustive.
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2.3.14. For example, for S = Spec(R) where R is a complete discrete valuation ring and K its function field, a 1-motive over K with good reduction (resp. potentially good reduction) is defined (in [53]) by the property of yielding a 1-motive over R (resp. after a finite extension of K). To any 1motive M = [L → G] over K is canonically associated (see [53] for details) a strict 1-motive over K, i.e., M 0 = [L0 → G0 ], such that G0 has potentially 0 →M good reduction, a quasi-isomorphism Mrig rig in the derived category of bounded complexes of fppf-sheaves on the rigid site of Spec(K), producing a canonical isomorphism T` (M 0 ) ∼ = T` (M ) between the `-adic realizations, for any prime `. For a strict 1-motive M = [L → G] over K, the geometric monodromy µ : L × T ∨ → Q (where T ∨ is the character group of the torus T ⊆ G) is defined by valuating the trivialization of the Poincar´e biextension. The geometric monodromy is zero if and only if M has potentially good reduction. This theme is further investigated in [14]. 2.3.15. The use of 1-motives in arithmetical geometry is well testified, e.g., see [20], [54], [25], [33] and [37]. The Tate conjectures for abelian varieties over number fields (which have been proven by Faltings) do have a similar formulation for 1-motives. The proof of this result is sketched in [36, §4] where an ´etale version of the equivalence in 2.2.1 is also provided. Note that in [7] we also investigate L-functions with respect to Mordell-Weil and Tate-Shafarevich groups of 1-motives. Also the theme of 1-motivic Galois groups is afforded. For M a 1-motive over a field k of zero characteristic let M ⊗ be the Tannakian subcategory generated by M in suitable mixed realisations (hopefully mixed motives). The motivic Galois group of M , denoted Galmot (M ), is the fundamental group of M ⊗ . The group Galmot (M ) has an induced weight filtration W∗ and the unipotent radical W−1 Galmot (M ) has a nice characterisation (see [15]for details). Furtehrmore, Fontaine’s theory relating p-adic mixed Hodge structures over a finite extension K of Qp to mixed motives would provide categories of 1-motives over the p-adic field K (see [27]). 2.3.16. Passing from 1-motives to 2-motives is conceivable but (even conjecturally) harmless. A general guess is that there should be abelian categories M0 ⊆ M 1 ⊆ · · · ⊆ M where M0 = Artin motives, M1 = 1-motives and further on we have categories of n-motives Mn which can be realized as Serre subcategories of cohomological dimension ≤ n of the abelian category M of mixed motives. Assuming the existence of M a source of inspiration is [29], [58] and [12]: such Mn would be somehow ‘generated’ by motives of varieties of dimen-
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sion ≤ n and M (X), the motive of X smooth and projective, decomposes as ⊕M i (X)[−i] where M i (X) ∈ Mi such that M i (X) = M 2d−i (X) for d = dim(X).
Acknowledgement I would like to thank all my co-authors, in particular: F. Andreatta, A. Bertapelle, M. Bertolini, J. Ayoub, B. Kahn and M. Saito for useful discussions and a keen interest in these matters.†
References [1] Andreatta, F. and L. Barbieri-Viale: Crystalline realizations of 1-motives, Math. Ann. 331 N. 1 (2005) 111-172 [2] Ayoub, J. and L. Barbieri-Viale: On the category of 1-motivic sheaves, in preparation. [3] Barbieri-Viale, L.: Formal Hodge theory, (most up-to-date info at my home page) preprint http://arxiv.org/abs/math.AG/0511560 [4] Barbieri-Viale, L.: A pamphlet on motivic cohomology, Milan Journal of Math. 73 (2005) 53-73 [5] Barbieri-Viale, L.: On algebraic 1-motives related to Hodge cycles, in Algebraic Geometry – A Volume in Memory of P. Francia Walter de Gruyter, Berlin/New York, 2002, 25-60. [6] Barbieri-Viale, L. and A. Bertapelle: Sharp De Rham realization, in preparation. [7] Barbieri-Viale, L. and M. Bertolini: Values of 1-motivic L-functions, in preparation. [8] Barbieri-Viale, L. and B. Kahn: On the derived category of 1-motives, in preparation. [9] Barbieri-Viale, L., A. Rosenschon and M. Saito: Deligne’s conjecture on 1-motives, Annals of Math. 158 N. 2 (2003) 593-633. [10] Barbieri-Viale, L., A. Rosenschonand V. Srinivas: The N´eronSeveri group of a proper seminormal complex variety, preprint http://arxiv.org/abs/math.AG/0511558 [11] Barbieri-Viale, L. and V. Srinivas: Albanese and Picard 1-motives, M´emoire SMF 87, Paris, 2001. [12] Beilinson, B.: Remarks on n-motives and correspondences at the generic point, in Motives, polylogarithms and Hodge theory , Part I (Irvine, CA, 1998) 35–46, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002. [13] A. Bertapelle: Deligne’s duality on de Rham realizations of 1-motives, preprint http://arxiv.org/abs/math.AG/0506344 [14] Bertapelle, M., M. Candilera and V. Cristante: Monodromy of logarithmic Barsotti-Tate groups attached to 1-motives, J. Reine Angew. Math. 573 (2004), 211–234. † Note that work in progress and preliminary versions of my papers are firstly published on the web and currently updated, e.g., browsing from my home page.
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[15] Bertolin, C.: Le radical unipotent du groupe de Galois motivique d’un 1motif, Math. Ann. 327 (2003), no. 3, 585–607. [16] Berthelot, P.: Cohomologie cristalline des sch´emas de caract´eristique p > 0, Springer LNM 407, 1974. [17] Bloch, S. and H. Esnault: An additive version of higher Chow groups, Ann. Sci. Ec. Norm. Sup. (4) 36 (2003), no. 3, 463–477. [18] Bloch, S. and V. Srinivas: Enriched Hodge Structures, in Algebra, arithmetic and geometry. Part I, II. Papers from the International Colloquium held in Mumbai, January 4–12, 2000. Edited by R. Parimala. Tata Institute of Fundamental Research Studies in Mathematics, 16, 171-184. [19] Bosch, S., W. Lutkebohmert and M. Raynaud: N´eron Models, Springer Ergebnisse der Math. 21 Heidelberg, 1990. [20] Brylinski, J.-L. : 1-motifs et formes automorphes (th´eorie arithm´etique des domaines de Siegel) in Conference on automorphic theory (Dijon, 1981), 43– 106, Publ. Math. Univ. Paris VII, 15, 1983. [21] Carlson, J.A.: The obstruction to splitting a mixed Hodge structure over the integers, I, University of Utah, Salt Lake City, 1979. [22] Carlson, J.A.: The one-motif of an algebraic surface, Compositio Math. 56 (1985) 271–314. [23] Deligne, P.: Th´eorie de Hodge III Publ. Math. IHES 44 (1974) 5–78. [24] Esnault, H., V. Srinivas and E. Viehweg: The universal regular quotient of the Chow group of points on projective varieties, Invent. Math. 135 (1999), no. 3, 595–664. [25] Faltings, G. and C.-L. Chai: Degeneration of abelian varieties. With an appendix by David Mumford. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 22 Springer-Verlag, Berlin, 1990. [26] Fontaine, J.-M.: Groupes p-divisibles sur les corps locaux, Ast´erisque 47-48 Soci´et´e Math´ematique de France, Paris, 1977. [27] Fontaine, J.-M. and K. Joshi: Notes on 1-Motives, in preparation (1996). ´ [28] Gabriel, P.: Expos´e VIIB : Etude infinitesimale des Sch´emas en Groupes, in SGA3 Sch´emas en Groupes I, Springer LNM 151, 1970. [29] Grothendieck, A.: Motifs, manuscript, 1965-70. [30] Grothendieck, A.: Dix expos´es sur la cohomologie des sch´emas, North Holland, 1968. [31] Grothendieck, A.: Expos´e VII: Biextensions des faisceaux de groupes - Expos´e VIII: Compl´ements sur les biextensions, propri´et´es g´en´erales des biextensions des sch´emas en groupes, in SGA7 - Groupes de monodromie en g´eom´etrie alg´ebrique (1967-68), Springer LNM 288 340, 1972-73. [32] Griffiths, P. A.: Some transcendental methods in the study of algebraic cycles, Springer LNM 185, Heidelberg, 1971, 1-46. [33] Harari, D. and T. Szamuely: Arithmetic duality theorems for 1-motives, J. Reine Angew. Math. 578 (2005) 93-128 [34] de Jong, A.J.: Smoothness, semistability and alterations Publ. Math., IHES 83 (1996) 51–93. [35] Jannsen, U.: Motivic sheaves and filtrations on Chow groups, in Motives,’ AMS Proc. of Symp. in Pure Math. 55 Part 1, 1994, 245–302. [36] Jannsen, U.: Mixed motives, motivic cohomology, and Ext-groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), Birkh¨ auser, Basel, 1995, 667–679. [37] Kato, K. and F. Trihan: On the conjectures of Birch and Swinnerton-Dyer
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in characteristic p > 0, Invent. Math 153 (2003) 537-592. [38] Illusie, L.: Complexe de De Rham-Witt et cohomologie cristalline, Ann. sci´ Norm. Sup. 4e s´erie t. 12 (1979) 501–661. ent. Ec. [39] Laumon, G.: Transformation de Fourier generalis´ee, http://arxiv.org/abs/alggeom/9603004 - Preprint IHES (Transformation de Fourier g´eom´etrique, IHES/85/M/52) 47 pages. [40] Lichtenbaum, S.: Suslin homology and Deligne 1-motives, in Algebraic Ktheory and algebraic topology NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 407 Kluwer Acad. Publ., Dordrecht (1993) 189–197. [41] Lieberman, D.: Intermediate jacobians, in Algebraic Geometry Oslo, WoltersNordhoff, 1972. [42] Manin, Yu: Correspondences, motifs and monoidal transformations, Math. USSR Sbornik 6 (1968), 439–470. [43] Mazur, B. and W. Messing: Universal extensions and one dimensional crystalline cohomology, Springer LNM 370, 1974. [44] Mazza,C., V. Voevodsky and C. Weibel: Notes on Motivic Cohomology, Voevodsky’s Lectures at IAS 1999/2000, Preprint. [45] Mumford, D.: Biextensions of formal groups, in International Colloquium on Algebraic Geometry (Bombay, 1968) Oxford University Press, 1969, 307–322. [46] Murre, J.: On contravariant functors from the category of preschemes over a field into the category of abelian groups (with application to the Picard functor), Publ. Math., IHES 23 (1964) 581–619. [47] Murre, J.: On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990) 190–204. [48] Murre, J.: Applications of algebraic K-theory to the theory of algebraic cycles, in Proc. Algebraic Geometry, Sitges, Springer LNM 1124, 1985. [49] Oda, T.: The first De Rham cohomology group and Dieudonn´e modules, Ann. ´ Norm. Sup. 4e s´erie t. 2 (1969) 63-135. scient. Ec. [50] Orgogozo, F.: Motifs de dimension inf´erieure ou ´egale `a 1, Manuscripta mathematica 115 (3), 2004. [51] Ramachandran, N.: Duality of Albanese and Picard 1-motives, K-Theory 22 (2001), 271–301. [52] Ramachandran,N.: One-motives and a conjecture of Deligne, J. Algebraic Geom. 13 no. 1, (2004) 29–80. [53] Raynaud, M.: 1-motifs et monodromie g´eom´etrique, Expos´e VII, Ast´erisque 223 (1994) 295–319. [54] Ribet, K. A.: Cohomological realization of a family of 1-motives, J. Number Theory 25 (1987), no. 2, 152–161. [55] A. J. Scholl: Classical motives, in Motives AMS Proc. of Symp. in Pure Math. 55 Part 1, 1994, 163–187. [56] J.-P. Serre: Morphismes universels et vari´et´es d’Albanese/Morphismes universels et diff´erentielles de troisi´eme esp´ece, in Vari´et´es de Picard, ENS S´eminaire C. Chevalley 3e ann´ee: 1958/59. [57] Spiess, M. and Szamuely, T.: On the Albanese map for smooth quasiprojective varieties, Math. Ann. 325 (2003), no. 1, 1–17. [58] Voevodsky, V.: Triangulated categories of motives over a field, in Cycles, Transfers, and Motivic Cohomology Theories Princeton Univ. Press, Annals of Math. Studies 143 2000. [59] Weil, W.: Courbes alg´ebriques et vari´et´es ab´eliens, Hermann, Paris, 1971.
3 Intersection Forms, Topology of Maps and Motivic Decomposition for Resolutions of Threefolds Mark Andrea A. de Cataldo † Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA,
[email protected]
Luca Migliorini ‡ Dipartimento di Matematica, Universit` a di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy,
[email protected]
3.1 Introduction This paper has two aims. The former is to give an introduction to our earlier work [7] and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps. The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y . This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y . We assume familiarity with Hodge theory and with the formalism of derived categories. On the other hand, we provide a few explicit computations of perverse truncations and intersection cohomology complexes which we could not find in the literature and which may be helpful to understand the machinery. We discuss in detail the case of surfaces, threefolds and fourfolds. In the surface case, our “intersection forms” version of the decomposition theorem stems quite naturally from two well-known and widely used theorems on surfaces, the Grauert contractibility criterion for curves on a surface and the so called “Zariski Lemma,” cf. [1]. The following assumptions are made throughout the paper Assumption 3.1.1. We work with varieties over the complex numbers. A map f : X → Y is a proper morphism of varieties. We assume that X is smooth. All (co)homology groups are with rational coefficients. These assumptions are placed for ease of exposition only, for the main results remain valid when X is singular if one replaces the cohomology of X † Partially supported by N.S.F. Grant DMS 0202321 ‡ Partially supported by GNSAGA
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with its intersection cohomology, or the constant sheaf QX with the intersection cohomology complex of X. It is a pleasure to dedicate this work to J. Murre, with admiration and respect.
3.2 Intersection forms 3.2.1 Surfaces S
Let D = Dk ⊆ X be a finite union of compact irreducible curves on a smooth complex surface. There is a sequence of maps PD
r
r∗
∗ H2 (D) −→ H2BM (X) ' H 2 (X) −→ H 2 (D).
(3.1)
The group H2 (D) is freely generated by the fundamental classes [Dk ]. The group H 2 (D) ' H2 (D)∨ and, via Mayer-Vietoris, it is freely generated by the classes associated with points pk ∈ Dk . The map cl
H2 (D) −→ H 2 (X),
cl := P D ◦ r∗
is called the class map and it assigns to the fundamental class [Dk ] the cohomology class c1 (OX (Dk )). The restriction map r, or rather r ◦ P D, assigns to a Borel-Moore 2-cycle meeting transversely all the Dk , the points of intersection with the appropriate multiplicities. The composition H2 (D) −→ H 2 (D) gives rise to the so-called refined intersection form on D ⊆ X: ι : H2 (D) × H2 (D) −→ Q
(3.2)
with associated symmetric intersection matrix ||Dh · Dk ||. If X is replaced by the germ of a neighborhood of D, then X retracts to D so that all four spaces appearing in (3.1) have the same dimension b2 (D) =numbers of curves in D. In this case the restriction map r is an isomorphism: the Borel-Moore classes of disks transversal to the Dk map to the point of intersection. On the other hand, cl may fail to be injective, e.g. (C × P1 , {0} × P1 ). We recall two classical results results concerning the properties of the intersection form ι, dealing respectively with resolutions of normal surface singularities and one dimensional families of curves. They are known as the Grauert’s Criterion and the Zariski Lemma (cf. [1, p.90]). Theorem 3.2.1. Let f : X → Y be the contraction of a divisor D to a normal surface singularity. Then the refined intersection form ι on H2 (D) is negative definite. In particular, the class map cl is an isomorphism.
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Theorem 3.2.2. Let f : X → Y be a surjective proper map of quasiprojective smooth varieties, X a surface, Y a curve. Let D = f −1 (y) be any P fiber. Then the rank of cl is b2 (D) − 1. More precisely, let F = k ak Dk , ak > 0, be the cycle-theoretic fiber. F · F = 0 and the induced bilinear form H2 (D) H2 (D) × −→ Q h[F ]i h[F ]i is non degenerate and negative definite. Remark 3.2.3. Theorem 3.2.1 can be interpreted in terms of the topology of the “link” L of the singularity. Let N be a small contractible neighborhood of a singular point y and L be its boundary. Choose analytic disks ∆1 , · · · , ∆r cutting transversally the divisors D1 , · · · , Dr at regular points. The classes of these disks, generate the Borel-Moore homology H2BM (f −1 (N )) ' H 2 (f −1 (N )). The statement 3.2.1 implies that each class ∆i is homologous to a rational linear combination of exceptional curves. Equivalently, for every index i some multiple of the 1-cycle ∆i ∩ L bounds in the link L of y. This is precisely what fails in the aforementioned example (C × P1 , {0} × P1 ). A similar interpretation is possible for the “Zariski lemma.” In view of the important role played by these theorems in the theory of complex surfaces it is natural to ask for generalizations to higher dimension. We next define what is the analogue of the intersection form for a general map f : X → Y (cf. 3.1.1)
3.2.2 Intersection forms associated to a map General theorems, due to J. Mather, R. Thom and others (cf. [14]) ensure that a projective map f : X → Y can be stratified, i.e. there is a decom` position Y = S` of Y by locally closed nonsingular subvarieties S` , the strata, so that f : f −1 (S` ) → S` is, for any `, a topologically locally trivial fibration. Such stratification allows us, when X is nonsingular, to define a sequence of intersection forms. Let L be the pullback of an ample bundle on Y . The idea is to use sections of L to construct transverse slices and reduce the strata to points, and to use a very ample line bundle η on X to fix the ranges: Let dim S` = `, let s` a generic point of the stratum S` and Ys a complete intersection of ` hyperplane sections of Y passing through s` , transverse to S` ; as for surfaces, we consider the maps: I`,0 : Hn−` (f −1 (s` )) × Hn−` (f −1 (s` )) −→ Q.
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obtained intersecting cycles supported in f −1 (s) in the smooth (n − `)dimensional ambient variety f −1 (Ys ) : Hn−` (f −1 (s` )) → Hn−` (f −1 (Ys )) ' H n−` (f −1 (Ys )) → H n−` (f −1 (s` )). We can define other intersection forms, in different ranges, cutting the cycles in f −1 (s` ) with generic sections of η. The composition: Hn−`−k (f −1 (s)) → Hn−`−k (f −1 (Ys )) ' H n−`+k (f −1 (Ys )) → H n−`+k (f −1 (s)). gives maps I`,k : Hn−`−k (f −1 (s` )) × Hn−`+k (f −1 (s` )) −→ Q. Let us denote by \
η k : Hn−`+k (f −1 (s` )) → Hn−`−k (f −1 (s` )),
the operation of cutting a cycle in f −1 (s` ) with k generic sections of η. Composing this map with I`,k , we obtain the intersection forms we will consider: \ I`,k ( η k ·, ·) : Hn−`+k (f −1 (s` )) × Hn−`+k (f −1 (s` )) −→ Q. Remark 3.2.4. These intersection forms depend on η but not on the particular sections used to cut down the dimension. They are independent of L. In fact we could define them using a local slice of the stratum S` and its inverse image, without reference to sections of L. Example 3.2.5. Let f : X → Y be a resolution of singularities of a three` ` fold Y , with a stratification Y0 C y0 , defined so that f is an isomorphism over Y0 , the fibers are one-dimensional over C, and there is a divisor S D = Di contracted to the point y0 . We have the following intersection forms: • let c be a general point of C and s ∈ H 0 (Y, O(1)) be a generic section vanishing at c; there is the form H2 (f −1 (c) × H2 (f −1 (c)) −→ Q which is nothing but the Grauert-type form on the surface f −1 ({s = 0}); • similarly, over y0 , there is the form on H4 (D) given by η ∩[Di ]·[Dj ]; it is a Grauert-type form, computed on a hyperplane section of X with respect to η; • finally, we have the more interesting H3 (D) × H3 (D) −→ Q. One of the dominant themes of this paper is that Hodge theory affords non degeneracy results for these forms and that this non degeneration has strong cohomological consequences.
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To see why Hodge theory is relevant to the study of the intersection forms, let us sketch a proof of Theorem 3.2.1, under the hypothesis that X and Y are projective. The proof we give is certainly not the most natural or economic. Its interest lies in the fact that, while the original proof seems difficult to generalize to higher dimension, this one can be generalized. It is based on the observation that the classes [Di ] of the exceptional curves are “primitive” with respect to the cup product with the first Chern class of any ample line bundle pulled back from Y . Even though such a line bundle is certainly not ample, some parts of the “Hodge package,” namely the Hard Lefschetz theorem and the Hodge Riemann bilinear relations, go through. To prove this, we introduce a technique, which we call approximation of L-primitives, which plays a decisive role in what follows. Proof of 3.2.1 in the case X and Y are projective. Let L be the pullback to X of an ample line bundle on Y . Since the map is dominant, L2 6= 0, and we get the Hodge-Lefschetz type decomposition: H 2 (X, R) = Rhc1 (L)i ⊕ Ker{c1 (L)∧ : H 2 (X) → H 4 (X)}. Denote the kernel above by P 2 . This decomposition is orthogonal with respect to the Poincar´e duality pairing which, in turn, is non degenerate when restricted to the two summands. The decomposition holds with rational coefficients. However, real coefficients are more convenient because we take limits. Consider a sequence of Chern classes of ample Q-line bundles Ln , converging to the Chern class of L, e.g. Ln = L + (1/n)η, η ample on X. 2 Define P1/n = Ker{c1 (Ln ) : H 2 (X) → H 4 (X)}. These are (b2 − 1)2 dimensional subspaces of H 2 (X). Any limit point of the sequence P1/n in Pb2 (R) gives a codimension one subspace W ⊆ H 2 (X), contained in Ker{c1 (L) : H 2 (X) → H 4 (X)} = P 2 . Since dim W = b2 − 1 = dim P 2 , we 2 = P 2. must have limn P1/n 2 The Hodge Riemann Bilinear Relations hold on P1/n by classical Hodge 2 theory. The duality pairing on the limit P is non degenerate. It follows that the Hodge Riemann Bilinear Relations hold on P 2 as well. The classes of the exceptional curves Di are in P 2 , since we can choose a section of the very ample line bundle on Y not passing through the singular point and pull it back to X. The fact that these classes are independent is known classically. Let us briefly mention here that if there is only one component Di then 0 6= [Di ] ∈ H 2 (X) in the K¨ahler X. In general, one may also argue along the following lines (cf. [5], [4, §8]: use the Leray spectral sequence over an affine neighborhood V of the singularity y to show that
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H 2 (f −1 (V )) → H 2 (f −1 (y)) is surjective; use the basic properties of mixed Hodge structures to deduce that H 2 (X) → H 2 (f −1 (y)) is also surjective; conclude by dualizing and by Poincar´e Duality. The classes [Di ] are real of type (1, 1) and for such classes α ∈ P 2 ∩ H 1,1 the Hodge Riemann bilinear relations give Z α∧α < 0 X
whence the statement of 3.2.1.
3.2.3 Resolutions of isolated singularities in dimension 3 In this section we study the intersection forms in the case of the resolution of three-dimensional isolated singularities. Many of the features and techniques used in the general case emerge already in this case. Besides motivating what follows, we believe that the statements and the techniques used here are of some independent interest. We prove all the relevant Hodge-theoretic results about the intersection forms associated to the resolution of an isolated singular point on a threefold. This example will be reconsidered in the last section, where we give a motivic version of the Hodge theoretic decomposition proved here. As is suggested in the proof of Theorem 3.2.1 sketched at the end of the previous section, in order to draw conclusions on the behaviour of the intersection forms, we must investigate the extent to which the Hard Lefschetz theorem and the Hodge Riemann Bilinear Relations hold when we consider the cup product with the Chern class of the pullback of an ample bundle by a projective map. In order to motivate what follows let us recall an inductive proof of the Hard Lefschetz theorem based on the Hodge Riemann relations: Hard Lefschetz and Hodge-Riemann relations in dimension (n−1) and Weak Lefschetz in dimension n imply Hard Lefschetz in dimension n. Let X be projective nonsingular and XH be a generic hyperplane section with respect to a very ample bundle η. Consider the map c1 (η) : H n−1 (X) → H n+1 (X). The Hard Lefschetz theorem states it is an isomorphism. By the Weak Lefschetz Theorem i∗ : H n−1 (X) → H n−1 (XH ) is injective, and its dual i∗ : H n−1 (XH ) → H n+1 (X), with respect to Poincar´e duality on X
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and XH , is surjective. The cup product with c1 (η) is the composition i∗ ◦ i∗ H n−1 (X)
c1 (η)
NNN NNNi∗ NNN NN&
/ H n+1 (X) p8 i∗ pppp ppp ppp
H n−1 (XH )
R and is therefore an isomorphism if and only if the bilinear form XH remains non degenerate when restricted to the subspace H n−1 (X) ⊆ H n−1 (XH ). This inclusion is a Hodge substructure. The Hodge Riemann relations on XH imply that the Hodge structureR H n−1 (XH ) is a direct sum of Hodge structures polarized by the pairing XH . It follows that the restriction of R the Poincar´e form XH to H n−1 (X) is non degenerate, as wanted. The other cases of the Hard Lefschetz Theorem (i.e. c1 (η)k for k ≥ 2) follow immediately from the weak Lefschetz theorem and the Hard Lefschetz theorem for XH . Assumption 3.2.6. Y is projective with an isolated singular point y, dimY = 3. X is a resolution and f : X → Y is an isomorphism when restricted to f −1 (Y − y). Suppose D = f −1 (y) is a divisor and let Di be its irreducible components. As usual in this paper, we will denote by η a very ample line bundle on X, and by L the pullback to X of a very ample line bundle on Y . Of course L is not ample. We want to investigate whether the Hard Lefschetz theorem and the Hodge-Riemann relations hold if we consider cup-product with c1 (L) instead of with an ample line bundle. Remark 3.2.7. Since c1 (L)3 6= 0 we have an isomorphism c1 (L)3 : H 0 (X) → H 6 (X). Remark 3.2.8. Clearly the classes [Di ] ∈ H 2 (X) are killed by the cup product with c1 (L), since we can pick a generic section of OY (1) not passing through y and its inverse image in X will not meet the Di . Since [Di ] 6= 0, it follows that c1 (L) : H 2 (X) → H 4 (X) is not an isomorphism. We now prove that in fact the subspace Im{H4 (D) → H 2 (X)} generated by the classes [Di ] is precisely Ker c1 (L) : H 2 (X) → H 4 (X). Theorem 3.2.9. Let s ∈ Γ(Y, OY (1)) be a generic section and Xs = f −1 ({s = i 0}) → X. Then: a) i∗ : H 1 (X) → H 1 (Xs ) is an isomorphism. b) i∗ : H 3 (Xs ) → H 5 (X) is an isomorphism.
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c) i∗ : H 2 (X)/(Im{H4 (D) → H 2 (X)}) → H 2 (Xs ) is injective. d) i∗ : H 2 (Xs ) → Ker{H 4 (X) → H 4 (D)} is surjective. e) The map H3 (D) → H 3 (X) is injective. Proof Set X0 = X − Xs and Y0 = Y − {s = 0} and let us consider the Leray spectral sequence for f : X0 → Y0 . Since Y0 is affine, we have H k (Y0 ) = 0 for k > 3. O
H 4 (D) UU
UUUU UUUUd2 UUUU UUUU UUUU U* 3 H (D) UU UUUU UUUUd2 UUUU UUUU UUUU U* 2
H (D)
H 1 (D)
H 0 (Y0 )
H 1 (Y0 )
H 2 (Y0 )
H 3 (Y0 ) /
The sequence degenerates so that we have surjections H 3 (X0 ) → H 3 (D) and H 4 (X0 ) → H 4 (D). But from [12, Proposition 8.2.6], H 3 (X) → H 3 (D) → 0 and H 4 (X) → H 4 (D) → 0 are also surjective. We have the long exact sequence Hc1 (X0 )
H 5 (X0 )∗ = {0}
/ H 1 (X)
/ H 1 (Xs ) EE EE EE EE E"
0
/ H 2 (X0 )
/ H 2 (X) B H4 (X0 ) 1
H4 (D)
The other statements are obtained applying duality. Since on H 2 (Xs ) the bilinear relation of Hodge-Riemann hold, the argument given at the beginning of this section shows that c1 (L)2 : H 1 (X) −→ H 5 (X) is an isomorphism
/ H 2 (Xs )
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and c1 (L) : H 2 (X)/H4 (D) −→ Ker {H 4 (X) → H 4 (D)} is an isomorphism. The Hodge Riemann relations hold for P 1 := H 1 (X), P 2 := Ker c1 (L)2 : H 2 (X)/H4 (D) → H 6 (X) since, by the weak Lefschetz Theorem, they follow from those for Xs . The Hodge Riemann relations for P 3 = Ker{c1 (L) : H 3 (X) → H 5 (X)}, of which H3 (D) is a subspace, must be considered separately: the main technique to be used here is the approximation of primitives introduced in the previous section to prove Theorem 3.2.1. R Theorem 3.2.10. The Poincar´e pairing X is a polarization of P 3 Proof Since c1 (L)2 : H 1 (X) → H 5 (X) is an isomorphism, there is a decomposition, orthogonal with respect to the Poincar´e pairing, H 3 (X) = P 3 ⊕ c1 (L)H 1 (X) and, in particular, dim P 3 = b3 − b1 , just as if L were ample. The Poincar´e pairing remains non-degenerate when restricted to P 3 . The classes c1 (L)+(1/n)c1 (η) are Chern classes of ample line bundles, hence 3 = Ker{c (L) + (1/n)c (η) : H 3 (X) → H 5 (X)} are b − b -dimensional P1/n 1 1 3 1 subspaces of H 3 (X) . As in the proof of 3.2.1 a limit point of the sequence 3 , considered as points in the real Grassmannian Gr(b − b , b ), gives a P1/n 3 1 3 3 3 5 subspace of H (X), contained in Ker{c1 (L) : H (X) → H (X)} = P 3 and, 3 by equality of dimensions, lim P1/n = P 3 . The Hodge Riemann relations must then hold on the limit P 3 as explained in the proof of Theorem 3.2.1. Finally, let us remark that the cup-productRwith η gives an isomorphism c1 (η) : H4 (D) → H 4 (D) via the bilinear form X c1 (η) ∧ [Di ] ∧ [Dj ] which is negative definite. As we remarked in 3.2.5 this form is just the intersection form on the exceptional curves of the restriction of f to a hyperplane section (with respect to η) of X. Summarizing: c1 (L)2
H4 (D) 5 8
H0
H1
;
c1 (L)
?
,
H 2 /H4 (D) C F H 3 J
M
Ker{H 4 → H 4 (D)}
(
H5
c (L)3
P 1R
c1 (η)
U W Y [. 4 H (D)
the groups in the central row behave, with respect to L, as the cohomology of a projective nonsingular variety on which L is ample.
3 H6
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We are now in position to prove the first non-trivial fact on intersection forms which generalizes 3.2.1: Corollary 3.2.11. H 3 (D) has Hodge structure which is pure of weight 3 and the Poincar´e form is a polarization. In particular, the (skew-symmetric) intersection form H3 (D) × H3 (D) → Q is non degenerate. Proof This follows because H3 (D) → H 3 (X), is injective and identifies H 3 (D) with a Hodge substructure of the polarized Hodge structure P 3 ⊆ H 3 (X). This is a non-trivial criterion for a configuration of (singular) surfaces contained in a nonsingular threefold to be contractible to a point. See [7, Corollary 2.1.11] for a generalization to arbitrary dimension. For example, the purity of the Hodge structure implies that H 3 (D) = ⊕H 3 (Di ). We will see that the non degeneracy statement of 3.2.11 also plays an important role in the motivic decomposition of X described in section 3.5. Remark 3.2.12. The same analysis can be carried on with only notational changes for an arbitrary generically finite map from a nonsingular threefold X, e.g. assuming that there is also some divisor which is blown down to a curve etc. In this case the Hodge structure of X can be further decomposed, splitting of a piece corresponding to the contribution to cohomology of this divisor. Remark 3.2.13. The classical argument of Ramanujam [15], [13], to derive the Akizuki-Kodaira-Nakano Vanishing Theorem from Hodge theory and Weak Lefschetz can be adapted to give the following sharp version: if L is a line bundle on a threefold X, with L3 6= 0, a multiple of which is globally generated, then H p (X, ΩqX ⊗ L−1 ) = 0 for p+q < 2, and for p+q = 2 but (p, q) 6= (1, 1). More precisely H 1 (X, Ω1X ⊗ L−1 ) 6= 0 if and only if some divisor is contracted to a point.
3.2.4 Resolutions of isolated singularities in dimension 4 Let us quickly consider another similar example in dimension 4, Assumption 3.2.14. f : X → Y , where Y still has a unique singular point y and X is a resolution. As before, η will denote a very ample bundle on X, and L the pull-back of a very ample bundle on Y . Set D = f −1 (y).
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An argument completely analogous to the one used in the previous example shows that the sequence of spaces H 0 (X), H 1 (X), H 2 (X)/H6 (D), H 3 (X)/H5 (D), H 4 (X), Ker{H 5 (X) → H 5 (D)}, Ker{H 6 (X) → H 6 (D)}, H 7 (X), H 8 (X) satisfies the Hard Lefschetz Theorem with respect to the cup product with L. The corresponding primitive spaces P 1 , P 2 , P 3 , are endowed with pairing satisfying the Hodge Riemann bilinear relations. The new fact that we have to face shows up when studying the Hodge Riemann bilinear relations on H 4 (X). The “approximation of primitives” technique here must be modified, since the dimension of P 4 = Ker{c1 (L) : H 4 (X) → H 6 (X)} is greater than b4 − b2 . Hence, if we introduce the primitive spaces 4 = Ker{c (L) + (1/n)c (η) : H 4 (X) → H 6 (X)} with respect to the amP1/n 1 1 ple classes c1 (L) + (1/n)c1 (η), their limit is a proper subspace, of dimension b4 − b2 , of P 4 . We can determine the exact dimension of P 4 : Lemma 3.2.15. dim Ker{c1 (L) : H 4 (X) → H 6 (X)} = b4 −b2 +dim H6 (D). c1 (L)
c1 (L)
Proof Since c1 (L)2 : H 2 (X)/H6 (D) → H 4 (X) → Ker{H 6 (X) → H 6 (D)} is an isomorphism, we have an orthogonal decomposition H 4 (X) = P 4 ⊕ Im{c1 (L) : H 2 (X) → H 4 (X)}. The statement follows from: Ker{c1 (L) : H 2 (X) → H 4 (X)} = H6 (D). The “excess” dimension of P4 is thus dim H6 (D). On the other hand P 4 contains an obvious subspace of this dimension, namely c1 (η)H6 (D), the subspace generated by the classes obtained intersecting the irreducible components of the exceptional divisor with a generic hyperplane section. R Remark 3.2.16. The intersection form X c1 (η)2 ∧ [Di ] ∧ [Dj ] is negative definite, as it is just the intersection form on the exceptional curves of a double hyperplane section of X. This last remark implies the following orthogonal decomposition H 4 (X) = c1 (η)H6 (D)⊕(c1 (η)H6 (D))⊥ ∩P 4 ⊕Im{c1 (L) : H 2 (X) → H 4 (X)}. and (c1 (η)H6 (D))⊥ ∩ P 4 has dimension b4 − b2 . This subspace turns out to be the subspace of “approximable L-primitives” we are looking for, as shown in the following Theorem 3.2.17. lim Ker {c1 (L) +
n→∞
1 c1 (η) : H 4 (X) → H 6 (X)} = (c1 (η) H6 (D))⊥ ∩ P 4 . n
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Proof The two subspaces have the same dimension, so it is enough to prove that 1 Ker[{c1 (L) + c1 (η) : {H 4 (X) → H 6 (X)}}] ⊆ (c1 (η)H6 (D))⊥ . n If (c1 (L) + (1/n)c1 (η)(α)) = 0, then, using c1 (L)[Di ] = 0 : Z Z c1 (η) ∧ [Di ] ∧ α = −n c1 (L) ∧ [Di ] ∧ α = 0. X
X
Corollary 3.2.18. The Poincar´e pairing is a polarization of the weight 4 pure Hodge structure (c1 (η)H6 (D))⊥ ∩ P 4 . Let us spell out the consequences of this analysis for the intersection form H4 (D) × H4 (D) → Q. First notice that the same argument used in the proof of 3.2.9.e shows that the map H4 (D) → H 4 (X) is injective. It follows that H4 (D) has a pure Hodge structure which is the direct sum of two substructures polarized (with opposite signs) by the Poincar´e pairing. The next result shows that in fact H4 (D) is the direct sum of two substructures, polarized (with opposite signs) by the Poincar´e pairing. This gives a clear indication of what happens in general: Corollary 3.2.19. The intersection form H4 (D) × H4 (D) → Q is non degenerate. There is a direct sum decomposition: H4 (D) = c1 (η)H6 (D) ⊕ (c1 (η)H6 (D))⊥ orthogonal with respect to the intersection form, which is negative definite on the first summand and positive on the second.
3.3 Intersection forms and Decomposition in the Derived Category We now show how the results we quoted at the beginning of the first section can be translated in statements about the decomposition in the derived category of sheaves of the direct image of the constant sheaf. We will freely use the language of derived categories. In particular we will use the notion of a constructible sheaf and the functors Rf∗ , Rf! , f ∗ , f ! . In section 3.3.3 we briefly review the classical E2 -degeneration criterion of Deligne [9], [10] in order to motivate the construction of the perverse cohomology complexes. These complexes are a natural generalization of the higher direct image local systems for a smooth map. The construction of perverse cohomology is carried out in section 3.4.
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We denote by S(Y ) the abelian category of sheaves of Q-vector spaces on Y , and by Db (Y ) the corresponding derived category of bounded complexes. We shall make use of the following splitting criterion in the derived category. We state it in the form we need it in this paper. For a more general statement and a proof the reader is referred to [5] and [7]. Let (U, y) be a germ of an isolated n-dimensional singularity with the ` obvious stratification = V y, j : V → U ← y : i be the obvious maps, P be a self-dual complex on U with P|V = L[n], L a local system on V , and P ' τ≤0 P. We wish to compare P , ICU (L) := τ≤−1 Rj∗ L[n] and the stalk H0 (P )y . Lemma 3.3.1. The following are equivalent: 1) there is a canonical isomorphism in the derived category P ' ICU (L) ⊕ H0 (P )y [0]; 2) the natural map H0 (P ) −→ H0 (Rj∗ j ∗ P ) = Rn j∗ L is zero.
3.3.1 Resolution of surface singularities For a normal surface Y , let j : Yreg → Y be the open embedding of its regular points. The intersection cohomology complex, which we will consider in much more detail in the next section, is ICY = τ≤−1 Rj∗ QYreg [2]. The following, which we will prove as a consequence of 3.2.1, is the first case of the Decomposition theorem which needs to be stated in the derived category and not just in the category of sheaves. Theorem 3.3.2. Let f : X → Y be a proper birational map of quasiprojective surfaces, X smooth, Y normal. There is a canonical isomorphisms '
Rf∗ QX [2] −→ ICY ⊕ R2 f∗ QX [0]. Proof We work locally on Y . Let (Y, p) be the germ of an analytic normal surface singularity, f : (X, D) → (Y, p) be a resolution. The fiber D = f −1 (p) is a connected union of finitely many irreducible compact curves Dk .
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Note that j ∗ Rf∗ Q ' QY −p . Consider the following diagram r / Rj∗ j ∗ Rf∗ QX [2] O RRR RRR RRR `0 RRRR)
= Rj∗ QY −p [2]
Rf∗ QX [2]
τ≤0 Rj∗ j ∗ Rf∗ QX [2] O
`−1
'
`−2
τ≤−1 Rj∗ j ∗ Rf∗ QX [2] O
=: ICY
τ≤−2 Rj∗ j ∗ Rf∗ QX [2]
= QY [2]
'
We are looking for the obstructions to the existence of the lifts `0 , `−1 and `−2 . Since Rk f∗ QX = 0, k ≥ 3, we have that τ≤0 Rf∗ QX [2] ' Rf∗ QX [2]. In particular, `0 exists and it is unique. From the exact triangle · · · → τ≤−1 Rj∗ j ∗ Rf∗ QX [2] → τ≤0 Rj∗ j ∗ Rf∗ QX [2] → +1
→ H0 (Rj∗ j ∗ Rf∗ QX [2]) ' R2 j∗ QY −p → · · · the map `−1 exists if and only if the natural map ρ : R2 f∗ QX → R2 j∗ QY −p is trivial. Using the isomorphisms (R2 f∗ QX )p ' H 2 (X) and (R2 j∗ QY −p )p ' H 2 (X −D), the map ρ can be identified with the restriction map ρ appearing in the long exact sequence of the pair (X, D) : cl
ρ
. . . −→ H2 (D) −→ H 2 (X) ' H 2 (D) −→ H 2 (X − D) −→ . . . where have identified H2 (D) ' H 2 (X, X − D) via Lefschetz Duality. By Theorem 3.2.1, cl is an isomorphism and ρ is trivial. This lift `−1 is unique and splits by Lemma 3.3.1. Remark 3.3.3. It can be shown easily that a lift Rf∗ QX [2] −→ τ≤−2 Rj∗ QU [2] = QY [2] exists if and only if H 1 (f −1 (p), Q) = {0}, i.e. if and only if ICY ' QY [2], if and only if (Y, p) is a rational homology manifold. It follows that, in general, the natural map QY → Rf∗ QX does not split and Rf∗ QX does not decompose as a direct sum of its shifted cohomology sheaves as in (3.6). Example 3.3.4. Let f : X = C × P1 → Y be the real algebraic map contracting precisely D := {0} × P1 to a point p ∈ Y . One has a non split exact sequence in the category P (Y ) of perverse sheaves on Y : 0 −→ ICY −→ Rf∗ QX [2] −→ H 2 (P1 )p [0] −→ 0.
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It is remarkable that while the lift `−2 does not exist in general, the lift `−1 always exists. While looking for a non-trivial map Rf∗ QX [2] → QY [2], one ends up finding another more interesting map to ICY . Recall that the dualizing sheaf ωX ' QX [2n]. Dualizing the canonical isomorphism of Theorem 3.3.2 and, keeping in mind that ICY and H2 (D)p [0] are simple objects in P (Y ) (cf. section 3.4.2), we get Corollary 3.3.5. There are canonical isomorphisms '
PD
'
H2 (D)p [0] ⊕ ICY∗ −→ Rf∗ ωX [−2] ' Rf∗ QX [2] −→ ICY ⊕ H 2 (D)p [0], such that the composition is a direct sum map and induces the intersection form ι on H2 (D) and the Poincar´e-Verdier pairing on the self-dual ICY . In particular, if X is compact, then the induced splitting injection IH • (Y ) ⊆ H • (X) exhibits the left hand side as the pure Hodge substructure of the right hand side orthogonal to the space cl(H2 (D)) ⊆ H 2 (X) with respect to the Poincar´e pairing on X. 3.3.2 Fibrations over curves Let f : X → Y be a map of from a smooth surface onto a smooth curve. ˆ → Yˆ the smooth part of the map f , by j : Yˆ → Y the Denote by fˆ : X open immersion, by T i := Ri fˆ∗ QXˆ = Ri f∗ QX |Yˆ . For ease of exposition we assume that f has connected fibers. Fix an ample line bundle η on X. The isomorphism stated in the next proposition will depend on η. Proposition 3.3.6. There is an isomorphism Rf∗ QX [2] ' j∗ T 0 [2] ⊕ P ⊕ j∗ T 2 [0], with P a suitable self-dual (with respect to the Verdier duality functor) object of Db (Y ). Proof We work around one critical value p ∈ Y and replace Y by a small disk centred at p, X by the preimage of this disk, etc. Since the fibers are connected, QY ' f∗ QX ' j∗ T 0 ' j∗ T 2 . Since η is f -ample, η : T 0 ' T 2 , which in this case implies that η : j∗ T 0 ' j∗ T 2 . There are the natural truncation maps f∗ QX → Rf∗ QX → R2 f∗ QX [−2].
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There is the natural adjunction map R2 f∗ QX → j∗ T 2 . It is splitting/surjective in view of the presence of η. Putting together, there is a sequence of maps η
c
π
σ
c
j∗ T 0 [2] → Rf∗ QX [2] → Rf∗ QX [4] → j∗ T 2 [2] → j∗ T 0 [2] → . . .
(3.3)
where the composition ηπc is the isomorphism mentioned above η : j∗ T 0 ' j∗ T 2 and σ := (ηπc)−1 . The reader can verify that the composition j∗ T 0 [2] ⊕ j∗ T 2 [0]
γ:=(c+ηcσ)[−2]
−→
Rf∗ QX [2]
σπη⊕π[−2]
−→
j∗ T 0 [2] ⊕ j∗ T 2 [0]
is the identity, i.e. γ splits. Let P := Cone(γ). There is a direct sum decomposition Rf∗ QX [2] ' j∗ T 0 [2] ⊕ P ⊕ j∗ T 2 [0].
(3.4)
The self-duality of P follows from the self-duality of Rf∗ QX [2] and of j∗ T 0 [2]⊕ j∗ T 2 [0]. The object P introduced in the previous proposition has a simple structure: Proposition 3.3.7. Assumptions as in 3.3.6. The object P splits in Db (Y ) as P = V ⊕ j∗ T 1 [1], where V = Ker R2 f∗ QX → j∗ T 2 is a skyscraper sheaf supported at Y − Yˆ . This decomposition is canonical and compatible with Verdier duality. Proof By inspecting cohomology sheaves we see that Hi (P ) = 0 for i 6= 0, 1, that H−1 (P ) = R1 f∗ QX and that H0 (P ) = V . In view of Lemma 3.3.1, we need to show that r0 : H0 (P ) → R1 j∗ T 1
(3.5)
is the zero map. We now show that this is equivalent to the Zariski Lemma. By applying adjunction to (3.4), we obtain the a commutative diagram Rf∗ QX [2]
'
j∗ T 0 [2] ⊕ P ⊕ j∗ T 2 [0]
/ Rj∗ j ∗ Rf∗ QX [2]
'
/ Rj∗ T 0 [2] ⊕ Rj∗ P ⊕ Rj∗ T 2 [0]
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The associated map of spectral sequences Hp (Y, Hq (−)) =⇒ Hp+q (Y, −) gives a commutative diagram H2 (D)
cl
/ H 2 (X)
/ H 2 (X − D)
r
'
H0 (P ) ⊕ (j∗ T 2 )p
(r 0 ,id)
'
/ H1 (Rj∗ T 1 ) ⊕ (j∗ T 2 )p .
It follows that, using the identifications above, Im(cl) = Ker r = Ker r0 ⊆ H0 (P ). In particular, r0 = 0, if and only if dim Ker r = dim H0 (P ). Note that dim H0 (P ) = b2 (f −1 (p)) − 1. It follows that r0 = 0 if and only if dim Im(cl) = b2 (f −1 (p)) − 1. The latter is implied by Theorem 3.2.2. We conclude by Lemma 3.3.1. Finally, we have: Theorem 3.3.8. There is an isomorphism Rf∗ QX [2] ' j∗ T 0 [2] ⊕ j∗ T 1 [1] ⊕ V [0] ⊕ j∗ T 2 [0]. Remark 3.3.9. From 3.3.8 follows that R1 f∗ QX ' j∗ R1 fˆ∗ QXˆ . Note that this implies the Local Invariant Cycle Theorem. Since R2 f∗ QX = j∗ R2 fˆ∗ QXˆ ⊕ V , we have the coarser decomposition Rf∗ QX [2] ' R0 f∗ QX [2] ⊕ R1 f∗ QX [1] ⊕ R2 f∗ QX [0]. In particular, the Leray spectral sequence degenerates at E2 . It is easy to see that the Leray filtration on the cohomology of X is by Hodge substructures: L2 = f ∗ H(Y ),
L1 = Ker {f∗ : H(X) → H(Y )}.
3.3.3 Smooth maps Even in the case of a smooth fibration f : X → Y of a surface over a curve, the study of the complex Rf∗ QX is non-trivial, without any projectivity assumptions. Example 3.3.10. Let X be a Hopf surface. There is a natural holomorphic smooth fibration f : X → P1 with fibers elliptic curves. Since b1 (X) = 1, one sees easily that the Leray Spectral Sequence for f is not E2 -degenerate. In particular, Rf∗ QX is not isomorphic to ⊕i Ri f∗ QX [−i]. Let us briefly list some of the important properties of a smooth projective map f : X → Y of smooth varieties.
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The sheaves Ri f∗ QX are locally constant over Y , i.e. they are local systems. In fact, f is differentiably locally trivial over Y in view of Ehresmann’s theorem. The model for a general decomposition theorem for Rf∗ QX is the following Theorem 3.3.11. Let f : X n → Y m be a smooth projective map of smooth quasi-projective varieties of the indicated dimensions and η be an f -ample line bundle on X; Rf∗ QX 'D(Y ) ⊕Ri f∗ QX [−i] i
η :R
n−m−i
f∗ QX ' R
n−m+i
(3.6)
f∗ QX ,
∀ i ≥ 0;
The local systems Rj f∗ QX on Y are semisimple.
(3.7) (3.8)
The first Chern class of the line bundle η ∈ H 2 (X, Q) = HomD(X) (QX , Q[2]), defines maps η : QX −→ QX [2],
η : Rf∗ QX −→ Rf∗ QX [2],
η r : Rf∗ QX −→ Rf∗ QX [2r]
and finally η r : Ri f∗ QX −→ Ri+2r f∗ QX . Formula (3.7) is then just a re-formulation of the Hard Lefschetz Theorem for the fibers of f and can be named the Relative Hard Lefschetz Theorem for smooth maps. We remind the reader that a functor T : D(Y ) → A, A an abelian category, is said to be cohomological (cf. [20, II, 1.1.5]) if, setting T i (K) = T 0 (K[i]), to a distinguished triangle +1
K→L→M → corresponds a long exact sequence in A → T i (K) → T i (L) → T i (M ) → T i+1 (K) → ... The cohomology sheaf functor H0 : D(Y ) → S(Y ) is cohomological. Noting that Hi (Rf∗ ) = Ri f∗ , (3.6) can be re-phrased by saying that Rf∗ QX is decomposable with respect to the functor H0 . It is important to note that (3.7) implies (3.6) by the Deligne-Lefschetz Criterion. Formula (3.8) states that every local subsystem L ⊆ Rj f∗ QX admits a complement, i.e. a local system L0 such that L ⊕ L0 = Rj f∗ QX . Let us note some of the important consequences of Theorem 3.3.11. The 0 H -decomposability (3.6) of Rf∗ QX implies immediately the E2 -degeneration
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of the Leray spectral sequence, i.e. of the spectral sequence associated with the cohomological functor H0 : H p (Y, Rq f∗ QX ) =⇒ H p+q (X, Q) -degenerates atE2 .
(3.9)
This degeneration implies the surjection H k (X, Q) −→ H 0 (Y, Rk f∗ QX ) = H k (Xy , Q)π1 (Y,y) ,
(3.10)
i.e. the so-called Global Invariant Cycle Theorem. The theory of mixed Hodge structures allows to show, using a smooth compactification of X, that in fact the monodromy invariants are a Hodge substructure of H k (Xy , Q), which as a polarized mixed Hodge structures is independent of y ∈ Y (Theorem of the Fixed Part). In fact, (3.8) is a consequence of this fact. In general, if f is not smooth, Theorem 3.3.11 fails completely. The Relative Hard Lefschetz Theorem (3.7) fails due to the presence of singular fibers, i.e. fibers along which the differential of f drops rank. The sheaves Rj f∗ QX are no longer locally constant. Moreover, they are not semisimple in the category of constructible sheaves: e.g. j! QC∗ → QC . The following examples shows that the H0 -decomposability (??) fails in general and so does the E2 -degeneration of the Leray Spectral Sequence (3.9). Example 3.3.12. Let X be the blowing up of CP2 , along ten points lying on an irreducible cubic C 0 and C be the strict transform of C 0 on X. Since C 2 = −1 the curve contracts to a point under a birational map f : X → Y . We leave to the reader the task to verify that 1) the Leray Spectral Sequences for H 2 (X, Q) and for IH 2 (Y, Q) are not E2 -degenerate and that, though the Leray spectral sequence always degenerates over suitably small Euclidean neighborhoods on Y , 2) the complex ICY does not split as a direct sum of its shifted cohomology sheaves. The following more general class of examples shows that the failure of the E2 -degeneration is very frequent. Example 3.3.13. Let f : X → Y be a projective resolution of the singularities of a projective and normal variety Y such that there is at least one index i such that the natural mixed Hodge structure on H i (Y, Q) is not pure (e.g. i = 2 in 3.3.12). Then the Leray Spectral Sequence for f does not degenerate at E2 . If it were, then the edge sequence would give an injection of mixed Hodge structures f ∗ : H j (Y, Q) → H j (X, Q), forcing the mixed Hodge structure of such a Y to be pure. However, not everything is lost.
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3.4 Perverse sheaves and the Decomposition Theorem One of the main ideas leading to the theory of perverse sheaves is that Theorem 3.3.11, which holds for smooth maps, can be made to hold for arbitrary proper algebraic maps provided that it is re-formulated using the perverse cohomology functor pH0 in place of the cohomology sheaf functor H0 . Just as this latter is τ≤0 τ≥0 , with τ the standard truncation functors of a complex, the perverse cohomology functor will be expressed as pH0 = pτ≤0 pτ≥0 , where pτ is the so called perverse truncation functor. Roughly speaking, the perverse truncation functor (with respect to middle perversity, which is the only case we will consider) is defined by gluing standard truncations on the strata, shifted by a term which depends on the dimension of the stratum. The choice of the shifting is dictated by the behavior of the standard truncation with respect to duality, as we suggest in 3.4.1. In this context and keeping this in mind, perverse truncation becomes quite natural. We believe it can be useful to give a few details of its construction and an example of computation, related to the examples given in section 3.2.2. In analogy with the cohomology sheaf functor H0 , the perverse cohomology functor pH0 will be a cohomological functor which takes values in an abelian subcategory of Db (Y ), whose object are the so called perverse sheaves. For a general proper map these objects play the role played by local system for smooth maps.
3.4.1 Truncation and Perverse sheaves Db (Y
Let ) be the bounded derived category of the category S(Y ) of sheaves of rational vector spaces on Y . We are interested in the full subcategory D(Y ) of those complexes whose cohomology sheaves are constructible. This means that, given an object F of D(Y ), there is an algebraic Whitney ` stratification Y = S` , depending on F , such that Hj (F )|S` is a finite rank local system. By the Thom Isotopy Lemmata, Rf∗ QX , and in fact any other complex appearing in this paper, is an object of D(Y ). One is interested in direct sum decompositions of this complex, in the geometric meaning of the summands and in the consequences, both theoretical and practical, of such splittings. We now define the t-structure on D(Y ) associated with the middle perversity. Instead of insisting on its axiomatic characterization (cf. [2]), we give the explicit construction of the perverse truncations p τ≤m : D(Y ) −→ D(Y ), and p τ≥m : D(Y ) −→ D(Y ). These come with natural morphisms pτ p ≤m F −→ F and F −→ τ≥m F . We start with the following: Lemma 3.4.1. Let Z be nonsingular of complex dimension r, and
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F ∈ D(Z) with locally constant cohomology sheaves. With D the Verdier duality operator, there are natural isomorphisms: τ≤k DF ' Dτ≥−k−2r F
τ≥k DF ' Dτ≤−k−2r F
Proof Since the dualizing complex is in this case isomorphic to QZ [2r], it is enough to prove that there are natural isomorphisms τ≤k Rhom(F, QZ ) ' Rhom(τ≥−k F, QZ )
τ≥k Rhom(F, QZ ) ' Rhom(τ≤−k F, QZ ).
We prove the first statement. The proof of the second is analogous. Applying Rhom and τ≤k to the map F → τ≥−k F , we get: Rhom(τ≥−k F, QZ ) −→ Rhom(F, QZ ) ↑ ↑ τ≤k Rhom(τ≥−k F, QZ ) −→ τ≤k Rhom(F, QZ ). To prove the statement it is enough to show that the three complexes Rhom(τ≥−k F, QZ ), τ≤k Rhom(τ≥−k F, QZ ) and τ≤k Rhom(F, QZ ) have the same cohomology sheaves. Since F and QZ have locally constant cohomology sheaves, there are natural isomorphisms of complexes of vector spaces Rhom(F, QZ )y ' Rhom(Fy , Qy ) ' ⊕i Hom(H−i Fy , Qy )[−i]. The cohomology sheaves of the three complexes, are, therefore, equal to Hom(H−i Fy , Qy ) for i ≤ k and vanish otherwise. The construction of the perverse truncation is done by induction on the strata of Y starting from the shifted standard truncation on the open stratum Ud . In the sequel we will indicate by U` the union of strata of dimension bigger than or equal to `. With a slight abuse of notation, we will write ` U`+1 = U` S` , with S` now denoting the union of strata of dimension `. ` Let F ∈ Ob(D(Y )) be Y-constructible for some stratification Y = S` . All the constructions below will lead to Y-constructible complexes. We Ud Ud define p τ≤0 = τ≤− dim Y and p τ≥0 = τ≥− dim Y . U
U
Suppose that p τ≤0`+1 : D(U`+1 ) −→ D(U`+1 ) and p τ≥0`+1 : D(U`+1 ) −→ U` U` D(U`+1 ) have been defined. We proceed to define p τ≤0 and p τ≥0 on U` = ` U`+1 S` . Let i : S` → U` ←− U`+1 : j be the inclusions: the exact triangles U
[1]
00 τ≤0 F → F → i∗ τ>−dimS i∗ F −→
[1]
00 i! τ<− dim S i! F −→ F −→ τ≥0 F −→
0 τ≤0 F → F → Rj∗ p τ>0`+1 j ∗ F −→
[1]
and U
0 Rj! p τ<0`+1 j ! F −→ F −→ τ≥0 F −→
[1]
define four functors (cf. [2, 1.1.10, 1.3.3 and 1.4.10]), i.e. the four objects
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0 F , τ 0 F , τ 00 F and τ 00 F which make the corresponding triangles exact, τ≥0 ≤0 ≥0 ≤0 are determined up to unique isomorphism. Define p U` τ≤0
p U` τ≥0
00 0 τ≤0 , := τ≤0
00 0 := τ≥0 τ≥0 .
Define: p
U0 , τ≤0 := p τ≤0
p
U0 . τ≥0 := p τ≥0
We have the following compatibility with respect to shifts. p
τ≤m (F [`]) ' p τ≤m+` (F )[`],
p
τ≥m (F [`]) ' p τ≥m+` (F )[`].
These formulas hold for the ordinary truncation functors as well and we symbolically summarize them as follows (τm ([`]))[−`] = τm+` . The perverse truncations so defined have the following properties: • By the construction above, if F is Y-cohomologicaly constructible, then so are p τ≤m F and p τ≥m F . • Let P (Y ) be the full subcategory of complexes Q such that dim Supp (H−i (Q) ≤ i for every i ∈ Z and the same holds for D(Q), the Verdier dual of Q. P (Y ) is an abelian category. The functor H0 (−) : D(Y ) −→ P (Y ),
p
H0 (F ) := p τ≤0 p τ≥0 F ' p τ≥0 p τ≤0 F,
p
is cohomological. Define Hm (F ) := pH0 (F [m]).
p
These functors are called the perverse cohomology functors. Any distin[1]
guished triangle F −→ G −→ H −→ in D(Y ) gives rise to a long exact sequence in P (Y ): . . . −→ pHi (F ) −→ pHi (G) −→ pHi (H) −→ pHi+1 (F ) −→ . . . . If F is Y-cohomologicaly constructible, then so are pHm (F ), ∀m ∈ Z. • Poincar´e- Verdier Duality induces functorial isomorphisms for F ∈ Ob(D(Y )) p
τ≤0 DF ' D p τ≥0 F,
p
τ≥0 DF ' D p τ≤0 F
D( pHj (F )) ' pH−j (D(F )).
This can be seen from the construction above. In fact, by Lemma 3.4.1, Ud Ud the isomorphisms hold for U = Ud , since p τ≤0 = τ≤− dim Y and p τ≥0 = τ≥− dim Y .
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U D ' D p τ U and p τ U D ' D p τ U for U = U Suppose that p τ≤0 `+1 . It ≥0 ≥0 ≤0 then follows that the same isomorphisms hold for U = U` . In fact, 0 DF , and the inducapplying the functor D to the triangle defining τ≤0 p U p U 0 F , so tive hypothesis τ≤0 D ' D τ≥0 , we get the triangle defining τ≥0 0 DF ' τ 0 F . The argument for τ 00 is identical. We get that Dτ≤0 ≥0 ≤0 00 00 F . It follows that Dτ 00 τ 0 ' τ 00 Dτ 0 ' τ 00 τ 0 D and the Dτ≤0 DF ' τ≥0 ≤0 ≤0 ≥0 ≤0 ≥0 ≥0 first wanted isomorphism follows. The second is equivalent to the first one. The third one follows formally: D( pHm (F )) ' D p τ≤0 p τ≥0 (F [−m]) ' p p −m (DF ). pτ ≥0 τ≤0 (D(F )[m]) ' H • For every F and m one constructs, functorially, a distinguished triangle p
[1]
τ≤m F −→ F −→ p τ≥m+1 F −→ .
The objects of the abelian category P (Y ) are called perverse sheaves. An object F of D(Y ) is perverse if and only if the two natural maps p τ≤0 F −→ F and F −→ p τ≥0 F are isomorphisms. Example 3.4.2. Let f : X −→ Y a surjective proper map of surfaces, X smooth. The direct image Rf∗ Q[2] is a perverse sheaf. Example 3.4.3. To give an example of how the truncation functors can be computed from the construction given above, let us examine the example of section 3.2.3. The assumptions in 3.2.6 are in force and we use the same notation. We show that: p
τ≤0 Rf∗ QX [3] ' τ≤0 Rf∗ QX [3],
p
τ≤−1 Rf∗ QX [3] = H4 (D)y [1].
Y −y 0 Rf Q [3] = = τ>−3 and j ∗ Rf∗ QX [3] = QY −y [3], we have τ≤0 Since pτ>0 ∗ X p 00 0 Rf∗ QX [3]. The perverse truncation τ≤0 Rf∗ QX [3] = τ≤0 τ≤0 Rf∗ QX [3] = 00 Rf Q [3] is computed by the triangle τ≤0 ∗ X +1
00 τ≤0 Rf∗ QX [3] −→ Rf∗ QX [3] −→ i∗ τ>0 i∗ Rf∗ QX [3] −→ .
Since i∗ Rf∗ QX [3] = ⊕j H 3−j (D)y [j], we have i∗ τ>0 i∗ Rf∗ QX [3] = H 4 (D)y [−1], so that p
τ≤0 Rf∗ QX [3] ' Cone {Rf∗ QX [3] → H 4 (D)y [−1]} ' τ≤0 Rf∗ QX [3].
Keeping in mind the truncation rules, we have the triangle +1
Y −y ∗ 0 τ≤−1 Rf∗ QX [3] −→ Rf∗ QX [3] −→ Rj∗ pτ>−1 j QY [3] = Rj∗ j ∗ QY [3] −→
from which we deduce that 0 τ≤−1 Rf∗ QX [3] ' i! i! Rf∗ QX [3] ' ⊕j Hj (D)y [j − 3].
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00 0 00 The truncation p τ≤−1 Rf∗ QX [3] = τ≤−1 τ≤−1 Rf∗ QX [3] = τ≤−1 (⊕j Hj (D)y [j − 3]) is computed by the triangle +1
00 τ≤−1 (⊕j Hj (D)y [j − 3]) −→ ⊕j Hj (D)y [j − 3] → i∗ τ>−1 (⊕j Hj (D)y [j − 3]) −→
from which the conclusion follows. It is to be observed that the category of perverse sheaves is Artinian and Noetherian, that its simple object can be completely characterized and have an important geometric meaning: they are the intersection cohomology complexes.
3.4.2 The simple objects of P (Y ) Goresky and MacPherson introduced the intersection cohomology groups of Y for an arbitrary perversity. Here we deal with the case of middle perversity. These groups were first defined as the homology of a chain subcomplex of the complex of geometric chains with twisted coefficients on Y . Later, following a suggestion by Deligne, they realized these groups as the hypercohomology of what they called the intersection cohomology complexes with twisted coefficients of Y . These complexes are the building blocks of P (Y ). They are special examples of perverse sheaves and every perverse sheaf can be exhibited as a finite series of non trivial extensions of objects of this kind supported on closed subvarieties of Y . Let Z ⊆ Y be a closed subvariety, Z o ⊆ Zreg ⊆ Z be an inclusion of Zariski-dense open subsets and L be a local system on Z o . GoreskyMacPherson associate with this data the intersection cohomology complex ICZ (L) in P (Z). Up to isomorphism, this complex is independent of the choice of Z o : if L and L0 are local systems on Z o and Z o0 respectively and L|Z o ∩Z o 0 ' L0|Z o ∩Z o 0 , then the associated intersection cohomology complexes on Z are canonically isomorphic. The complex ICZ (L), when viewed as a complex on Y , is perverse on Y . The intersection cohomology complex of Y is defined to be ICY := ICY (QYreg ). If Y is smooth, or a rational homology manifold, then ICY ' QY [dim Y ]. If Z is smooth and L is a local system on Z, then ICZ (L) ' L[dim Z]. Proposition 3.4.4. The simple objects in P (Y ) are precisely the ones of the form ICZ (L), L simple on Z o . In particular, if L is simple, then ICZ (L) does not decompose into non-trivial direct summands in D(Y ). The semisimple objects of P (Y ) are finite direct sums of such intersection cohomology complexes on possibly differing subvarieties.
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Every perverse sheaf Q ∈ P (Y ) is supported on a finite union of closed subvarieties of Y . Let Z be any one of them. There is a Zariski-dense open subset Z o ⊆ Zreg , such that Q|Z o ' L[dim Z], where L is a local system on Z o . The object Q admits a finite filtration where one of the quotients is ICZ (L) and all the others are all supported on supp (Q)−Z o . It follows that Q admits a finite filtration where the quotients are intersection cohomology complexes supported on closed subvarieties of Y . An intersection cohomology complex ICZ (L) is characterized by its not admitting subquotients supported on smaller dimensional subspaces of Z. Its eventual splitting is entirely due to a corresponding splitting of L. Let us define the intersection cohomology complexes. Assume Y is a stratification and L is a local system on the open stratum Ud . We start by defining ICUd (L) := L[dim Y ]. Now suppose inductively that ICU`+1 (L) has been defined on U`+1 and we define it on U` by ICU` (L) := τ≤−`−1 Rj∗ ICU`+1 (L). Let us give formulae for ICY (L) when Y and L have isolated singularities. It suffices to work in the Euclidean topology. Let (Y, p) be a germ of an isolated singularity, j : U := Y − p → Y be the open embedding and L be a local system on U . We have ICY (L) = τ≤−1 (Rj∗ L[dim Y ]).
(3.11)
If dim Y = 1, then ICY (L) = j∗ L[1]. The stalk at p are the invariants of L i.e. H 0 (U, L). In general, when dim Y ≥ 2, then ICY (L) is a complex, not a sheaf. If L is simple, then ICY (L) is simple and does not split nontrivially in D(Y ). The cohomology sheaves Hj (ICY (L)) are non trivial only for j ∈ [− dim Y, −1] and we have H− dim Y (ICY (L)) = j∗ L, (3.12) H− dim Y +` (ICY (L)) = H ` (U, L)p , 1 ≤ ` ≤ dim Y − 1, where Vp denotes a skyscraper sheaf at p ∈ Y with stalk V . In order to familiarize ourselves with these complexes, we compute two important examples: Example 3.4.5. We consider a threefold Y with an ordinary double point y and with associated link L. Let j : Y − y → Y be the open embedding, so that (3.11) gives ICY = τ≤−1 Rj∗ Q[3]. The cohomology sheaves at y are Hk (ICY ) = H k+3 (L)p , for k ≤ −1,
Hk (ICY ) = 0 otherwise.
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The singularity is analytically equivalent to a cone over a smooth quadric in projective space, hence its link is homeomorphic to the S 1 -bundle over S 2 ×S 2 with Chern class (1, 1). The long exact sequence for this S 1 -fibration gives H k (L) = Q for k = 0, 2, 3, 5
H k (L) = 0 otherwise,
which in turn implies that Hk (ICY ) = Q for k = −3, −1,
Hk (ICY ) = 0 otherwise.
We have a triangle in D(Y ), (not of perverse sheaves) +1
QY [3] → ICY → Qy [1] → The fact that H−1 (ICY ) = Qy should be compared with the existence of a small resolution with fiber a projective line over the singular point, and the statement of the Decomposition Theorem 3.4.11. Example 3.4.6. Let Y = C2 , and L be a local system on C2 − (x1 x2 = 0) defined by the two monodromy operators T1 and T2 acting on the vector space V = Lp , the stalk of L at p = (1, 1) ∈ C2 . We first determine the intersection cohomology complex over C2 −{0}. Denoting by j : C2 −{x1 x2 = 0} → C2 − {0} the natural map, we have ICC2 −{0} (L) = τ≤−2 Rj∗ L[2] = (j∗ L)[2]. Denoting by j 0 : C2 − {0} → C2 the natural map, we have ICC2 (L) = τ≤−1 Rj∗0 ICC2 −{0}(L) = τ≤−1 Rj∗0 (j∗ L[2]). In order to determine the cohomology sheaves of ICC2 (L), we compute H i (C2 − {0}, j∗ L) for i = 0, 1. More precisely, we should determine these groups for a fundamental system of neighborhoods of the origin; however the cohomology groups are in fact constant. Set N1 = T1 −Id, N2 = T2 −Id. We have H 0 (C2 −{0}, j∗ L) = H 0 (C2 −{x1 x2 = 0}, L) = Ker N1 ∩Ker N2 = V π1 . Since j∗ L = τ≤0 Rj∗ L, and fundamental deleted neighborhoods around the axes are homotopic to circles, so that Hi (Rj∗ L) = 0 for i ≥ 2, we have the following exact triangle in D(C2 − {0}) +1
j∗ L = τ≤0 Rj∗ L −→ Rj∗ L −→ H1 (Rj∗ L)[−1] −→ . ` The sheaf H1 (Rj∗ L) is the local system on (x1 x2 = 0) − {(0, 0)} = D1 D2 , with fiber coker N1 and monodromy T2 on D1 , and fiber coker N2 and monodromy T1 on D2 . Since C2 − {x1 x2 = 0} retracts to a torus T 2 , the cohomology of L is isomorphic to the group cohomology of Z2 with values
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in V as a Z2 -module via the monodromy operators T1 , T2 , which can be computed by the Koszul complex (see for instance [21] ) φ
ψ
0 −→ V −→ V ⊕ V −→ V −→0, with ψ(v1 , v2 ) = N2 (v1 ) − N1 (v2 ).
φ(v) = (N1 (v), N2 (v))
The long exact sequence associated to the exact triangle above gives H−1 (ICY (L))0 ' H 1 (C2 −{0}, j∗ L) =
{(N1 (v1 ), N2 (v2 )) | N1 N2 (v1 − v2 ) = 0} . (N1 (v), N2 (v))
More generally, a similar recipe holds for the cohomology sheaves of the intersection cohomology complex of a local system defined on the complement of a normal crossing, see [3].
3.4.3 Decomposability, E2 -degenerations and filtrations Definition 3.4.7. Let H = H0 be the sheaf cohomology functor. We say that F in D(Y ) is H-decomposable if M F 'D(Y ) Hi (F )[−i] i
We say that F in D(Y ) is p H-decomposable if M p i F 'D(Y ) H (F )[−i]. i
If F is H-decomposable, then the spectral sequence H p (Y, Hq (F )) =⇒ Hp+q (Y, F ) is E2 -degenerate. This spectral sequence is the Leray Spectral Sequence when F = Rf∗ (G). In this case the corresponding filtration is called the Leray filtration. The analogous statements holds for pH-decomposability. The corresponding spectral sequence is called the Perverse Leray Spectral Sequence: Hp (Y,p Hq (Rf∗ G)) =⇒ Hp+q (Y, G) and the corresponding filtration is called the perverse filtration. Definition 3.4.8. Let f : X → Y be a map, n = dim X. The perverse filn+j tration H≤b (X) ⊆ H n+j (X), b, j ∈ Z is defined to be the perverse filtration on Hj (Y, Rf∗ QX [n])
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It coincides, up to a shift, with the Leray filtration, when f is smooth. If these decomposing isomorphisms exist, they are seldom unique. We now give the statement (not in the most general form) of one of the more general criteria for decomposability, see [9] and [10]: Theorem 3.4.9 (Deligne degeneration criterion). Let K be an object of Db (Y ), and let η ∈ H 2 (X). Suppose that η ` : pH−` (K) → pH` (K) is an isomorphism for all `. Then K is pH-decomposable. The same statement holds if we consider the functor H. Example 3.4.10. By the computation done in 3.4.3, we have the following description of the perverse filtration for the resolution of a threefold: i H≤−2 (X) = {0}, 2 i H≤−1 (X) = Im{H4 (D) → H 2 (X)}, H≤−1 (X) = 0otherwise, 4 i H≤0 (X) = Ker{H 4 (X) → H 4 (D)}, H≤0 (X) = H i (X) otherwise ,
H i (X)≤1 = H i (X) for all i. The condition 3.4.9, that η : pH−1 (Rf∗ QX [3]) = H4 (D)y −→ H 4 (D)y = pH1 (Rf∗ QX [3]) be an isomorphism, is just the non degeneracy of the intersection form η ∩ [Di ] · [Dj ]. Note that in this case, the explicit description makes it clear that the perverse filtration is given by Hodge substructures.
3.4.4 The Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber We can now state the generalization of Deligne’s Theorem 3.3.11 to the case of arbitrary proper maps. Recall that if X is smooth, then ICY = QX [n]. Theorem 3.4.11. Let f : X → Y be a proper map of algebraic varieties. Then the complex M p i Rf∗ ICX ' H (Rf∗ ICX )[−i] (3.13) i
is pH-decomposable. The complexes pHj (Rf∗ ICX ) are semisimple i.e. there is a canonical isomorphism Hj (Rf∗ ICX ) 'P (Y ) ⊕ICZa (La )
p
(3.14)
for some finite collection, depending on j, of semisimple local systems La on smooth distinct varieties Zao ⊆ Z ⊆ Y .
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Let η be an f -ample line bundle on X. Then η r : pH−r (Rf∗ ICX ) ' pHr (Rf∗ ICX ).
(3.15)
The Verdier Duality functor is an auto-equivalence D : D(Y ) → D(Y ) which preserves P (Y ) and for which one has D ◦ pH−j ' pHj ◦ D. This fact implies that the summands appearing in the semisimplicity statement for j are pairwise isomorphic to the ones appearing for −j and that the local systems L are self-dual. Theorem 3.4.11 is the deepest known fact concerning the homology of algebraic maps. The original proof uses algebraic geometry in positive characteristic in an essential way. M. Saito has given a transcendental proof of a more general statement concerning his mixed Hodge modules in the series of papers [17], [18], [17]. We give a proof for the push-forward of intersection cohomology (with constant coefficients) first in the case of semismall maps (cf. [5]) and then for arbitrary maps in [7]. Though at present our methods do not afford results concerning the push-forward with more general coefficients, they give new and precise results on the perverse filtration and on the refined intersection forms. C. Sabbah [16] has recently proved a decomposition theorem for push-forwards of semisimple local systems. Remark 3.4.12. It is now evident that the computations in 3.3.1 and 3.3.2 establish the Decomposition Theorem for maps from a smooth surface. In the case of the proper birational map f : X → Y of 3.3.1, in fact, the complex Rf∗ QX [2] is perverse, as observed in 3.4.2, and 3.3.2 states that it splits into ICY and R2 f∗ QX [0]. In the case of the family of curves treated in 3.3.2 we have that j∗ T 0 [2] = pH−1 (Rf∗ QX [2])[1], and j∗ T 2 [0] = pH1 (Rf∗ QX [2])[−1], and we showed in 3.3.6 that η : j∗ T 0 → j∗ T 2 is an isomorphism. The perverse sheaf P splits, see 3.3.7, in j∗ T 1 [1] = ICY (T 1 ), and V , concentrated on points. Remark 3.4.13. For the case of the resolution of a threefold with isolated singularities, whose Hodge theory has been treated in 3.2.3, we have, as seen in 3.4.3, H−1 (Rf∗ QX [3]) ' H4 (D)y ,
p
H1 (Rf∗ QX [3]) ' H 4 (D)y ' η ∧ H4 (D)y ,
p
and we have the splitting H0 (Rf∗ QX [3]) ' ICY ⊕ H3 (D)y .
p
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Similarly, for the 4-fold with isolated singularities, see 3.2.4, H−2 (Rf∗ QX [4]) ' H6 (D)y ,
p
H−1 (Rf∗ QX [4]) ' H5 (D)y ,
p
H2 (Rf∗ QX [4]) ' H 6 (D)y ' η 2 ∧ H6 (D)y ,
p
H1 (Rf∗ QX [4]) ' H 5 (D)y ' η ∧ H5 (D)y ,
p
and we have the splitting H0 (Rf∗ QX [4]) ' ICY ⊕ H4 (D)y .
p
3.4.5 Results on intersection forms In this section we list some of the results of [7] which are related to the theme of this paper. For simplicity, we state them in the special case when f : X → Y is a map of projective varieties, X smooth. Let η and A be ample line bundles on X and Y respectively, L := f ∗ A. Theorem 3.4.14. For ` ≥ 0 and b ∈ Z, the subspaces given by the perverse filtration (cf. 3.4.3) ` H≤b (X) ⊆ H ` (X)
are pure Hodge sub-structures. The quotient spaces ` ` (X) (X)/H≤b−1 Hb` (X) = H≤b
inherit a pure Hodge structure of weight `. ` (X) ⊆ H `+2 (X) and induces maps, The cup product with η verifies η H≤a ≤a+2 `+2 (X). The cup product with L is compatible still denoted η : Ha` (X) → Ha+2 with the Decomposition Theorem 3.4.11 and induces maps L : Ha` (X) → Ha`+2 (X). These maps satisfy graded Hard Lefschetz Theorems (cf. [7, Theorem 2.1.4]). −j n−i−j −j Define P−i := Ker η i+1 ∩ Ker Lj+1 ⊆ H−i (X), i, j ≥ 0 and P−i := 0 otherwise. In the same way in which the classical Hard Lefschetz implies the Primitive Lefschetz Decomposition for the cohomology of X, the graded Hard Lefschetz Theorems imply the double direct sum decomposition of
Theorem 3.4.15. Let i, j ∈ Z. There is a Lefschetz-type direct sum decomposition into pure Hodge sub-structures of weight (n − i − j), called the (η, L)-decomposition: M n−i−j j−2m H−i (X) = η −i+` L−j+m Pi−2l . `, m ∈Z
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ηL n−i−j One can define bilinear forms Sij on H−j (X) by modifying the Poincar´e pairing Z ηL η i ∧ Lj ∧ α ∧ β Sij ([α], [β]) := X
and descending it to the graded groups. These forms are non degenerate. In fact their signature can be determined in the following generalization of the Hodge Riemann relations. ηL Theorem 3.4.16. The 3.4.15 is orthogonal with respect to Sij . The forms ηL Sij induce polarizations of each (η, L)-direct summand.
The homology groups H∗BM (f −1 (y)) = H∗ (f −1 (y)), y ∈ Y , are filtered by virtue of the decomposition theorem (one may call this the perverse BM (f −1 (y)) → H n+∗ (X) is filtration). The natural cycle class map cl : Hn−∗ filtered strict. The following generalizes the Grauert’s contraction criterion. Theorem 3.4.17. Let b ∈ Z, y ∈ Y . The natural class maps BM (f −1 (y)) −→ Hbn+b (X) c`b : Hn−b,b BM (f −1 (y)) ⊆ Ker L ⊆ H n+b (X) with a pure is injective and identifies Hn−b,b b Hodge substructure, compatibly with the (η, L)-decomposition. Each (η, L)BM (f −1 (y)) is polarized up to sign by S ηL . direct summand of Hn−b,b −b,0 ηL BM (f −1 (y)) is non degenerate. to Hn−b,b In particular, the restriction of S−b,0
By intersecting in X cycles supported on f −1 (y), we get the refined inBM (f −1 (y)) → H n+∗ (f −1 (y)) which is tersection form (see section 3.2.2) Hn−∗ filtered strict as well. Theorem 3.4.18 (The Refined Intersection Form Theorem). Let b ∈ Z, y ∈ Y . The graded refined intersection form BM Hn−b,a (f −1 (y)) −→ Han+b (f −1 (y))
is zero if a 6= b and it is an isomorphism if a = b. We have seen in earlier sections how these results can be made explicit in the case of surfaces, threefolds and fourfolds. For more applications in any dimension see [7]. In fact, the method of proof of the results stated in this section is inspired by the low dimensional examples of surfaces, threefolds and fourfolds.
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3.4.6 The decomposition mechanism It is quite hard to describe what kind of geometric phenomena are expressed by the Decomposition Theorem. The complex Rf∗ QX essentially describes H ∗ (f −1 (U )) for any neighborhood U of a point y. We gain some geometric insight if we represent cohomology classes by Borel-Moore cycles in f −1 (U ). Let S be the stratum containing y. By the exact sequence i
j∗
∗ H∗BM (f −1 (U ∩ S)) → H∗BM (f −1 (U )) → H∗BM (f −1 (U − S)),
the Borel Moore cycles in f −1 (U ) are of two kind: those in Im i∗ , which are homologous to cycles supported on the inverse image of the stratum S, and those whose restriction to f −1 (U − S) is not trivial. Neither i∗ is injective nor j ∗ is surjective: there are non trivial cycles in f −1 (U ∩ S) which become homologous to zero in f −1 (U ), and there are cycles in f −1 (U − S) which cannot be closed to cycles in f −1 (U ). The Decomposition Theorem gives strong information on both types. The first deep aspect of the Theorem is that the subspace Im i∗ , has a uniform behavior for all projective maps, related to the non degeneracy of the intersection forms. For instance, we already noticed in 3.2.3, how the Grauert Theorem 3.2.1 implies that the classes of disks transverse to exceptional curves are homologous to linear combinations of the classes of these curves. Such non degeneracy results, see 3.4.17, 3.4.18, are peculiar to algebraic maps and stem from “weight” considerations, either in characteristic 0 (Hodge Theory) or in positive characteristic (weights of Frobenius, cf. [2]). The Decomposition Theorem, though, contains other deep information. Since Rf∗ QX splits as a direct sum of terms associated with the strata, we have a splitting map, which can be made canonical after an ample line bundle η on X has been chosen, from the subspace Im j ∗ ⊆ H∗BM (f −1 (U −S) to H∗BM (f −1 (U ) : i.e. the following is split exact 0 −→ Im i∗ −→ H∗BM (f −1 (U ) −→ Im j ∗ −→ 0. The image of the splitting defines a subspace of H∗BM (f −1 (U ) which is complementary to Im i∗ and consists of classes which are closures of some Borel Moore cycles in f −1 (U − S). The deep fact here, is that these cycles are governed by the intersection cohomology complex construction on Y ; each stratum having S in its closure contributes to H∗BM (f −1 (U ) via the intersection cohomology of a local system on the stratum.
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3.5 Grothendieck motive decomposition for maps of threefolds We assume we are in the situation 3.2.6. Again, this is for ease of exposition only. See Remark 3.5.6. We have already shown that 2 H−1 (X) = Im H4 (D) → H 2 (X) =: Im i∗ ,
H04 (X) = Ker {H 4 (X) → H 4 (D)} =: Ker i∗ . The choice of an ample line bundle η allows to split the perverse filtration: H 2 (X) = Im i∗ ⊕ (c1 (η) ∧ Im i∗ )⊥ ; H 4 (X) = Ker i∗ ⊕ (Im i∗ )⊥ . We have, canonically, that H 3 (X) = Im H3 (D) ⊕ (Im H3 (D))⊥ . so that H i (Y ) = H i (X) for i = 0, 1, 5, 6, IH 3 (Y ) = (Im H3 (D))⊥ ,
IH 2 (Y ) = (c1 (η) ∧ Im H4 (D), )⊥
IH 4 (Y ) = (Im H4 (D))⊥ ;
here we are using the convention for intersection cohomology compatible with singular cohomology: IH i (Y ) := Hi−n (Y, ICY )). We want to realize these splittings by algebraic cycles on X × X, in order to find a Grothendieck motive for the intersection cohomology of Y . These cycles will be supported on D × D. We start with the following simple lemma. Lemma 3.5.1. Let X be a projective n-fold, and Y ⊆ X be a subvariety. Let W ⊆ Im {Hs (Y ) → H 2n−s (X)} ⊆ H 2n−s (X) be a vector subspace on which the restriction of the Poincar´e pairing remains non degenerate, i.e. H(X) = W ⊕ W ⊥ . Then the projection PW ∈ End(H(X)) ' H(X × X) on W relative to the above splitting can be represented by a cycle supported on Y ×Y. Proof Let {e1 } be a basis for H(X) such that e1 , · · · , ek ∈ W and ek+1 , · · · , eN ∈ W ⊥ . For i = 1, · · · , k, we can represent ei by a cycle γi contained in Y . In force of the hypothesis, the dual basis {eiˇ} is of the form eiˇ=
k X j=1
aij ej
for 1 ≤ i ≤ k
eiˇ=
N X j=k+1
aij ej
for k + 1 ≤ i ≤ N.
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P In particular e1ˇ, · · · , ekˇ are represented by the cycles γiˇ= kj=1 aij γj supP ported on Y . The projector PW = ki=1 ei ⊗ eiˇ is thus represented by the Pk cycle i,j=1 aij γi × γj , which is supported on Y . The following is a standard but very useful application of “strictness” in Hodge Theory Lemma 3.5.2. Let Y ⊆ X be a codimension d subvariety of an n-fold and let π : Y˜ → Y a resolution of singularities. Suppose β ∈ Im{H2k (Y ) → H 2(n−k) (X)} ∩ H n−k,n−k (X). Then there is β˜ ∈ H n−k−d,n−k−d (Y˜ ) such ˜ = β. that (i ◦ π)∗ (β) Proof We consider the weights of the homology groups as given by their being dual of the cohomology groups. Thus H2k (Y ) has weights ≥ −2k. The map H2k (Y ) → H 2(n−k) (X) is of type (n, n). Since the Hodge structure on H 2(n−k) (X) is pure, the strictness of maps of Hodge structures implies that Im {H2k (Y ) → H 2(n−k) (X)} = Im {W−2k H2k (Y ) → H 2(n−k) (X)}. It follows that β = i∗ β 0 for some β 0 ∈ W−2k H2k (Y ). On the other hand this group coincides with Im {π∗ : H2k (Y˜ ) → H2k (Y )} for any resolution, whence the statement. Theorem 3.5.3. Let f : X → Y , D as before. Then there exist algebraic 3-dimensional cycles Z−1 , Z0 , Z1 , supported on D × D such that: • Z1 defines the projection of H(X) onto H14 (X) = c1 (η) ∧ Im{H4 (D) → H 2 (X)} ⊆ H 4 (X); 2 (X) = Im{H (D) → H 2 (X)} ⊆ • Z−1 defines the projection of H(X) onto H−1 4 2 H (X); • Z0 defines the projection of H(X) on Im{H3 (D) → H 3 (X)} ⊆ H 3 (X). Proof Let Λ be the inverse of the negative-definite intersection matrix Iij = R c (η) ∧ [Di ] ∧ [Dj ]. We denote by η ∩ Di the curve obtained intersecting X 1 the divisor Di with a general section of η. Set: X X Λij [(η ∩ Di ) × Dj ] Z1 = Λij [Di × (η ∩ Dj )]. Z−1 = It is immediate to verify that Z−1 and Z1 define the sought-for projectors. The construction of Z0 is not so direct: Since, by 3.2.10, the Poincar´e paring is non degenerate on Im{H3 (D) → H 3 (X)}, by 3.5.1 we can represent the projection on H3 (D) by a cycle supported on D × D. Furthermore, the projection is a map of Hodge structures, hence its representative cycle P3 ∈ H 6 (X × X) has type (3, 3). By 3.5.2 we have P3 = i∗ π∗ β for some
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^ ^ β ∈ H 1,1 (D × D), where D × D is any resolution of D × D. By Lefschetz’ ˜ It Theorem on (1, 1)-classes there is an algebraic cycle Z˜ such that β = [Z]. is clear that Z0 = i∗ π∗ Z does the job. The following follows immediately. Corollary 3.5.4. The Grothendieck motive, (X, ∆X − Z0 − Z1 − Z−1 ) is a Betti realization of the Intersection cohomology of Y . We can be more specific: the projector ∆X − Z0 − Z1 − Z−1 is supported on the fiber product X ×Y X, therefore defines a relative motive over Y in the sense of [8] (see also [6]). By [8, Lemma 2.23], the isomorphism of algebras End(Rf∗ QX [3]) = H6BM (X ×Y X) ensures that the Betti realization of this relative motive is the projector associated with the splitting for Rf∗ QX [3] we have used in this section. Remark 3.5.5. From the construction of the cycles it is evident that Z−1 and Z1 define in fact Chow motives, not only Grothendieck motives. Under some hypothesis it is possible to construct a Chow projector for Z0 as well. For instance, if D is smooth irreducible, and its conormal bundle ID /I2D is ample. In this case, let Z0 the cycle in D × D representing the Hodge Λ operator with respect to the polarization given by the conormal bundle. It is immediate to verify that Z0 defines the Chow motive we need. In general some knowledge of the nature of the resolution may allow one to find a Chow motive whose Betti realization is intersection cohomology. Remark 3.5.6. It is not difficult to modify the proofs to produce a Grothendieck motive for the intersection cohomology of Y for an arbitrary threedimensional variety Y , (e.g. with non-isolated singularities). If, for example, some divisor D0 is blown down to a curve C, then one needs to construct a further projector, represented by a cycle which is a linear combination of the components of D0 ×C D0 . This projector splits off the contribution of D0 to the cohomology of X. We leave this task to the reader. Remark 3.5.7. If Y is a fourfold with isolated singularities, then the computations in 3.2.4 express its intersection cohomology as a Hodge substructure of the cohomology of a resolution X. The method developed in this section does not apply in general since we do not know whether the classes of the projectors, which are push forward of classes of type (p, p) on a resolution of the product of the exceptional divisor with itself, are represented by algebraic cycles. On the other hand, this can be achieved in the presence of
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supplementary information on the singularities of Y or on the exceptional divisor. For example: if the singularities are locally isomorphic to toric singularities. This allows to define a motive for the intersection cohomology in several interesting cases.
References [1] Barth, W., C. Peters and A. Van de Ven: Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Springer-Verlag, Berlin, 1984. [2] Beilinson, A. A., J.N. Bernstein and P. Deligne: Faisceaux pervers, Ast´erisque 100, Paris, Soc. Math. Fr. 1982. [3] Cattani, E., A. Kaplan and W.Schmid: L2 and intersection cohomologies for a polarizable variation of Hodge structure, Invent. Math. 87 (1987), no. 2, 217–252. [4] de Cataldo, M.: Lectures on the Hodge theory of projective manifolds, in Hodge Theory and Its Related Topics, Proceedings of the Algebraic Geometry Summer School, Byeonsan, June 2003, Ed. J. Keum and B. Jun, KAIST, Daejon, Korea [5] de Cataldo, M. and L. Migliorini: The Hard Lefschetz Theorem and the ´ Topology of semismall maps, Ann. Sci. Ecole Norm. Sup. (4) 35 (2002), no. 5, 759–772 [6] de Cataldo, M. and L. Migliorini: The Chow Motive of semismall resolutions, Math. Res. Lett. 11, pp.151-170 (2004) [7] de Cataldo, M. and L. Migliorini: The Hodge Theory of Algebraic maps, ´ arXiv:math.AG/0306030 v2. To appear in Ann. Sci. Ecole Norm. Sup. [8] Corti, A. and M. Hanamura: Motivic decomposition and intersection Chow groups. I, Duke Math. J. 103 (2000), no. 3, 459–522 [9] Deligne, P.: Th´eor`eme de Lefschetz et crit`eres de d´eg´en´erescence de suites spectrales, Publ.Math. IHES 35 (1969), 107–126 [10] Deligne, P.: D´ecompositions dans la cat´egorie D´eriv´ee, in Motives (Seattle, WA, 1991), 115–128, Proc. Sympos. Pure Math., 55,Part 1, Amer. Math. Soc., Providence, RI, 1994. [11] Deligne, P.: Th´eorie de Hodge, II, Publ.Math. IHES 40 (1971), 5–57 [12] Deligne, P.: Th´eorie de Hodge, III, Publ.Math. IHES 44 (1974), 5–78 [13] Esnault, H. and E. Viehweg: Vanishing and Non-Vanishing Theorems, in Actes du Colloque de Th´eorie de Hodge, Asterisque 179-180 1989, 97–112 [14] Goresky, M. and R. MacPherson: Stratified Morse Theory, Ergebnisse der Mathematik, 3.folge 2, Springer-Verlag, Berlin Heidelberg 1988 [15] Ramanujam, C. P.: Remarks on the Kodaira vanishing Theorem, J. Indian Math.Soc. 36 (1972), pp.41–51 [16] Sabbah, C.: Polarizable twistor D-modules, preprint. [17] Saito, M.: Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) [18] Saito, M.: Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333 [19] Saito, M.: Decomposition theorem for proper K¨ahler morphisms, Tohoku Math. J. (2) 42, no. 2, (1990), 127–147
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[20] Verdier, J.-L.: Des Cat´egories D´eriv´ees des Categories Ab´eliennes, Asterisque 239, 1996 [21] Weibel, C.: An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994
4 Finite Correspondences and Transfers over a Regular Base ´ Fr´ed´eric DEGLISE Dpartement de Mathmatiques Institut Galile Universit Paris 13 Universit Paris 13, 99, avenue Jean-Baptiste Clment 93430 - Villetaneuse
[email protected]
To Jacob Murre for his 75th birthday
Abstract In this survey we establish the foundations for the theory of motivic complexes of V. Voevodsky, more precisely, for the theory of sheaves with transfers as exposed in [FSV00, chap. 2 and chap. 4]. We intended to give detailed proofs with a minimum amount of references. The purpose of this paper is twofold. First, we establish the theory over a general regular noetherian base using the Tor formula for intersection multiplicities of [Ser58]. Secondly, we give all the proofs of [FSV00] in the case of a perfect field. Although this paper relies on the ideas of [FSV00], the exposition differs notably as we consider only the Nisnevich topology (instead of the Zariski topology) and we work directly with correspondences up to homotopy.
General Notation, and Conventions All schemes in this paper are implicitly assumed to be noetherian. We simply say smooth (resp. ´etale) where one should say smooth (resp. ´etale) of finite type for a scheme over a base or a morphism. For any scheme S, we will denote by SmS the category of smooth S-schemes. We shall also consider the category Smcor S of smooth S-schemes equipped with finite Scorrespondences, together with its canonical graph functor γ : SmS → Smcor S (see def. 4.1.19). Every presheaf (or sheaf) considered in this paper is assumed to be a presheaf of abelian groups, unless explicitly stated otherwise. Let S be a scheme and F be a presheaf over SmS . We will extend the presheaf F to the category of pro-smooth S-schemes in the obvious way: if 151
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F. D´eglise
X• = (Xi )i∈I is such a pro-object we put F (X• ) = lim F (Xi ), −→ op i∈I
the colimit being computed in the category of abelian groups. We identify isomorphic pro-smooth S-schemes. This implies that a proobject X• which admit a limit in the category of S-schemes is determined uniquely by this limit (because the terms of the pro-object are of finite presentation over S). If for example this limit is affine over S, we can consider X• as a pro-object of affine S-schemes. We often adopt the suggestive notation ˜ X X• = lim ←− i i∈I
to denote a pro-smooth scheme (Xi )i∈I where I is a right filtering essentially small category. This “lim”-symbol indeed denotes the projective limit taken tautologically in the category of pro-smooth schemes. ˇ The topology used to compute associated sheaves, cohomology or Cech cohomology, is the Nisnevich topology, unless explicitly stated otherwise. We will denote the category of sheaves with transfers over S by NStr (see def. 4.2.3) and the subcategory of sheaves with transfers which are homotopy invariant by HNStr . Such sheaves will simply be called “homotopy sheaves”. This terminology is inspired by the theory of perverse sheaves. Indeed, at least in the case of a perfect base field k, the category HNktr is the heart of the category DM−eff (k) with respect to the homotopy t-structure†. Acknowledments: This text is based on the first part of the author thesis which was written under the direction of F. Morel. It is another occasion to thank him for all the mathematical ideas which he transmitted to me. The rewriting of this part was widely supported by the advice and encouragement of J. Wildeshaus for which I am very grateful. Finally, working with D.-C. Cisinski has been a constant motivation for me and, as he is also one of the few readers of my thesis, I seize the opportunity to convey my heartfelt mathematical friendship to him. † Note that the homotopy t-structure is also defined by Morel on the stable homotopy category of schemes over k. Its heart is indeed the category of homotopy invariant sheaves but these do not have transfers in general, in the sense given here. Thus the correct terminology for the tr should be ”homotopy sheaves with transfers” or ”homotopy oriented sheaves”, object of HNS but there is no risk of confusion here.
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Introduction The generalisation of the theory of sheaves with transfers over a regular base is presented in the first two sections. In § 4.1 we prove all the basic facts concerning finite correspondences over a regular base, using only Serre’s Tor formula for intersection multiplicities. New here (§ 4.1.5) is the treatment of functoriality for finite correspondences, as well as the base change and “forget the base” functors. Next, in § 4.2, we develop the theory of sheaves with transfers over a regular base. The arguments are very close to but more precise than those of [FSV00, chap. 4]. However, the treatment of the functoriality is new. In the remainder of the paper, we prove the fundamental facts about homotopy invariant sheaves with transfers over a perfect field. These proofs follow the main ideas of [FSV00, chap. 2], but seem to us more accurate for two reasons. First, in § 4.3 we interpret the constructions about pre-theories in the framework of correspondences up to homotopy, thereby showing how Voevodsky’s arguments directly lead to constructing cycles. We draw in particular attention to Proposition 4.3.21, a corrected version of a result of [FSV00, chap. 2]– valid only in the case of an infinite field, and Theorem 4.3.24, a generalisation of a result in loc. cit., a result which allows us to use the Nisnevich topology in what follows. In this part, we use the functoriality of finite correspondences. Secondly, we only use the Nisnevich topology. This leads to another proof of a fundamental result for sheaves with transfers: assigning the Nisnevich sheaf functor is a homotopy invariant (cf. Corollary 4.4.14). The strategy of the proof is to use the ˇ Nisnevich Cech cohomology functor together with the computation used to prove Proposition 4.4.10, a slightly more precise version of Proposition 5.4 of loc. cit.). Another fundamental theorem states that a homotopy invariant sheaf with transfers over a perfect field has homotopy invariant cohomology. We prove this important theorem of Voevodsky in the last section. The argument of the proof is due to Voevodsky. Our contribution consists of establishing the preliminary facts in a clear and fully general fashion, specifically (§ 4.5.3) the construction of the localisation long exact sequence for homotopy sheaves. These facts are slightly different from their analogs in loc. cit. since we use the Nisnevich topology. In particular, we have to use our generalisation of the theory as stated in Theorem 4.3.24. We hope that the reader appreciates Voevodsky’s beautiful argument, explained in § 4.5.4 which proves localisation and homotopy invariance for cohomology at the same time.
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Future Developments The author has established the theory in this generality in order to construct the category of motivic complexes over a general regular base and, more importantly, to give full functoriality for this category. This will be done in a future paper by D.-C. Cisinski and the author (cf. [CD]). For the full functoriality, we intend to use the recent work of J. Ayoub on cross functors to get the six functors formalism for motivic complexes. First of all, we have to construct a non effective version of motivic complexes; this can be done by considering symmetric spectra of complexes. Still, there is a technical problem as the theory developed here allows only regular base schemes. However, though this is not obvious, one can do the work of Ayoub using only those regular bases. Nonetheless, the author’s aim is to construct motivic complexes over an arbitrary base using the theory of relative cycles of [FSV00] - surely this was the original intention of Voevodsky.
4.1 Finite Correspondences 4.1.1 Relative Cycles In this section we use a particular case of the notion of a relative cycle from [FSV00, chap. 2], by restricting to the case of a regular base. We set up the foundations of the theory using [Ser58] and [Ful98]. We begin by recalling a few facts about equidimensionality: Definition 4.1.1. Let f : X → S be a morphism. One says that f is equidimensional if i) f is of finite type. ii) the relative dimension of f is constant. iii) Every irreducible component of X dominates an irreducible component of S. If S is regular, normal or more generaly geometrically unibranch, one obtains an equivalent set of conditions by replacing the third property with the stronger one of being universally open (cf. [EGA4, 14.4.4]). The following lemma allows us to simplify this notion in the particular case which we are interested in. Lemma 4.1.2. Let X, S be irreducible schemes, S geometrically unibranch. The following conditions on a morphism f : X → S are equivalent: 1) f is finite equidimensional. 2) f is finite onto.
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3) f is proper, equidimensional of dimension 0. Proof Conditions (1) and (2) are equivalent because a finite morphism is of constant relative dimension 0 and universally closed. The equivalence between (1) and (3) follows from the Stein factorisation. We will use the following notion from the general theory of [SV00], which is all what we need for our constructions: Definition 4.1.3. Let S be a regular scheme and X an S-scheme. We define the abelian group c0 (X/S) as the subgroup of the group of cycles on X generated by points x whose closure in X is finite equidimensional over S. The elements of this group are called finite relative cycles on X/S. Remark 4.1.4. In loc. cit., the cycles defined above are denoted by cequi (X/S, 0), and are called equidimensional relative cycles of relative dimension 0 on X/S. A cycle α on X is a finite relative cycle on X/S if its support is finite equidimensional over S (Lemma 4.1.2). Remark also that if S = S1 t S2 , c0 (X/S) = c0 (X/S1 ) ⊕ c0 (X/S2 ). This allows us to reduce to the case S irreducible. 4.1.5. Let S be a regular scheme and X an S-scheme. Consider a closed subscheme Z of X which is finite equidimensional over S. The irreducible components of Z which dominate an irreducible component of S are finite equidimensional over S. Let (zi ){i=1,...,n} be the generic points of Z which dominate an irreducible component of S. One associates to Z a finite relative cycle on X/S: X [Z]X/S = lg(OZ,zi ).zi . i
The Pullback. Let S and T be regular schemes and consider a cartesian square Y q
T
f
/X
∆
/S
g
p
with p smooth. Let α be a finite relative cycle on X/S and U be its support. We show that the pullback cycle f ∗ (α) is well defined in the sense of [Ser58],
156
F. D´eglise
that is, the hypothesis of V.C.7 (b) is satisfied†. As U is finite equidimensional over S, V = f −1 (U ) is again finite equidimensional over T . Suppose that U is irreducible. Considering the irreducible component of X containing U and its image on S, we can suppose that X and S are irreducible. As the morphism p is smooth, it is equidimensional of dimension n and the codimension of U in X is n. Moreover, the morphism q is equidimensional of dimension n and the codimension of V in Y is n. This proves that f ∗ (α) is well defined. Moreover, it is a finite relative cycle on Y /T . Definition 4.1.6. With the preceding notations, we will put ∆∗ (α) = f ∗ (α) as a cycle in c0 (Y /T ). Using [Ser58, V.C.7 exercice 1], we obtain functoriality of this pullback with respect to composition of cartesian squares. We note also the following lemma: Lemma 4.1.7. Let S, T be regular schemes. Consider a cartesian square Y q
T
f
/X
∆
/S
g
p
such that p is smooth and g is flat. Then, for any closed subscheme Z in X which is finite equidimensional over S, ∆∗ ([Z]X/S ) = [Z ×S T ]Y /T . Proof Put α = [Z]X/S and let Γf be the graph of f as a closed subscheme of Y × X. As Y is regular, using [Ser58, V.C.8], ∆∗ (α) = [Γf ].[Y × α]. But Γf is isomorphic to Y , hence it is a regular scheme. Its local rings are all Cohen-Macaulay local rings and the result now follows from [Ful98, Prop. 7.1], (See also [Ser58, V.C.1 Th. 1] for the identification between Serre’s and Samuel’s intersection multiplicities.) We next going to discuss the compactness of relative cycles Consider (Ti )i∈I , a pro-object of affine regular noetherian S-schemes. It admits a limit T in the category of affine S-schemes. Indeed, Ti is the spectrum over S of a coherent OS -algebra Ai . The family (Ai )i∈I op , together with its natural transition morphisms, forms an ind-OS -algebra. It admits a limit, and we put † More precisely, we consider the extension of the theory presented in [Ser58] to the case of arbitrary noetherian regular schemes, as described in V.C.8. Moreover, we don’t need Theorem 1 of V.B.3 as the correct equality of dimension is already true in our case. Besides, the positivity of intersection multiplicities for arbitrary regular noetherian local rings has been proved recently by O.Gabber.
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A = lim Ai . −→ op i∈I
Then T = SpecS (A). Note that T need not be noetherian nor regular†. Proposition 4.1.8. We adopt the notation above and suppose T is regular noetherian. Let X be a smooth S-scheme. We put Xi = X ×S Ti , XT = X ×S T and consider the cartesian square XT
T
fi
/ Xi
∆i
/ Ti .
gi
Then the morphism δ = lim ∆∗i : lim c0 (Xi /Ti ) → c0 (XT /T) is an iso−→ −→ op op i∈I
i∈I
morphism. Proof Let us first prove that δ is surjective. Let Z be a closed integral subscheme of XT , finite equidimensional over T. Then Z is defined by a quasi-coherent ideal of OXT . As XT is noetherian this ideal is coherent, generated by a finite number of sections f1 , . . . , fn . The sheaf OXT is the inductive limit of the OXi , and we can assume that there exists i ∈ I such that f1 , . . . , fn lift to OXi . Let Zi be the closed subscheme of Xi defined by the equations f1 = 0, . . . , fn = 0. Note that the square Z
/ XT
Zi
pi ×S 1X
/ Xi
is cartesian. Denote by I/i the category whose objects are the arrows j → i. Considering such an arrow, we let Zj be the pullback of Zi along the corresponding morphism. This defines a pro-object (Zj )j∈I/i such that the canonical morphism Z → lim Zj ←− j∈I/i
is an isomorphism. Using [EGA4] there exists an arrow j → i such that: 1) Zj is integral because the transition morphism of the pro-object (Zj )j≥i are dominant (use Cor. 8.4.3 of loc.cit.). 2) Zj is finite surjective over a component of Tj , using 8.10.5 of loc.cit.. † If T is noetherian, the author is aware essentially of two hypotheses that imply that T is regular: (1) the transition morphisms of (Ti )i∈I are flat, or (2) S is of equal characteristic.
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F. D´eglise
Thus the cycle [Zj ] of Xj associated to Zj is a finite relative cycle on Xj /Tj . As fj−1 (Zj ) = Z is an integral scheme, we obtain fj∗ ([Zj ]) = [Z], that is, ∆∗j ([Zj ]) = [Z]. We finally show that δ is injective. Let i ∈ I and αi ∈ c0 (Xi /Ti ) such that δ(αi ) = 0. For any j → i in I with corresponding transition morphism ∗ (α). Then (Z ) fji : Xj → Xi , let Zj be the support of fji j j∈I/i is a pro∗ object. As fi (α) = 0 this pro-object has the empty scheme as limit. This means that the canonical morphism ∅ → lim Zj ←− j∈I/i
is an isomorphism. From the first point 8.10.5 of loc. cit. there exists j → i ∗ (α) = 0, which shows α is 0 in the colimit such that Zj = ∅. Thus fji i lim c0 (Xi /Ti ). −→ op i∈I
We treat now the general pushout. One of the advantages of relative cycles is that pushout by any morphism is always defined and functorial. Lemma 4.1.9. Let S be an irreducible scheme and f : X → Y be a morphism of finite type S-schemes. Let Z be a closed integral subscheme of X. If Z is finite and surjective over S then f (Z), equipped with its reduced structure of subscheme in Y , is closed and finite surjective over S. The morphism Z → f (Z) is finite surjective. Proof Indeed, as Z/S is proper, f (Z) is closed in Y . With its induced structure of reduced subscheme of Y it is proper over S, as can be seen for example from the valuative criterion of properness (cf. [Har77]). Moreover using [EGA3, 4.4.2] f (Z) is finite over S because its fibers are finite. Thus the induced morphism Z → f (Z) is finite. Definition 4.1.10. Let S be a scheme, X and Y be finite type S-schemes and f : X → Y be an S-morphism. For Z a closed integral subscheme of X which is finite and equidimensional over S, we set according to the preceding lemma f∗ ([Z]) = d.[f (Z)] ∈ c0 (X/S) where d is degree of the extension of function fields induced by f . By linearity, this defines a morphism f∗ : c0 (Y /S) → c0 (X/S). 4.1.11. This pushout coincides with the one of [Ser58, V-27.6]. As it is always reduced to a pushout by a proper morphism according to the preceding
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lemma, it is functorial in f . This is easily seen directly using the transitivity of degree extensions. Proposition 4.1.12. Let S, S 0 be regular schemes and q : S 0 → S be a flat morphism. Consider the two following cartesian squares of schemes: X0 g
Y
t
Θ 0 p
S0
∆ q
/X f /Y / S.
We denote by ∆ • Θ the cartesian square defined by the external arrows of this diagram. Then for all finite relative cycle α ∈ c0 (X/S), we have the relation g∗ (∆ • Θ)∗ (α) = ∆∗ f∗ (α). Proof Using linearity, we can assume that α is a closed integral subscheme Z of X. The cycle f∗ (α) is supported in f (Z). Thus we can assume Y equals f (Z) and f is proper. Finally, using Lemma 4.1.7 we reduce to the classical projection formula of [Ful98, 1.7]. We state another projection formula involving intersection products that will be useful in the sequel. Proposition 4.1.13. Let X, X 0 and S be regular schemes and consider the diagram y< S aCC yyy f CC /Y Y0 ∆ p / 0 X X g
where ∆ is cartesian and p smooth. Let σ ∈ c0 (Y /X), ∈ c0 (Y 0 /S). Then, the following equation holds when the intersections involved are proper: f∗ ∆∗ (σ). = σ.f∗ (). Proof Let V be the support of . As V is proper over S, the restriction f |V : V → Y is proper using the arguments preceding Definition 4.1.10. Thus the formula is simply formula (10) of [Ser58, V.C.8.],
160
F. D´eglise
4.1.2 Composition of Finite Correspondences Let S be a regular scheme. Generalising the definition of [Voe00a], we introduce finite S-correspondences. Definition 4.1.14. Let X and Y be two smooth S-schemes. We define the group of finite S-correspondences from X to Y as the abelian group cS (X, Y ) = c0 (X ×S Y /X) . We adopt the following notation. If X and Y are two S-schemes, we put XY = X ×S Y and we denote the canonical projection by pXY X : XY → X. If X, X 0 , Y , Y 0 are smooth schemes, we denote (XX 0 )Y Y 0
p(X)Y
:= XX 0 Y Y 0
/ XY , /X
XX 0
the cartesian square induced by the canonical projections. The following lemma shows that the composition law of finite correspondences is well defined. Lemma 4.1.15. Let X, Y , Z be smooth S-schemes, α ∈ cS (X, Y ) and β ∈ cS (Y, Z). (X)Y (Z) ∗
(XY )Z ∗
(α) of XY Z intersect properly. 1) The cycles p(Y )Z (β) and p(X)Y 2) According to the first point, the intersection product in XY Z (XY )Z ∗
p(Y )Z
(X)Y (Z) ∗
(β).p(X)Y
(α)
is well defined. It is a finite relative cycle on XY Z/X. Proof We can assume that α and β are closed integral subschemes. Then we can assume also that X, Y and Z are irreducible, considering the particular component dominated by α respectively β. Consider the following diagram: β ×Y α
α
X.
/β
/Y
/Z
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The vertical arrows are all finite equidimensional. In particular, β ×Y α is finite equidimensional over X. But there is a canonical isomorphism β ×Y α → (Xβ) ×XY Z (αZ) = (Xβ) ∩ (αZ). In particular, every component of (Xβ) ∩ (αZ) is finite equidimensional over X. Thus, they are all of codimension dim(Y ) + dim(Z) in XY Z, which proves the first point. The second point then follows from the preceding remark. Definition 4.1.16. Let X, Y and Z be smooth S-schemes, and α ∈ cS (X, Y ) and β ∈ cS (Y, Z). Consider the pushout (X)Y Z p(X)Z : c0 (XY Z/X) → c0 (XZ/X) defined in 4.1.10. We put ∗ (X)Y Z (XY )Z ∗ (X)Y (Z) ∗ β ◦ αp(X)Z = p(Y )Z (β).p(X)Y (α) , ∗
which is a well defined finite S-correspondence from X to Z by the preceding lemma. Example 4.1.17. Let f : X → Y be an S-morphism between smooth Sschemes. As Y /S is separated, the S-graph Γf of f is a closed subscheme of XY . As the canonical projection Γf → X is an isomorphism, the cycle [Γf ]XY /X belongs to cS (X, Y ). This allows us to define a map HomSmS (X, Y ) → cS (X, Y ) , which is obviously injective. We will use the same letter f to denote the finite S-correspondence [γf ]XY /X . This identification is justified by following lemma, which shows that composition of S-morphisms coincides with composition of finite S-correspondences. Lemma 4.1.18. Let X, Y , Z be smooth S-schemes. Then the following relations hold: 1) For all α ∈ cS (X, Y ), β ∈ cS (Y, Z), γ ∈ cS (Z, T ), γ ◦ (β ◦ α) = (γ ◦ β) ◦ α. 2) For all α ∈ cS (X, Y ) and all S-morphisms f : Y → Z, f ◦ α = (1X ×S f )∗ (α) using Definition 4.1.10 for the pushout. 3) For all β ∈ cS (Y, Z) and all S-morphisms f : X → Y , considering the cartesian square fZ : XZ / Y Z , one has
X
f
/Y
β ◦ f = fZ∗ (β)
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F. D´eglise
using definition 4.1.6 for the pullback. 4) For all S-morphisms f : X → Y and g : Y → Z, [Γg ] ◦ [Γf ] = [Γg ◦ f ]. Proof (1). The idea to prove this relation is to show that one can compose the three correspondences by pulling them all back to XY ZT , forming the intersection product in that scheme and then pushing down the result to XT . We carry out this procedure in detail: (XY )Z ∗ (X)Y (Z) ∗ Z γ ◦ (β ◦ α) = γ ◦ pXY p (β).p (α) XZ ∗ (Y )Z (X)Y ∗ ∗ (XZ)T (X)Z(T ) (XY )Z ∗ (X)Y (Z) ∗ XY Z = pXZT p (γ).p p p (β).p (α) XT ∗ XZ ∗ (Z)T (X)Z (Y )Z (X)Y ∗ ∗ (XY )Z ∗ (XZ)T (X)Y Z(T ) (X)Y (Z) ∗ ZT = pXZT (γ).pXY p(Y )Z (β).p(X)Y (α) (a) XT ∗ p(Z)T XZT ∗ p(X)Y Z (XY Z)T ∗ (XZ)T ∗ (X)Y Z(T ) ∗ (XY )Z ∗ (X)Y (Z) ∗ XY ZT = pXZT |p(Z)T (γ).p(X)Y Z p(Y )Z (β).p(X)Y (α) (b) XT ∗ pXZT ∗ p(XZ)T ∗ ∗ ∗ (X)Y (ZT ) (XY Z)T (XY )Z(T ) ZT (α) (c) = pXY (γ). p(Y )Z (β).p(X)Y XT ∗ p(Z)T ∗ ∗ ∗ (XY Z)T (XY )Z(T ) (X)Y (ZT ) ZT = pXY (γ).p(Y )Z (β).p(X)Y (α) (d) XT ∗ p(Z)T
with the following justifications: / XY Z . / XZ
a) This is Proposition 4.1.12 for the cartesian squares XY ZT
XZT
/X
XT X dJJ r8
b) This is the projection formula 4.1.13 for
r rrr
XY ZT
JJ / XZT
/ XZ. XY Z c) This is the functoriality of pullback and pushout, and the compatibility of intersection product with pullback. d) This notation is valid using the associativity of intersection product (cf. [Ser58, V.C.3.b]).
A similar computation works for the right hand side of the equality. This finishes the proof of (1). (2). We can assume that α is a closed integral subscheme of XY . Then we must compute the following cycle: Z XY Z ∗ XY Z ∗ f ◦ α = pXY p [Γ ].p (α) f XZ ∗ YZ XY Z = pXY XZ ∗ ([XΓf ].[αZ]) .
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The intersection involved in this cycle is particularly simple: XΓf ×XY _ Z αZ VVVV * b
XΓf × XY α
XΓf VV V i V VV V VV V
_
/ αZ _ JJJ JJ% a / α
_
/ XY Z iα JJJ J% ∼ / XY
XΓf
p
where p is induced by the canonical isomorphism Γf → Y , and i, iα are the canonical closed immersions. The front square in this cube is cartesian, so a is an isomorphism. But the back and right squares are both cartesian, which implies that the left square is also cartesian. Hence b is an isomorphism. This implies that W = XΓf ×XY Z αZ is isomorphic to α. Thus it is an integral scheme. In particular, the intersection of XΓf and αZ is reduced to the single component W . Moreover, the intersection multiplicity of W is 1 using [Ful98, Prop. 7.2]. Indeed, a base field is not needed here: we only need the comparison between Serre’s and Samuel’s intersection multiplicities (the Tor formula). This is [Ser58, V.C.4.]. We have obtained: [XΓf ]XY Z .[αZ]XY Z = [XΓf ×XY Z αZ]XY Z = i∗ ([XΓf ×XY Z αZ]XΓf ) = i∗ ([XΓf ×XY α]XΓf ) = i∗ p∗ (α). We now use the factorization
XY
p ulll
XΓf TTi T)
1X × S γf
/ XY Z
where γf : Y → Y Z is the graph morphism. Thus i∗ = (1X ×S γf )∗ p∗ , which means (1X ×S γf )∗ = i∗ p∗ as p is an isomorphism. And finally: Z XY Z pXY XZ ∗ ([XΓf ].[αZ]) = pXZ ∗ (1X ×S γf )∗ (α) = (1X ×S f )∗ α. (3). We compute (XY )Z∗ (XY )Z β ◦ f = p(X)Z p(Y )Z (β).[Γf Z]XY Z We consider the cartesian square Γf Z ι
XY Z
p Z pXY YZ
/ XZ Using the definition f × 1 S Z
/ Y Z.
of [Ser58, V.C.7], (XY )Z
∗
Z p(Y )Z (β).[Γf Z]XY Z = ι∗ ι∗ |pXY (β) = ι∗ p∗ (f ×Z 1Z )∗ (β). YZ Z As p is an isomorphism, p∗ = (p∗ )−1 . The result follows since pXY XZ ◦ ι = p.
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F. D´eglise
(4). This follows either from (2) and functoriality of pushout or (3) and functoriality of pullback. The preceding lemma implies that the product ◦ of finite S-correspondences is associative, and that for any smooth S-scheme X the S-morphism 1X , seen as a correspondence, is the neutral element. Definition 4.1.19. Let S be a regular scheme. We denote by Smcor S the category whose objects are smooth separated S-schemes of finite type and whose morphisms are the finite S-correspondences. We denote by γ : SmS → Smcor S the faithful functor which is the identity on objects and sends an Smorphism to its graph (cf. ex. 4.1.17). If X is a smooth S-scheme, we denote by [X] the corresponding object of Smcor S . This category is additive and for all smooth S-schemes X and Y , [X] ⊕ [Y ] = [X t Y ]. 4.1.3 Monoidal Structure Lemma 4.1.20. Let X, X 0 , Y , Y ’ be smooth S-schemes. Then for any ∗ ∗ X0Y 0 α ∈ cS (X, Y ) and β ∈ cS (X 0 , Y 0 ), the cycles pXY XY X 0 Y 0 (α) and pXY X 0 Y 0 (β) intersect properly, and the intersection cycle is a finite relative cycle on XY X 0 Y 0 /XX 0 . Proof Let assume α et β are closed integral subscheme. Consider the diagram: Xβ ×X α
/α
/X
Xβ
XX 0 . All vertical arrows are finite equidimensional. But Xβ ×X α is isomorphic to XY β ∩ αX 0 Y 0 . Thus this scheme is finite equidimensional on XX 0 , which implies that the corresponding intersection is proper and concludes the proof. Definition 4.1.21. Let X, X 0 , Y , Y 0 be smooth S-schemes. For all α ∈ cS (X, Y ) and β ∈ cS (X 0 , Y 0 ) we put XY X α ⊗tr S β = pXY
0Y 0 ∗
X (α).pXY X0Y 0
0Y 0 ∗
(β).
From the preceding lemma, this is a well defined cycle and an element of cS (XX 0 , Y Y 0 ).
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Lemma 4.1.22. Suppose we are given finite S-correspondences: α : X → Y, α0 : Y → Z, β : X 0 → Y 0 , β 0 : Y 0 → Z 0 . Then, 0 0 tr 0 tr (α0 ◦ α) ⊗tr S (β ◦ β) = (α ⊗S β ) ◦ (α ⊗S β).
Proof As for the associativity of composition of finite correspondences, the proof consists of showing that one can first pull back all cycles to XY ZX 0 Y 0 Z 0 , then take the intersection and push down the result to XZX 0 Z 0 . As in the proof of 4.1.18 1), we use the two projection formulas 4.1.12 and 4.1.13 and the functoriality of pushout and pullback. We leave the details to the reader. Proposition 4.1.23. The category Smcor S is monoidal symetric with tensor product [X] ⊗tr S [Y ] = [X ×S Y ] for smooth S-schemes X and Y , and where the tensor product of finite Sγ correspondences is given by definition 4.1.21. The functor SmS − → Smcor S of definition 4.1.19 is monoidal; the tensor product on SmS is the cartesian product over S. Proof Commutativity is obvious using commutativity of the intersection product (cf. [Ser58, V.C.3.a]. Associativity is proved in the same way as associativity of composition product: using the projection formulas (cf. prop. 4.1.12 and 4.1.13), one reduces to associativity of the intersection product. For the last assertion, it suffices to prove the equality f ⊗tr S 1Y = f ×S 1Y 0 for smooth S-schemes X, X , Y and an S-morphism f : X → X 0 . Let ∆Y be the diagonal of Y /S and Γf the S-graph of f . Following now the same line as in the proof of Lemma 4.1.18,3) we note that the intersection of Γf Y Y and XX 0 ∆Y is isomorphic to the graph of f ×S 1Y which is isomorphic to XY and hence reduced. This implies that the intersection multiplicities are 1 and gives the result.
4.1.4 A Finiteness Property Let (Xi )i∈I be a pro-object of affine smooth S-schemes. As we have seen (Prop. 4.1.8), this pro-object admits a limit X in the category of affine Sschemes. We assume that X is regular noetherian. In this case, for any smooth S-scheme Y , extending definition 4.1.14, we put
166
F. D´eglise
¯cS (X, Y ) = c0 (X ×S Y /X) . Moreover, the projection morphisms pi : X → Xi induce by pullback a morphism p∗i : cS (Xi , Y ) → ¯cS (X, Y ). These morphisms are obviously natural in i ∈ I. Then Proposition 4.1.8 admits immediately the following corollary: Proposition 4.1.24. Consider the hypotheses and notations above. Then the morphism lim p∗i : lim cS (Xi , Y ) → ¯cS (X, Y ) −→op −→ op
i∈I
(4.1)
i∈I
is an isomorphism. Remark 4.1.25. This proposition is implicitly used in the proof of in [SV00, Chap.5, Prop. 3.1.3]. 4.1.26. Note that we can define a product ¯cS (X, Y )⊗Z cS (Y, Z) → ¯cS (X, Z), (¯ α, β) 7→ β¯◦ α ¯ , using the arguments of Lemma 4.1.15. Then, using the argument of the proof of Lemma 4.1.18, 1), we obtain the relation (γ ◦ β)¯◦ α ¯= γ¯◦ (β¯◦ α ¯ ). In particular, the abelian group ¯cS (X, Y ) is functorial with respect to finite S-correspondences in Y . Finally, considering this functoriality, the isomorphism (4.1) is natural in Y with respect to finite S-correspondences. 4.1.27. Suppose now that we are given a second pro-object (Xi0 )i∈I of smooth affine S-schemes and a family of S-morphisms fi : Xi0 → Xi which are compatible with transition morphisms. Let X0 be the projective limit of (Xi0 )i∈I , p0i : X0 → Xi0 the canonical projection and f : X0 → X the projective limit of the (fi )i∈I . By considering suitable pullbacks of relative cycles, we obtain the commutative diagram cS (Xi , Y ) fi∗
cS (Xi0 , Y )
p∗i
/¯ cS (X, Y )
p0i ∗
f∗
/¯ cS (X0 , Y ) .
This commutative diagram shows that the isomorphism (4.1) is natural in X with respect to morphisms of pro-S-schemes. 4.1.5 Functoriality Base change. Let τ : T → S be a morphism of regular schemes. If X and Y are smooth
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S-schemes, we identify the schemes XT ×T YT and X ×T Y via the canonical isomorphism; we denote both schemes by XYT . Given smooth S-schemes X, X 0 , Y , Y 0 we consider the following cartesian squares: XX 0 Y Y 0
/ XY
(XX 0 )Y Y 0
p(X)Y
XX 0
/X
XX 0 Y YT0
XXT0
/ XYT
(XX 0 )Y Y 0
q(X)Y
/ XT
/ XY
XYT
XT
τXY
/X
with the obvious projections. For every finite S-correspondence α : X → Y , ∗ (α). using Definition 4.1.6 we put αT = τXY Lemma 4.1.28. Let X and Y be smooth schemes in SmS . Then, for all α ∈ cS (X, Y ) and β ∈ cS (Y, Z), we have βT ◦ αT = (β ◦ α)T . Proof (XY )Z ∗ ∗ (X)Y (Z) ∗ ∗ ∗ XY Z (τY∗ Z β) ◦ (τXY α) = qXZ (τY Z β).q(X)Y (τXY α) ∗ q(Y )Z (X)Y (Z) ∗ (XY )Z ∗ ∗ XY Z ∗ α) (1) β).τ (p = qXZ τ (p XY Z (X)Y ∗ XY Z (Y )Z (XY )Z ∗ (X)Y (Z) ∗ XY Z ∗ β).(p(X)Y α) (2) = qXZ ∗ τXY Z (p(Y )Z ! (XY )Z ∗ (XY )Z ∗ ∗ Z = τXZ pXY β).(p(X)Y α) . (3) XZ ∗ (p(Y )Z where (1) follows from the functoriality of pullback, (2) is compatibility of pullback with intersection product (cf. [Ser58, V.C.7]0, and (3) is Proposition 4.1.12. Definition 4.1.29. Let τ : T → S be a morphism of regular schemes. Using the preceding lemma, we define the base change functor τ ∗ : Smcor → Smcor S T X/S 7→ XT /T cS (X, Y ) 3 α 7→ αT . We summarise the basic properties of base change for correspondences. Lemma 4.1.30.
1) The functor τ ∗ is symmetric monoidal.
2) Let τ0∗ : Smcor → Smcor be the classical base change functor on S T
168
F. D´eglise
smooth schemes. The following diagram is commutative. SmS τ0∗
SmT
γS
/ Smcor S ∗
γT
τ / Smcor . T
3) If σ : T 0 → T is a morphism of regular schemes, we have a canonical isomorphism of functors (τ ◦ σ)∗ ' σ ∗ ◦ τ ∗ . Proof (1). Let α ∈ cS (X, Y ), β ∈ cS (X 0 , Y 0 ). Then, ∗ XX 0 Y Y 0 ∗ XX 0 Y Y 0 ∗ (α ⊗tr β)T = τXX p (α).p (β) 0Y Y 0 XY X0Y 0 ∗ ∗ XX 0 Y Y 0 ∗ XX 0 Y Y 0 ∗ p (α) . τ p = τXX (β) 0 Y Y 0 XY XX 0 Y Y 0 X 0 Y 0 ∗ 0 0 0 0 XX Y Y XX Y Y ∗ = qXY (αT ) . qX (β) . 0Y 0 (2). This point follows from the fact that for any S-morphism f : X → Y , there is a canonical isomorphism ΓfT → Γf ×S T . (3). Indeed there is a canonical isomorphism XT 0 ' (XT )T 0 . Its naturality with respect to finite correspondences follows from the functoriality of pullback on cycles. Restriction Let τ : T → S be a smooth morphism of regular schemes and let X, Y be smooth T -schemes. We denote by δXY : X ×T Y → X ×S Y the canonical regular closed immersion, obtained from the diagonal immersion of T /S by base change. Let α ∈ cT (X, Y ). We consider, using Definition 4.1.10, the cycle δXY ∗ (α) as an element of cS (X, Y ). Lemma 4.1.31. Let X, Y and Z be smooth T -schemes. The following relations hold: 1) For all T -morphisms f : X → Y , δXY ∗ [Γf ]T = [Γf ]S . 2) For all α ∈ cT (X, Y ) and β ∈ cT (Y, Z), δXZ ∗ (β ◦ α)(δY Z ∗ (β)) ◦ (δXY ∗ (α)). Proof In this proof, we stop mentioning the extensions of the schemes involved to simplify the notation. The first assertion is obvious.
Correspondences and Transfers
169
For the second assertion, consider the diagram X ×T Y o
xxx xxxx xxxxx X ×T FY o FF FF δXY F#
p
XY Z qXY
XY Z qY Z
/ Y ×T Z X ×T Y ×TQ Z FFFFF Q n Q n FFFF QQQ nn n FFF Q n Q( vnn a b q / δXY Z X ×S Y ×T Z Y ×T Z X ×T Y ×SP Z
X ×S Y o
PPPc PPP P(
Z pXY XY
dnnnn n n n vn
X ×S Y ×S Z
Z pXY YZ
xx xx |xx δY Z
/ Y ×S Z,
where all horizontal arrows are canonical projections and all the other arrows are canonical closed immersions. The equality is obtained in the following way: XY Z ∗ XY Z XY Z ∗ XY Z XY Z ∗ XY Z ∗ δXZ∗ qXZ q (β).q (α) = p δ q (β).q (α) XY XZ ∗ XY Z∗ Y Z XY ∗ YZ Z ∗ ∗ ∗ ∗ = pXY d b b q (β).a p (α) XZ ∗ ∗ ∗ Z ∗ ∗ ∗ = pXY d q (β).b a p (α) ∗ XZ ∗ ∗ Z ∗ ∗ ∗ = pXY d q (β).d c p (α) ∗ XZ ∗ ∗ Z ∗ ∗ = pXY d q (β).c p (α) ∗ ∗ XZ ∗ Z XY Z ∗ XY Z ∗ = pXY p δ (β).p δ (α) Y Z∗ XY ∗ XZ ∗ YZ XY
using the functoriality of pullback and pushout and the projection formulas 4.1.13, 4.1.12. Definition 4.1.32. Let τ : T → S be a smooth morphism of regular schemes. Using the preceding lemma, we define a functor τ] : Smcor → Smcor T S τ (X → T ) 7→ (X → T − → S) cT (X, Y ) 3 α 7→ δXY ∗ (α). 4.1.33. Note that from the first point of the preceding lemma, the restriction of τ to SmT is the classical functor “forgetting the base”. Moreover, σ τ for a sequence of smooth morphisms R − →T − → S between regular schemes, we clearly have (τ ◦ σ)] = τ] ◦ σ] . Proposition 4.1.34. Let τ : T → S be a smooth morphism of finite type between regular schemes.
170
F. D´eglise
1) The functor τ] is left adjoint to the functor τ ∗ . 2) For every smooth algebraic T -scheme X (resp. S-scheme Y ), the obvious morphism obtained by adjunction τ] (τ ∗ X ⊗T Y ) → X ⊗S τ] Y is an isomorphism. Proof For the first assertion, we only remark that for a smooth T -scheme X (resp. S-scheme Y ), (τ] X) ×S Y ' X ×T (τ ∗ Y ). The second assertion is clear for the case of τ ∗ and τ] (for morphisms of schemes) and we only have to apply 4.1.30, 2) and 4.1.33, 1). 4.2 Sheaves with Transfers In this section S will be a regular scheme, unless stated otherwise. 4.2.1 The Nisnevich Topology We will consider the Nisnevich topology on the site SmS . Recall that a cover for the Nisnevich topology is a family of ´etale maps pi : Yi → X such that for any x ∈ X, there exists yi ∈ Yi satisfying pi (yi ) = x and the induced map between the residue fields κ(x) → κ(yi ) is an isomorphism. Among such covers, there is the family of covers induced by the distinguished squares of [MV99] which are cartesian squares W q
U
/V p
k j
/X
such that j is an open immersion, p is an ´etale morphism and the induced morphism p−1 (X − U )red → (X − U )red is an isomorphism. The family (p, j) is indeed a Nisnevich cover. Moreover, recall from [MV99, prop. 1.4] that a presheaf F on SmS is a Nisnevich sheaf if and only if for any disinguished square as above, the square F (X) j∗
F (U )
p∗
/ F (V )
−q ∗
k∗
/ F (W )
is cartesian. Let X be a smooth S-scheme and x a point of X. A Nisnevich neighbourhood of x in X is a pair (V, y) where V is an ´etale X-scheme and y a point of V over x such that the induced morphism κ(x) → κ(y) is an
Correspondences and Transfers
171
isomorphism. We let Vhx (X) be the category of Nisnevich neighbourhoods of x in X with arrows the morphisms of pointed schemes. This category is non empty, essentially small and left filtered. We define the h-localisation of ˜ V . Following the general notation X in x as the pro-scheme Xxh = lim ←− h V ∈Vx (X)
used in this article, we define the fiber of F at the point x of X as the F (V ). Standard arguments show that abelian group F (Xxh ) = lim −h→ op V ∈Vx (X)
the functor from Nisnevich sheaves to abelian groups F 7→ F (Xxh ) is exact and commutes with arbitrary sums. Moreover, the family of fiber functors induced by a pointed smooth S-scheme (X, x) is conservative for the category of Nisnevich sheaves over SmS . Remark 4.2.1. Let OhX,x be the henselisation of the local ring of X at x. Then Spec OhX,x is the limit of the pro-object Xxh . Definition 4.2.2. We will denote by PS (resp. NS ) the category of presheaves (resp. sheaves for the Nisnevich topology) on SmS . 4.2.2 Definition and Examples Recall the canonical map γ : SmS → Smcor S of definition 4.1.19. Definition 4.2.3. A presheaf F with transfers over S is an additive presheaf tr of abelian groups over Smcor S . We denote by PS the corresponding category. A sheaf with transfers over S is a presheaf F with transfers such that the functor F ◦ γ is a Nisnevich sheaf. We denote the full subcategory of Ptr S of tr sheaves with transfers by NS . Let X be a smooth S-scheme. We denote by LS [X] the presheaf on Smcor S represented by X. Lemma 4.2.4. Let X be a smooth S-scheme. The presheaf LS [X] restricted to SmS via γ is an ´etale sheaf. Proof Let Y be a smooth algebraic S-scheme. As Y is algebraic, it is sufficient to consider a surjective ´etale morphism f : V → Y . By additivity we may assume Y is irreducible. Let W = V ×X V and consider the canonical projections p, q : W → V . Using Lemma 4.1.18, 3) it is sufficient to show the exactness of the sequence f∗
p∗ −q ∗
X 0 → c0 (Y ×S X/Y ) −−X → c0 (V ×S X/V ) −− −−X → c0 (W ×S X/W ) .
∗ is injective. Let α ∈ c (V × X/V ) As fX is faithfully flat, the pullback fX 0 S
172
F. D´eglise
∗ (α). Write α as a linear combination be a cycle such that p∗X (α) = qX Pn α = i=1 λi .zi with zi a point of V ×S X whose closure is finite and surjective over Y . As pX and qX are ´etale, the assumptions imply n X i=1
λi .
X x∈p−1 X (zi )
x=
n X j=1
λj .
X
y.
−1 y∈qX (zj )
Denote the set {fX (z1 ), . . . , fX (zn )} by I. Note that if w ∈ I such that f (zi ) = w = f (zj ), i, j ∈ Z, then λi = λj . Indeed, the equality above shows that the coefficient of x = (zi , zj ) ∈ W ×S X is λi and λj . For w ∈ I, we put P λ(w) = λi for any i such that w = f (zi ). If we define β = w∈I λ(w).w, ∗ (β) = α as f then fX etale. Finally, Lemma 4.1.9 shows that β is an X is ´ element of c0 (Y ×S X/Y ).
4.2.3 Associated Sheaf with Transfers Let p : U → X be an S-morphism of smooth S-schemes. We denote by n the n-fold product of U over X. Using the convention S ˇn (U/X) = UX n+1 ˇ ˇ UX , consider the Cech simplicial scheme S∗ (U/X) associated to U/X. We denote the associated chain complex considered within the additive category generated by the category of schemes by Cˇ∗ (U/X). Applying the additive ˇ functor LS [.] to this complex, we get a complex LS C∗ (U/X) of sheaves with transfers, naturally augmented over LS [X]. The following proposition is an obvious generalisation of [Voe00b, prop. 3.1.3]. Proposition 4.2.5. Let X be a smooth S-scheme and p : U → X be a Nisnevich cover. The natural augmentation morphism LS Cˇ∗ (U/X) → LS [X] is a quasi-isomorphism in the category of Nisnevich sheaves over S. Proof We only need to check the assertion on the conservative family of points introduced in section 4.2.1. Let (Y, y) be a pointed smooth S-scheme. We consider OhY,y , the henselian local ring of Y at the point y and put h Y = Spec OY,y . Using Proposition 4.1.24 the canonical morphism cS Yyh , X → ¯cS (Y, X) defined on the fiber of LS [X] at (Y, y) (cf. section 4.2.1) is an isomorphism. Thus we need to show the exactness of the complex p∗ d0∗ dn∗ n+1 C∗ = . . . −− → ¯cS Y, UX → . . . −− → ¯cS (Y, U ) −→ ¯cS (Y, X) → 0.
Correspondences and Transfers
173
Let F be the set of reduced closed subschemes of Y ×S X, finite and equidimensional over Y, and ordered by inclusion,. Consider Z ∈ F and put n+1 n+1 Cn(Z) = ¯cS Y, Z ×X UX ⊂ ¯cS Y, UX . (Z)
(Z)
Then C∗ is a subcomplex of C∗ . Moreover the complex C∗ is increasing S (Z) with respect to Z and we have C∗ = Z∈F C∗ . Thus it suffices prove that (Z)
the complex C∗ is contractible. According to the hypothesis, Z is finite over Y. As Y is henselian, Z is a direct sum of local henselian schemes. Following [Ray70], the Nisnevich cover pZ : Z ×X U → Z admits a section ˇ s : Z → Z ×X U . It is now a classical fact that the augmented Cech complex ˇ C∗ (Z ×X U/Z) → Z is contractible in the additive category generated by the category of schemes. An explicit homotopy is given by the collection of morphism n+1 n+2 → Z ×X UX , sn = s ×X 1U n+1 : Z ×X UX
n ≥ −1.
X
The result follows by application of the functor cS (Y, .). Following the idea of the proof of [Voe00b, Lemma 3.1.6], we obtain ˇ 0 F the 0Lemma 4.2.6. Let F be a presheaf with transfers. Denote by H ˇ th Cech Nisnevich cohomology presheaf on SmS associated with F and let ˇ 0 F the canonical morphism. Then there exists a unique pair η : F → H 0 ˇ tr F, µ) such that (H ˇ 0 F is a sheaf with transfers satisfying H ˇ 0 F ◦γ = H ˇ 0F . 1) H tr tr 0 F of presheaves with transfers ˇ tr 2) µ is a natural transformation F → H that coincides with η. Proof As F is a presheaf with transfers, we have a canonical inclusion F (X) ' HomPtr (LS [X] , F ) ⊂ HomPS (LS [X] , F ) . S Conversely, a natural transformation of presheaves on SmS φ
F − → HomPS (LS [.] , F ) is equivalent to a structure of presheaf with transfers on F as soon as it respects the composition product (in which case, it is a monomorphism). The two structures are in one-to-one correspondence using the equation F (α).a = φX (a)Y .α,
∀α ∈ cS (Y, X) , a ∈ F (X).
(4.2)
We work with the natural transformation φ and not with the structure of a presheaf with transfers.
174
F. D´eglise
0 F . Consider ˇ tr 1) Suppose for the moment that we already have defined H α ∈ cS (Y, X) and a ∈ FNis (X). Recall that ˇ 0 F (X) = lim Ker F (U ) → F (U ×X U ) . H −→ U →X
As the colimit is filtered, there exists a Nisnevich cover U of X such that a can be lifted to an element aU ∈ F (U ). Applying Lemma 4.2.5, LS [U ] → LS [X] is an epimorphism. Hence there exist a Nisnevich cover V of Y and a correspondence αU ∈ cS (V, U ) such that p ◦ αU = α|V . We thus have obtained the following commutative diagram (the commutativity of the bottom square is the first condition appearing in the statement of the theorem) / HomP LS [X] , H ˇ 0F ˇ 0 F (X) H S O
/ HomP LS [U ] , H ˇ 0F S O
F (U )
/ HomP (LS [U ] , F ) . S
ˇ 0 F (U ) H η
This in turn can be translated into the following local equation which car0F ˇ tr acterizes H 0 ˇ tr H F (α|V ).a = ηY (F (αU ).aU ) ∈ F (V ).
2) Conversely, we need to prove that the above equality does not depend ˇ 0 F . Let U be a on the choice of the cover U because it then defines H tr ˇ 0 F . As H is left exact, Lemma Nisnevich cover of X and H = HomPS ., H 4.2.5 implies the following sequence is exact 0 → H(LS [X]) → H(LS [U ]) → H(LS [U ×X U ]). Let us consider the arrow η ˇ 0 F = H(LS [X]). F (X) ,→ HomPS (LS [X] , F ) −−X → HomPS LS [X] , H As it is natural, it induces the commutative diagram 0
0
/ H(LS [X]) O / KerU
/ H(LS [U ]) O
/ H(LS [U ×X U ]) O
/ F (U )
/ F (U ×X U ),
which is natural in U . Taking the limit over all Nisnevich covers U of X,
Correspondences and Transfers
175
we obtain the arrow ˇ0 φNis X : H F (X) = lim −→
ˇ 0F KerU −→ H(LS [X]) = HomPS LS [X] , H
U →X
ˇ 0 F according to equation (4.2): which in turn defines transfers on H 0 ˇ tr H F (α) : F (X) → F (Y ), a 7→ φNis X (a)Y .α.
ˇ 0 F (X) and α ∈ cS (Y, X). As in the first step of the proof, Consider a ∈ H choose a cover U (resp. V ) of X (resp. Y ) and liftings aU ∈ F (U ), αU ∈ cS (V, U ). Then tautologically, 0 ˇ tr H F (α|V ).a = ηY (F (αU ).aU ).
From this local equation we now easily deduce the compatibility of φNis with the product of correspondences. By the very construction, φNis extends φ. The uniqueness statement in the preceding lemma implies the natural0 F and the following ˇ tr ity with respect to F of the transformation F → H corollary. Corollary 4.2.7. With the notation of the previous lemma, the association tr ˇ0 atr : Ptr S → NS , F 7→ Htr F
defines a functor left adjoint to the forgetful functor Otr : NStr ,→ Ptr S. Moreover, the following diagram commutes Ptr S
atr /
aNis
PS
NStr
/ NS .
Recall that a Grothendieck abelian category is an abelian category which admits arbitrary direct sums, has a set of generators (any object is a quotient of a direct sums of the generators) and such that filtered inductive limits are exact. Proposition 4.2.8. The category NStr is an abelian Grothendieck category. It is complete (i.e. admits arbitrary small projective limits). The forgetful functor NStr → NS admits a left adjoint LS [.]. An essentially small family of generators for NStr is given by the sheaves LS [X] for a smooth S-scheme X.
176
F. D´eglise
Proof The existence of the right exact functor atr implies that NStr is cocomplete (as is the category Ptr S ). As in the classical case, an inductive limit of sheaves with transfers is constructed by first computing it in the category of presheaves with transfers, then taking the associated sheaf with transfers. This description shows that NStr is a Grothendieck abelian category. Consider a Nisnevich sheaf F . Then, classically F =
lim −→
ZS (X) ,
X/F ∈SmS /F
where ZS (X) is the free abelian sheaf represented by a smooth S-scheme X and the limit is taken over every morphism ZS (X) → F for such an X. We simply put LS [F ] = lim LS [X], the inductive limit being calculated −→ X/F ∈SmS /F
in the category of sheaves with transfers. The construction of the functor LS [.] implies the last assertion. Let X be a smooth S-scheme. The graph morphism induces a morphism of sheaves ηX : ZS (X) → Otr LS [X]. Using the description of the Ext groups for sheaves and for sheaves with transfers we deduce a canonical morphism, natural in X and the sheaf with transfers F i ηX : ExtiNtr (LS [X] , F ) → ExtiNS (ZS (X) , Otr F ) = H i (X; Otr F ). S
Proposition 4.2.9. Using the notation introduced above, for every smooth S-scheme X, every sheaf with transfers F over S, and every integer i ∈ N, i is an isomorphism. the morphism ηX Proof Using the Yoneda lemma, the property is clear for i = 0. Consider the case i > 0. The category NStr , being a Grothendieck abelian category, has enough injectives. In particular, the Ext groups with coefficients in F are calculated by choosing an injective resolution of F in NStr . Consequently, it suffices to prove that for any sheaf with transfers I which is injective in the category NStr , the sheaf Otr I is acyclic. Following [Mil80, prop. III.2.11], ˇ this property is in turn equivalent to the vanishing of all the positive Cech i ˇ cohomology groups H (X; Otr I). But this now follows from Proposition 4.2.5. Corollary 4.2.10. Let F be a presheaf with transfers. Then for all integers i (., F i ∈ N, the presheaf HNis Nis ) has a canonical structure of a presheaf with transfers.
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177
4.2.4 Closed Monoidal Structure Recall that by Proposition 4.1.23 there is a monoidal structure on Smcor S . Lemma 4.2.11. The category NStr admits a unique structure of a symmetric monoidal category with a right exact tensor product such that the graph functor tr Smcor S → NS is monoidal. Proof Let F and G be sheaves with transfers. Using Proposition 4.2.8, we can write F =
lim −→
LS [X] ,
X/F ∈SmS /F
G=
lim −→
LS [Y ] .
Y /F ∈SmS /G
Necessarily, the tensor product of sheaves with transfers must satisfy F ⊗tr S G=
lim −→
(LS [X] ⊗tr S LS [Y ]).
X/F,Y /G
The axioms of a symmetric monoidal category then follow from the corresponding properties of the category Smcor S and uniqueness is established as well. Definition 4.2.12. We denote the tensor product on NStr satisfying the conditions of the previous lemma by ⊗tr S. Remark 4.2.13. We can express the difference between the tensor product with transfers and the usual tensor product of abelian sheaves. Indeed, for any sheaf F with transfers we have an epimorphism of sheaves with transfers L F (X) ⊗Z LS [X] → F L X∈SmS F (X) ⊗Z cS (Y, X) → F (Y ) X∈SmS ρ ⊗ α 7→ ρ ◦ α, where we view ρ (resp. α) as a map LS [X] → F (resp. LS [Y ] → LS [X]). Thus, as ⊗tr S is right exact, we deduce an epimorphism of sheaves L 0 0 → F ⊗tr 0 ∈Sm (F (X) ⊗Z LS [X]) ⊗ (G(X ) ⊗Z LS [X ]) X,X S G S L 0 0 tr G)(Y ) F (X) ⊗ G(X ) ⊗ c (Y, X × X ) → (F ⊗ 0 Z Z S X,X ∈SmS S ρ ⊗ µ ⊗ α 7→ (ρ ◦ α) ⊗tr S (µ ◦ α). In particular, for any pointed scheme (Y, y), we have on the level of the fiber at Yyh (cf. section 4.2.1) an epimorphism of abelian groups L 0 h 0 h → (F ⊗tr X,X 0 ∈SmS F (X) ⊗Z G(X ) ⊗Z cS Yy , X × X S G)(Yy ) ρ⊗µ⊗α ¯ 7→ (ρ ◦ α ¯ ) ⊗tr ¯ ). S (µ ◦ α
178
F. D´eglise
Proposition 4.2.14. The monoidal category NStr is closed: the bifunctor tr (., .). ⊗tr S admits a right adjoint HomNS Proof Let F and G be sheaves with transfers. We put HomNStr (F, G) (X) = HomNStr F ⊗tr S LS [X] , G . As a sheaf with transfers is an inductive limit of representable presheaves with transfers (cf. Prop. 4.2.8), one easily obtains the expected adjoint property. 4.2.5 Functoriality We fix a morphism τ : T → S of regular schemes. The Abstract Case. Consider an abstract additive functor ϕ : Smcor → Smcor that sends a S T Nisnevich cover of an S-scheme to a Nisnevich cover of a T -scheme. In this situation, we will define the following two functors: 1) If F is a sheaf with transfers over S, we define over T the sheaf with transfers ϕ(F ) = lim LT [ϕ(X)]. −→ X/F
2) If G is a sheaf with transfers over T , we define over S the sheaf with transfers ϕ0 (G) = G ◦ ϕ. Note there is an abuse of notation in (1). This is justified by the fact that the functor ϕ on sheaves with transfers is an extension of the functor ϕ on schemes via the associated represented sheaf with transfers functor. The Yoneda lemma implies immediately that ϕ0 is right adjoint to ϕ. The same construction applies to the graph functor γS : SmS → Smcor S . Indeed this functor respects tautologically the Nisnevich coverings and we obtain an extension on sheaves γS : NS → NStr and a right adjoint which is tr the forgetful functor Otr S : N S → NS . Going back to the hypothesis of the beginning, we suppose given in addition the commutative diagram of functors SmS ϕ0
SmT
γS γT
/
/ Smcor
S ϕ cor SmT .
By hypothesis, ϕ0 respects Nisnevich coverings and the same process gives a pair of adjoint functors ϕ 0 : NS → NT ,
ϕ00 : NT → NS .
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179
It is now obvious that these functors are related by the commutative diagrams NS ϕ0
NT
γS
/ Ntr S
γT
/
ϕ tr NT ,
OS NStr NO S o
ϕ00
NT o
O
OT
ϕ0 tr NT .
Finally, suppose that ϕ is monoidal. Then the extension ϕ : NStr → NTtr is again monoidal. In addition, we have a canonical isomorphism HomNStr F, ϕ0 (G) ' HomNTtr (ϕ(F ), G) . The same remark applies to the pair of adjoint functors (γS , Otr S ). We first apply the abstract construction above to the monoidal functor cor τ ∗ : Smcor S → SmT
defined in 4.1.29. This yields the base change functor τ ∗ : NStr → NTtr and its right adjoint τ∗ = (τ ∗ )0 : NTtr → NStr . The first one is monoidal and the second one coincides with the usual pushout for sheaves without transfers. Suppose now that the morphism τ : T → S is a smooth morphism of regular schemes. We apply the abstract construction to the monoidal functor cor τ] : Smcor T → SmS
defined in 4.1.32. This yields the twisted exceptional direct image functor τ] : NTtr → NStr which is monoidal. Remark 4.2.15. When τ is ´etale, this functor is really the usual exceptional direct image τ! . Otherwise we need to twist this functor in order to get the fundamental equality τ! = τ∗ when τ is smooth projective. Lemma 4.2.16. If τ : T → S is smooth, there exists a canonical isomorphism of functors τ ∗ ' (τ] )0 . Proof Let F be a sheaf with transfers over S. Accordign to the definitions, τ ∗ F is the sheaf associated with the presheaf with transfers over T Y 7→ lim cT (Y, X ×S T ) . −→ X/F
The canonical isomorphism Y ×S X → Y ×T (X ×S T ) induces an isomorphism cS (τ] Y, X) → cT (Y, X ×S T ). The definition of composition product and base change for finite correspondences shows that this isomorphism is natural in X and Y with respect to finite correspondences (the projections involved in the two ways of computing products in the above isomorphic
180
F. D´eglise
groups coincide). As F = lim LS [X] in the category of sheaves with trans−→ X/F
fers, the result follows from the computation of inductive limits in the category of sheaves with transfers over S (cf. the proof of 4.2.8). In particular, when τ is smooth, τ ∗ is right adjoint to τ] . Thus τ ∗ is exact (and commutes with inductive and projective limits). Moreover, τ ∗ coincides with the usual base change functor on sheaves without transfers. The Projection Formula. Let F (resp. G) be a sheaf with transfers over S (resp. T ). We consider the adjunction morphism deduced from the previous lemma G → τ ∗ τ] G. Applying the functor (τ ∗ F ) ⊗tr S (.) to this morphism we get ∗ tr ∗ (τ ∗ F ) ⊗tr S G → (τ F ) ⊗S (τ τ] G).
Using the monoidal property of τ ∗ and adjunction we get a morphism tr φ : τ] ((τ ∗ F ) ⊗tr S G) → F ⊗S (τ] G).
Lemma 4.2.17. With the above hypotheses and notation, the morphism φ is an isomorphism. Proof The morphism φ is natural in F and G. As all functors involved commute with inductive limits, it is sufficient to check the isomorphism on representable sheaves F = LS [X], G = LT [Y ]. Then the morphism is reduced to the canonical isomorphism (X ×S T ) ×T Y → X ×S Y of S-schemes. Pro-smooth Morphisms. Let (Ti )i∈I be a pro-object of smooth affine S-schemes. As in Prop. 4.1.8, we write Ti = SpecS (Ai ) and put A = lim Ai . The scheme T = SpecS (A) −→op i∈I
is the projective limit of (Ti )i∈I in the category of affine S-schemes. We suppose it is regular noetherian. We denote by τ : T → S the canonical morphism†. First, we note that the functoriality constructed above for sheaves with transfers can also be constructed for presheaves with transfers. cor In particular, from the the functor τ ∗ : Smcor S → SmT we obtain the base change functor tr τˆ∗ : Ptr S → PT
(4.3)
† In general, τ is not necessarily formally smooth but only regular, that is, the fibers of τ are geometrically regular.
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tr and its right adjoint τˆ∗ : Ptr T → PS . In fact, when F is a sheaf with transfers ∗ ∗ over S, we have τ F = atr (ˆ τ F ) using the associated sheaf with transfers of Corollary 4.2.7. For a sheaf with transfers G over T we simply have τ∗ G = τˆ∗ G. Secondly, given a smooth scheme X over T, as it is in particular of finite presentation, there exists i ∈ I such that X/T descends to a finite presentation scheme Xi /Ti . That is, X = Xi ×Ti T. For any j → i, we put Xj = Xi ×Ti Tj . Using now [EGA4, 17.7.8], by enlarging i, we can assume Xi /Ti is smooth. We finally have
X = lim Xj ←− j∈I/i
where every Xj is smooth over S. Proposition 4.2.18. If the hypotheses described above are satisfied, we have a canonical isomorphism τˆ∗ F (X) ' lim F (Xj ) −→op j∈I/i
for any presheaf with transfers F over S. Proof According to the definition, τˆ∗ F (X) = lim cT (X, U ×S T) −→ U/F
where the limit runs over all morphisms LS [U ] → F of sheaves with transfers for a smooth S-scheme U . Note that using the notation introduced just before Prop. 4.1.8, we have a canonical isomorphism cT (X, U ×S T ) ' ¯cS (X, U ) . Proposition 4.1.24 now implies ! ¯cS (X, U ) = ¯cS
lim Xj , U ←−
j∈I/i
' lim cS (Xj , U ) , −→op j∈I/i
the isomorphism being functorial in U with respect to finite S-correspondences by 4.1.26. Finally, we can conclude as we have lim −→
lim cS (Xj , U ) = lim lim cS (Xj , U ) = lim F (Xj ). −→ −→op −→ −→op
U/F j∈I/iop
j∈I/i
U/F
j∈I/i
Suppose now that we are given a T-morphism f : X0 → X. This morphism is of finite presentation; hence there exists i ∈ I such that f descends to Ti . That is, there exist schemes of finite presentation Xi /Ti a nd Xi0 /Ti and a Ti -morphism fi : Xi0 → Xi such that f = fi ×Ti T. We put fj =
182
F. D´eglise
fi ×Ti Tj . Again using [EGA4, 17.7.8], we may assume Xi0 and Xi to be smooth over Ti . Then using § 4.1.27, the isomorphism of the preceding proposition is functorial with respect to (fj )j∈I/i . As a consequence, we obtain the following proposition. Proposition 4.2.19. Suppose that the hypotheses described before Proposition 4.2.18 are satisfied. Then for any sheaf with transfers F over S, τˆ∗ F is a sheaf with transfers. In particular, τ ∗ F (X) ' lim F (Xj ). −→op j∈I/i
Indeed, using the characterisation of a Nisnevich sheaf from 4.2.1, this is a consequence of the following lemma and the exactness of filtered inductive limits. Lemma 4.2.20. Consider a distinguished square of smooth T-schemes W
v
g
∆
U
u
/V f
/ X.
Then there exist i ∈ I and a distinguished square of smooth schemes over Ti Wi gi
vi
/ Vi
fi / Xi . ui
∆i
Ui such that ∆ = ∆i ×Ti T.
Proof We have already seen just before the above proposition that we can find i ∈ I and a square of smooth Ti -schemes Wi gi
Ui
vi
/ Vi
fi / Xi . ui
∆i
For any j → i, we put ∆j = ∆i ×Ti Tj , Zj = (Xj − Uj )red and Tj = (Vj ×Tj Zj )red . Then ∆ is the projective limit of the ∆j . By finding a suitable j → i, we can assume: a) this square is cartesian, that is the morphism Wj → Uj ×Xj Vj is an isomorphism (cf.. [EGA4, 8.10.5(i)]), b) the morphism Tj → Vj induced by fj is an isomorphism (cf. loc. cit.), c) the morphism uj is an open immersion (cf.. [EGA4, 8.10.5(i)]), d) the morphism fj is ´etale (cf.. [EGA4, 17.7.8(ii)]).
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183
In the course of § 4.4, we will need the following strengthening of the preceding proposition, relying on the same lemma: Lemma 4.2.21. Suppose that the hypotheses introduced before Proposition 0 the functor constructed in Lemma ˇ tr 4.2.18 are satisfied. We denote by H 4.2.6, either for presheaves with transfers over S or over T. Then we have tr a canonical isomorphism of functors Ptr S → PT : 0 0 ∗ ˇ tr ˇ tr 'H τˆ . τˆ∗ H
Proof Let F be a presheaf with transfers over S and X be a smooth Sscheme. Fix i ∈ I and a smooth Ti -scheme Xi such that X = Xi ×Ti T. We put Xj = Xi ×Ti Tj . For any (noetherian) scheme A, we let DA be the subcategory of A-schemes W such that there exists a distinguished square U ×X V
/V
/A
U
such that W = U t V as an A-scheme. This category is left filtered, as any Nisnevich covering admits a refinement of this form. Then 0 ˇ tr lim Ker F (W ) → F (W ×Xj W ) . τˆ∗ H F (X) lim −→op −→op j∈I/i
W ∈DX
j
Moreover the preceding lemma says precisely that the inclusion functor G DXj → DX , Wj 7→ Wj ×Xj X j∈I/i
is surjective, hence final. This implies that 0 ∗ ˇ tr H τˆ F (X) lim −→op j∈I/i
lim −→op
j∈I/i
lim −→
Ker τˆ∗ F (Wj ×Xj X) → τˆ∗ F (Wj ×Xj Wj ×Xj X)
lim −→
lim Ker F (Wj ×Xj Xk ) → F (Wj ×Xj Xk ) −→op
Wj ∈DXj
Wj ∈DXj k∈I/j
where the second equality follows from Proposition 4.2.18 and the exactness of filtered inductive limits. The lemma then follows.
4.3 Homotopy Equivalence for Finite Correspondences 4.3.1 Definition Consider a regular scheme S.
184
F. D´eglise
Definition 4.3.1. Let X and Y be smooth S-schemes. Consider two correspondences α, β ∈ cS (X, Y ). A homotopy from α to β is a correspondence H ∈ cS A1 × X, Y such that 1) H ◦ i0 = α 2) H ◦ i1 = β where i0 (resp. i1 ) is the closed immersion X → A1X corresponding to the point 0 (resp. the point 1) of A1X . The existence of a homotopy between two correspondences is obviously a reflexive and symmetric relation. However, transitivity fails. We thus adopt the following definition: Definition 4.3.2. Let X and Y be smooth S-schemes and α, β ∈ cS (X, Y ). We say α is homotopic to β, denoted by α ∼h β, if there exists a sequence of correspondences γ0 , . . . , γn ∈ cS (X, Y ) such that γ0 = α, γn = β and for every integer 0 ≤ i < n, there exists a homotopy from γi to γi+1 . The relation ∼h is obviously additive and compatible with the composition law of finite correspondences. Definition 4.3.3. For two smooth S-schemes X and Y , we denote the quotient of the abelian group cS (X, Y ) by the homotopy relation ∼h by πS (X, Y ) := cS (X, Y ) / ∼h We denote the category with objects smooth S-schemes and with morphisms the equivalence classes of finite S-correspondences with respect to the relation ∼h by πSmcor S .
4.3.2 Compactifications We present a tool (the good compactifications) which allows us to compute the equivalence classes of finite correspondences for the homotopy relation. Definition 4.3.4. Let S be a regular scheme and X be an algebraic Scurve. ¯ 1) A compactification of X/S is a proper normal curve X/S containing X as an open subscheme.
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185
¯ ¯ − X, seen as 2) Let X/S be a compactification of X/S. Put X∞ = X ¯ a reduced closed subscheme of X. We say that the compactification ¯ ¯ X/S of X/S is good if X∞ is contained in an open subscheme of X which is affine over S. ¯ When considering a given compactification X/S of a curve X/S, we will ¯ always put X∞ = X − X. ¯ Remark 4.3.5. If X/S is a good compactification of X/S, X∞ is finite over S as it is proper and affine over S. If S is irreducible, X∞ is surjective over S and Chevalley’s theorem (cf. [EGA2, II.6.7.1]) implies that S is affine. Definition 4.3.6. A pair(X, Z) such that X is a scheme and Z is a closed subscheme of X is called a closed pair. A morphism of closed pairs (f, g) : (Y, T ) → (X, Z) is a commutative diagram T g
Z
/Y f
/X
which is cartesian on the corresponding topological spaces. The morphism is said to be cartesian if it is cartesian as a square of schemes. Let (X, Z) be a closed pair such that X is an S-curve. A good compactification of (X, Z) ¯ which is a good compactification for both X/S over S is an S-scheme X and (X − Z)/S. —The Case of a Base Field. We suppose now S is the spectrum of a field k. Proposition 4.3.7. Let C/k be a quasi-affine regular algebraic curve. There exists a projective regular curve C¯ over k such that for all closed subschemes Z of C nowhere dense in C, C¯ is a good compactification of (X, Z) over k. Proof We can restrict to the case C is affine and integral. As C/k is algebraic, we can find a closed immersion C → Ank . Let C¯ be the reduced closure of C in Pnk . It is an integral projective curve over k. Consider the ¯ Then C˜ is finite over C. ¯ Is is then a proper algebraic normalisation C˜ of C. k-curve. As it is normal, it is then a projective regular curve over k from ¯ As [EGA2, 7.4.5 and 7.4.10]. The curve C is a dense open subscheme of C. ˜ Let Z be a closed it is normal it is again a dense open subscheme of C. ˜ subscheme of C of dimension 0. Then (C − C) t Z is a finite closed subset ˜ As C/k ˜ is projective, it admits an open affine neighbourhood in C. ˜ of C. —The Semi-local Case.
186
F. D´eglise
Theorem 4.3.8 (Walker). Let k be an infinite field. Let (X, Z) be a closed pair such that X is a smooth affine k-scheme and Z is nowhere dense in X. Let {x1 , . . . , xn } be a finite set of points of X. Then there exist 1) a smooth affine k-scheme S, 2) an open affine neighbourhood of x1 , . . . , xn in X, 3) a smooth k-morphism f : U → S of relative dimension 1 such that (U, U ∩ Z) admits a good compactification over S. Proof For the reader’s convenience, we include the following proof which follows the outline of [Wal96, Remark 4.13]. 1) Reduction: We can assume that X is irreducible. Moreover, we can assume all the xi are closed, taking specialisations if necessary. If we can find a good compactification in a neighbourhood Vi of each xi separately, we can define a good compactification in a neighbourhood of all the xi by first reducing the neighbourhoods Vi such that they become disjoint, then taking their disjoint union. We are thus reduced to the case of a single point x1 = x. Finally, as we can enlarge Z, we may assume that it is a divisor in X. 2) Construction of S: Let r be the dimension of Z. As X is an affine algebraic k-scheme, we can find a closed immersion X ,→ Ank . We identify X with its image in Ank under this embedding. Let us denote by: ¯ (resp. Z) ¯ the reduced closure of X (resp. Z) in Pn i) X k ¯ − X the intersection of X ¯ with the hyperplane at infinity, a ii) X˙ = X scheme of dimension less than r. 0
If necessary, we may increase n by considering an embedding Ank ,→ Ank . We find f by considering the orthogonal projection of Ank with center in general position among the linear subvarieties of Ank of codimension r. Parametrisation of the orthogonal projections Ank → Ark . These projections nr are parametrized by the points of Anr k . Indeed, let λ be a point of Ak nr and κ(λ) its residue field. It is in fact an element (λi,j ) 1≤i≤r of κ(λ) . 1≤j≤n
We associate to λ the linear projection pλ : Anκ(λ) → Arκ(λ) defined as the spectrum of the κ(λ)-linear morphism κ(λ)[t1 , . . . , tr ] → κ(λ)[X1 , . . . , Xn ] Pn ti 7→ j=1 Xj − λi,j . We denote by Lλ the center of this projection. It is the intersection of the r hyperplanes corresponding to the zeroes of each projection of Arκ(λ) to A1 composed with pλ . Moreover if L˙ λ denotes the boundary of Lλ κ(λ)
Correspondences and Transfers
187
as a reduced sub-scheme of Pnκ(λ) , we can extend the morphism pλ to a morphism p¯λ : Pn − L˙ λ → Pr . These notations being established, we κ(λ)
κ(λ)
state the following lemma which allows us to construct f : Lemma 4.3.9. Let Ωn be the open subset of Anr k defined by the points λ such that: 1) pλ |Zκ(λ) is finite, 2) X˙ κ(λ) ∩ L˙ λ is a finite set of closed points, 3) pλ is smooth at all the points of Xκ(λ) ∩ p−1 λ (pλ (x)). Then, for n large enough, Ωn is dense in Anr k . Proof It is easy to see that Ωn is open. To prove that it is dense, we proceed in two steps: i) Let us first assume that x is a rational point of X. Then we may further assume x = 0. The first condition defines a dense subset, as Z is closed in Ank of dimension r. The second condition defines a dense subset as the ˙ of dimension less than r, intersection in Pnk of the projective subvariety X, with a linear subvariety of codimension r in general position is finite. For the third condition we only need to assume that the intersection of Lλ and X is transversal in 0. We finally use [SGA4, expos´e XI, th´eor`eme 2.1]: Theorem 4.3.10. The intersection in Ank of X with r hypersurfaces of degree 2 containing 0 in general position is transversal. 2
Via the Veronese embedding of Ank in Ank , a linear subvariety of Ank corre2 sponds to a quadric in Ank and the preceding theorem can be applied to our case, replacing n by n2 . ii) General case. Let k 0 /k be a finite extension such that the fiber of x in X ⊗k k 0 consists of rational points x0i . For each i, the preceding lemma gives r r a dense open subset Ω0n,i of Anr k0 . As Ak0 /Ak is faithfully flat and he three conditions of the lemma satisfy faithfully flat descent, the direct image of ∩i Ω0n,i in Anr k is contained in Ωn . This implies that Ωn is dense. As k is infinite, Ωn admits a rational point λ. We set L˙ = X˙ ∩ L˙ λ , which is ¯ − L˙ → Pr ) be the restriction a finite k-scheme. Let p : X → Ark (resp. p¯ : X k of pλ (resp. p¯λ ). ˜ the closure of the To extend p¯ into a projective morphism we consider X, r ¯ ¯ ˙ ˜ and graph of p¯ in X ×k Pk . Then X − L is a dense open subscheme of X ˜ → Pr extends p¯. As X/k ¯ the canonical projection p˜ : X is projective, p˜ is k
188
F. D´eglise
projective. We have obtained the following diagram: X p
Ark
/X ¯ − L˙
/ ˜ xX x p¯ xx |xxx p˜
/ Pr . k
3) Construction of the compactification: As the square in the above diagram is cartesian and L is finite over k, the ˜ − X above Ar are finite. fibers of p˜ in X k Hence there exists an open affine neighbourhood S of p(x) in Ark such ˜ − X) is finite over S. Reducing S if necessary, we can that p˜−1 (S) ∩ (X −1 assume that p (S) → S is smooth using the third condition imposed on Ωn in the preceding lemma. Finally we put U = p−1 (S) and we denote by f : U → S the restriction of p to U . Then the morphism f is smooth of relative dimension 1. Moreover the restriction Z ∩ U → S of p is finite by ¯ = p˜−1 (S); the the first condition imposed on the points of Ωn . We set U ¯ → S of p˜ is projective. By the choice of S, U ¯ − U is finite restriction f¯ : U over S. To conclude the proof we use the following lemma, which shows ¯ − U ) t Z ∩ U admits that by reducing S near p(x) we may assume that (U an affine neighbourhood: ¯ → S be a projective curve and F be a closed Lemma 4.3.11. Let p¯ : U ¯ such that F/S is finite. Let x be a point of F and s = p¯(x). subscheme of U Then there exist an open affine neighbourhood S 0 of s in S and an effective ¯ such that: divisor D in X ¯S 0 − D. 1) FS 0 ⊂ U ¯S 0 − DS 0 is affine. 2) U ¯ /S is Proof Let Fs be the fiber of F over s. As a set, Fs is finite. As U ¯ projective there exists for i large enough a section f in Γ(U , OU¯ (i)) whose divisor D is disjoint from Fs . Thus there exists an open affine neighbourhood S 0 of s in S such that D is disjoint from FS 0 ; this gives the first condition. As S 0 is affine and DS 0 is the divisor associated to a global section of a very ¯S 0 , the scheme U ¯S 0 − DS 0 is affine. ample fiber bundle over U
4.3.3 The Relative Picard Group Definition 4.3.12. Let (X, Z) be a closed pair. We employ the notation
Correspondences and Transfers
189
Pic (X, Z) for the group of couples (L, s), where L is an invertible sheaf on X and ∼ → L|Z is a trivialisation of L over Z, modulo isomorphisms of s : OZ − invertible sheaves compatible with the trivialisation. The group structure is induced by the tensor product of OX -modules. There is a canonical morphism Pic (X, Z) → Pic(X). Definition 4.3.13. let X be a scheme and Z a closed subscheme of X. 1) If Z 0 is closed subscheme of Z, we define the restriction morphism from Z to Z 0 r
0
Z Pic (X, Z) −− → Pic (X, Z 0 ) (L, s) 7→ (L, s|Z 0 ).
2) Let (f, g) : (Y, W ) → (X, Z) be a morphism of closed pairs (cf. def. 4.3.6). We define the pullback morphism (f, g)∗ : Pic (X, Z) → Pic (Y, W ) (L, s) 7→ (f ∗ (L), g ∗ (s)) . 4.3.14. Let S be a regular affine scheme. We consider a smooth quasi-affine ¯ curve X/S and suppose that it admits a good compactification X/S. Let α be a relative cycle on X/S. Then, considered as a codimension 1 cycle ¯ it corresponds to an invertible sheaf L(α) on X ¯ whose isomorphism of X, class is unique. Moreover, if Z is the support of α, this sheaf has a canonical ¯ − Z. Let s(α) be its restriction to X∞ . We have thus trivialisation on X defined a canonical morphism λX/S ¯ X∞ , α 7→ (L(α), s(α)). c0 (X/S) −−−→ Pic X, Lemma 4.3.15. i) Let S 0 be a regular scheme and τ : S 0 → S be a flat morphism. ¯0 = X ¯ ×S S 0 , and let ∆ : X 0 /S 0 → X/S be the Put X 0 = X ×S S 0 , X morphism induced by τ . 0 ) → (X, ¯ 0 , X∞ ¯ X∞ ), the cartesian morphism We consider (f¯, f∞ ) : (X of closed pairs induced by τ . Then the following diagram is commutative λX 0 /S 0 ¯ X∞ / Pic X, c0 (X/S) ∆∗
c0
(X 0 /S 0 )
(f¯,f∞) 0 ¯ 0 , X∞ / Pic X . ∗
λX/S
190
F. D´eglise
¯ is a good comii) Let Z be a closed subscheme of X and suppose that X pactification of (X, Z) over S. Then the following diagram commutes c0 (X − Z/S) j∗
c0 (X/S)
λX−Z/S
¯ X∞ t Z / Pic X,
λX/S
r(Y ×S X∞ )
¯ X∞ . / Pic X,
Proof The second point is obvious by construction. For the first point, let f : X 0 → X be the flat morphism induced by τ . Let ¯ α be a finite relative cycle on X/S. Suppose that α is the class of a closed subscheme Z in X. Then by Proposition 4.1.7, ∆∗ α is the cycle associated to the closed subscheme f −1 (Z) of X 0 . Thus the conclusion follows from the construction of λ. Proposition 4.3.16. Consider the notation of the previous lemma and let Y be a smooth affine S-scheme. Then the morphism λY ×S X/Y factors through the homotopy relation. The induced morphism ¯ Y ×S X∞ πS (Y, X) → Pic Y ×S X, is an isomorphism. Proof Let i0 : Y → A1Y (resp. i1 : Y → A1Y ) be the zero section (resp. unit section) of A1Y /Y . Note that i0 and i1 are inclusions of a cycle associated to a principal Cartier divisor. Then the pullback maps c0 A1Y ×S X/A1Y → c0 (Y ×S X/Y ) induced by i0 and i1 coincide with the operation of intersecting with divisors defined in [Ful98, 2.3] (see also remark 2.3 of loc. cit.). This allows us to extend the first case of the previous lemma to the case where τ is i0 or i1 . Finally, using the homotopy invariance of the Picard group for regular schemes, λ indeed factors through the homotopy relation of S-correspondences. To show that the induced morphism is an isomorphism, we construct its inverse. It suffices to treat the case Y = S. Let (L, s) be an element ¯ X∞ . Consider an open affine neighbourhood V of X∞ in X. ¯ of Pic X, The trivialisation s of L then extends to a trivialisation s˜ of L over V . ¯ − V, s˜) is associated a unique Cartier divisor To the pseudo-divisor (L, X ¯ ¯ D(L, X − V, s˜) on X following [Ful98, Lemma 2.2]. Let α be the associated ¯ − V ). Moreover, as X/S is quasi-affine cycle. The support of α lies in (X ¯ ¯ and X/S is proper, V is dense in all the fibers of the curve X/S which ¯ implies X − V is finite over S. Finally, the support of α is finite over S and α is in fact a finite relative cycle on X/S.
Correspondences and Transfers
191
We prove now that the homotopy class of α in cS (S, X) does not depend on the choice of s˜. Suppose given two extensions s˜0 and s˜1 of s to V . Let α0 and α1 be the respective cycles obtained in the process described above. ¯ For Define L0 as the pullback of L along the morphism π : A1X¯ → X. i = 0, 1, we obtain a trivialisation π ∗ s˜i of L0 over A1V . Let H be the cycle associated to the pseudo-divisor D(L0 , A1X−V , tπ ∗ s˜0 + (1 − t)π ∗ s˜1 ), where t ¯ is the canonical parameter of A1X¯ . Then, using the beginning of the proof, we obtain H ◦ i0 = α0 and H ◦ i1 = α1 . Remark 4.3.17. The previous proposition is a particular case of the computation of the Suslin singular homology of the curve X/S in [SV96, Th. 3.1].
4.3.4 Constructing Useful Correspondences up to Homotopy —Factorisations. Proposition 4.3.18. Let S be an affine regular scheme, and (X, Z) a closed pair such that X is a smooth affine S-curve. Put U = X − Z and denote by i : U → X the canonical open immersion. Suppose that (X, Z) admits a good ¯ over S. Let L(1X ) be the invertible sheaf corresponding compactification X to 1X ∈ cS (X, X) in the notation of 4.3.14. The following conditions are equivalent: 1) For any smooth affine S-scheme Y , the morphism i◦
πS (Y, U ) −→ πS (Y, X) is surjective. 2) The morphism i◦
πS (X, U ) −→ πS (X, X) is surjective. 3) The invertible sheaf L(1X )|X×S Z is trivial. Proof Conditions (1) and (2) are equivalent to the existence of a section of i up to homotopy. Thus the proposition is implied by the following more precise lemma: Lemma 4.3.19. Consider the hypothesis of the preceding proposition. Let Y be a smooth affine S-scheme and β : Y → X a finite S-correspondence. The following conditions are equivalent:
192
F. D´eglise
1) There exists a finite S-correspondence α that makes the following diagram of S-correspondences commutative up to homotopy α k5 X − Z TTiT TTT) k k / X. β Y
2) The invertible sheaf L(β)|Y ×S Z is trivial; here we use the notation of 4.3.14. Moreover, the finite S-correspondences which satisfy condition (1) are in one-to-one correspondence with the trivialisations of L(β)|Y ×S Z . We use Proposition 4.3.16 applied first to the affine curve X/S and secondly to the quasi-affine curve U/S. (2) ⇒ (1): Consider a trivialisation s of L(β)|Y ×S Z . Then the class of the ¯ pair (L(β), s(β) ⊕ s) in Pic Y ×S X, Y ×S X∞ t Y ×S Z defines a finite S-correspondence α which, according to Lemma 4.3.15, satisfies i ◦ α = β as required. (1) ⇒ (2): Conversely, the finite S-correspondence α corresponds to an ¯ element of Pic Y ×S X, Y ×S X∞ t Y ×S Z which is the class of the pair (L(α), s(α)). Thus, as i ◦ α = β, there exists an isomorphism φ : L(β) → L(α) that makes the following diagram commutative L(β)|Y ×S X∞
φ|Y ×S X∞
WWWWW W+ s(β)
OY ×S X∞ .
/ L(α)|Y × X S ∞ g g g g sggs(α)| Y ×S X∞
Then s(α)|Y ×S Z ◦ φ−1 |Y ×S Z is indeed a trivialisation of L(β)|Y ×S Z . The last point of the lemma is clear from the proof. Example 4.3.20. As an easy application of this proposition, we consider two open subschemes X and U of the affine line A1k over a field k such that U ⊂ X. Put Z = (X − U )red . Then the open immersion i : U → X admits a 1 section in πSmcor k as Pk is a good compactification of (X, Z), X is affine and Pic (X ×k Z) = 0. Moreover, choosing a trivialisation of L(1A1 ) once and k for all, we define trivialisations for all open immersions i : U → X which are functorial with respect to open immersions in X and U . —Local Section of Open Immersions in πSmcor k . The following proposition is directly inspired by [Voe00a, Proposition 4.17]: Proposition 4.3.21. Let k be a field, X a smooth k-scheme, U a dense open subscheme of X, and x a point of X. Then there exist 1) an open neighbourhood V of x in X,
Correspondences and Transfers
193
2) a finite k-correspondence α : V → U , such that the following diagram is commutative up to homotopy wV j w {w / U i X, w
α
where i and j are the obvious open immersions. Proof Suppose first that k is infinite. Put Z = (X − U )red . Using Theorem 4.3.8, there exist an affine smooth k-scheme S, an affine open neighbourhood V of x in X, a smooth morphism f : V → S of relative dimension 1, and an ¯ such that X/S ¯ S-scheme X is a good compactification of (V, V ∩ Z). From the commutative diagram V ∩U
/V
/ ,
U
we see that the theorem holds for V , if it holds for X. Thus we can assume ¯ over X = V , which implies that (X, Z) has a good compactification X ¯ S. Let L(1X ) be an invertible sheaf over X ×S X which corresponds to 1X ∈ πS (X, X) according to Proposition 4.3.16. As Z is affine and closed ¯ in the proper curve X/S, it is finite over S. The scheme Spec (OX,x ) ×S Z is finite over the local scheme Spec (OX,x ), hence it is semi-local. This implies that Pic(Spec (OX,x ) ×S Z) = 0. In particular, L(1X ) is trivial over Spec (OX,x )×S Z. Thus there exists an open neighbourhood V of x in X such that L(1X ) is trivial over V ×S Z. From Lemma 4.3.19, applied to Y = V and j
to the finite S-correspondence V − → X, there exist a finite S-correspondence α : V → U which makes the following diagram commutative: V
α
U
i
j / X.
Let τ : S → k be the canonical morphism. As τ is smooth, the restriction functor τ˜] of definition 4.1.32 is well defined. Applying this functor to the preceding diagram, we see that the k-finite correspondence τ˜] (α) is appropriate. When k is finite, we consider L = k(t). We put XL = X ×k Spec (L) and similarly for any k-scheme. The point x corresponds canonically to a point of XL still denoted by x. Applying the preceding case to the open immersion iL : UL → XL and to the point x, we find a neighbourhood Ω of
194
F. D´eglise
x in XL and a finite L-correspondence α : Ω → UL such that iL ◦ α is the open immersion Ω → XL . As x comes from a point of X, we can always find an open neighbourhood V of x in X such that VL ⊂ Ω. The following diagram q VL qqq jL q q xq UL iL / XL . α|VL
is commutative in πSmcor L , with j : V → X the canonical immersion. Applying Proposition 4.1.8, we obtain a canonical isomorphism cL (VL , YL ) = lim ck (V ×k W, Y ) , −→ 1 W ⊂Ak
for any k-scheme Y , where the limit runs over the non empty open subschemes W of A1k . It is functorial in Y . In particular, we can lift both the finite L-correspondence α|VL and the homotopy making the above diagram commutative for a sufficiently small W in A1k . We thus obtain a finite k-correspondence α0 : V ×k W → U such that the diagram over k V ×k W
U
α0 mm mm mv mm i
j×k p /X
is commutative up to homotopy, with p : W → Spec (k) the canonical 1× p
j
k projection. Finally, we factor out j ×k p as V ×k W −−− →V − → X. Example 1 4.3.20 gives a section of the open immersion W → Ak , which shows 1 ×k p admits a section in πSmcor k and concludes the proof.
Corollary 4.3.22. Let k be a field, X a smooth k-scheme and U a dense open subscheme of X. Then there exist 1) an open covering p : W → X of X, 2) a finite k-correspondence α : W → U such that the following diagram is commutative up to homotopy W αp p p j wp / i X, U where i and j are the canonical open immersions. Proof We simply apply the preceding lemma to every point of X and use its quasi-compactness.
Correspondences and Transfers
195
4.3.23 (Homotopy Excision). The following proposition is one of the central points in our interpretation of Voevodsky’s theory. It is a generalisation of [Voe00a, Lemma 4.6]. Theorem 4.3.24. Let S be an affine regular scheme. Consider a distinguished square (cf. § 4.2.1) of smooth affine S-schemes W h
U
l
/V
j
f / C.
We put Z = C − U and T = V − W with their reduced structure and assume ¯ that there exist good compatifications C/S of (C, Z) and V¯ /S of (V, T ) which fit into the commutative square
/ V¯
f /C ¯
V f
C
¯
and satisfy V∞ ⊂ f¯−1 (C∞ ). Assume finally Pic (C ×S Z) = 0. Then the complex (j,f )
h−l
0 → [W ] −−→ [U ] ⊕ [V ] −−−→ [C] → 0 is contractible in the additive category πSmcor S . Proof In the following lemma, we will construct the chain homotopy between the complex above and the zero complex. Indeed, with the notations of this lemma, the chain homotopy is given by the two morphisms [U ] ⊕ [V ]
[C] oo o o o ow oo α
(γ,−β)oooo
[W ]
o wooo
[U ] ⊕ [V ].
The necessary relations are stated and proved in the lemma. Lemma 4.3.25. With the hypotheses and notation of the preceding theorem, there exists finite correspondences WO o
β
Uo
α
V
γ
C
196
F. D´eglise
that satisfy the following relations in πSmcor k : j ◦ α = 1C l ◦ β = 1V α◦f = h◦β l◦γ = 0 h ◦ γ = 1U − α ◦ j γ ◦h = 1 − β ◦l W
(1) (2) (3) (4) (5) (6)
We first apply Proposition 4.3.16 to the morphism 1C , as an element of ¯ CC∞ πS (C, C). It corresponds to the class of a pair(L(1C ), s(1C )) in Pic C C, †. By hypothesis Pic (C ×S Z) = 0, hence the invertible sheaf L(1C ) is trivial on C ×S Z. Let t be a trivialisation. We define πS (C, C) 3 α
←→
¯ (L(1C ), s(1C ) + t) ∈ picrC CCC ∞ t CZ.
Relation (1) simply follows from Lemma 4.3.19 as in the preceding applications. Using again Proposition 4.3.16, the morphism 1V , as an element of πS (V, V ), corresponds to the class of an element (L(1V ), s(1V )) in Pic V V¯ , V V∞ . By construction, the sheaf L(1C ) (resp. L(1V )) corresponds to the diagonal ∆C (resp. ∆V ) of C/k (resp. V /k) seen as a closed subscheme of C ×S C¯ (resp. V ×S V¯ ). Since the morphism g = f ×X Z : T → Z is an isomorphism, we obtain (f ×S g)−1 (∆X ∩ (X ×S Z)) = ∆V ∩ (V ×S T ) which finally gives (f ×S g)∗ (L(1C )|X×S Z ) = L(1V )|V ×S T . In particular, the section τ = (f ×S g)∗ (t) is a trivialisation of L(1V ) on V ×S T . Let us define β through πS (V, V ) 3 β ←→ (L(1V ), s(1V ) + τ ) ∈ Pic V V¯ , V V∞ t V T . Relation (2) is again a consequence of Lemma 4.3.19. It remains to construct γ. We consider the invertible sheaf M = (1C ×S f¯)∗ L(1C ) on C ×S V¯ . It corresponds to the divisor D = (1C ×S f¯)−1 (∆C ). Let u be the canonical trivialisation of M on C ×S V − D. As g is an isomorphism, v = (1C ×S g)∗ t is a trivialisation of M|C×S Z . Note that 1 + uv −1 is a regular invertible section of OC V¯ over CV∞ t CT . We define γ through πS (C, C) 3 γ ←→ (OC V¯ , 1 + uv −1 ) ∈ Pic C V¯ , CV∞ t CT . † In this proof, we sometimes omit the symbol ×S when it simplifies the notation
Correspondences and Transfers
197
By construction and Lemma 4.3.15, l ◦ γ corresponds to the pair (OC V¯ , 1), which is the zero correspondence. This is relation (4). ¯ Put Ω0 = Consider an open affine neighbourhood Ω of C∞ t Z in C. −1 f¯ (Ω) and let ν : Ω0 → Ω be the finite morphism induced by f¯. Then Ω0 is an open affine neighbourhood of V∞ t T . Thus the invertible regular function 1 + uv −1 admits an extension w to U ×S Ω0 . Following the computation of [Ful98, 1.4], we see that the correspondence h ◦ γ corresponds via the iso ¯ U C∞ t U T : morphism of prop. 4.3.16 to the following element in Pic U C, OU C¯ , N (w)|U C∞ tU T , where N is the norm associated to the extension ring corresponding to U Ω0 /U Ω. As w|U V∞ = 1, we easily obtain that N (w)|U C∞ = 1. A more detailed computation shows moreover N (w)|U T = s(1U ).t−1 , as g is an isomorphism and f is ´etale. The finite correspondence 1U − α ◦ j corresponds to the pair L(1U ) ⊗ (L(1C )|U C¯ )−1 , s(1U ).(s(1C ) + t)−1 . Thus relation (5) is now clear. Finally, using again Lemma 4.3.15, γ ◦ h corresponds to the pair OW ×S V¯ , 1 + s(1V ).τ −1 . Indeed, by definition, the pullback of v over W ×S V¯ is τ . Relation (6) now follows, since the finite correspondence 1W − β ◦ l corresponds to the pair L(1W ) ⊗ (L(1V )|W V¯ )−1 , s(1W ).(s(1V ) + τ )−1 . Only the relation (3) remains to be shown. We consider the trivialisation s(1V ) (resp. τ )) of the invertible sheaf L(1V ) over V V∞ (resp. V T ). As Ω0 is an affine neighbourhood of V∞ t T , the trivialisation s(1V ) (resp. τ ) admits an extension w1 (resp. w2 ) to V ×S Ω0 . Using a computation we have already seen, establishing relation (3) is equivalent to showing that the ¯ V C∞ t V Z are equal: following two elements of Pic V C, (f ×S 1C¯ )∗ L(1C ), (f ×S 1C¯ )∗ (s(1C ) + t) (1V ×S f¯)∗ L(1V ), N 0 (w1 + w2 )|V C∞ tV Z . We have denoted by N 0 the norm associated to the finite extension V Ω0 /V Ω. Using again that g is an isomorphism and f is ´etale, we obtain N 0 (w2 )|V Z =
198
F. D´eglise
(f ×S 1C¯ )∗ (t). But the equality 1C ◦ f = f ◦ 1V implies that the following pairs coincide (f ×S 1C¯ )∗ L(1C ), (f ×S 1C¯ )∗ (s(1C )) (1V ×S f¯)∗ L(1V ), N 0 (w1 )|V C∞ , and this concludes the proof. To finish, we give a simple example where we can construct compactifications that appear in the above theorem. Suppose we are only given only the distinguished square in the hypothesis of the preceding proposition, and assume that S is the spectrum of a field k. Then, according to Proposition ¯ which is a good compact4.3.7, there exists a smooth projective curve C/k f ification of (C, Z). The morphism V − → C → C¯ is quasi-affine. Applying Zariski’s main theorem (cf. [EGA3, chap. III, 4.4.3]), it can be factored as ˜ j f˜ V − → V˜ − → C¯ where ˜j is an open immersion and f˜ a finite morphism. As V˜ /k is algebraic, its normalisation V¯ is finite over V˜ , and still contains V as an open subscheme since V is normal. Thus V¯ is a good compactification of (V, Z) and we have the following commutative diagram V f
C
/ V¯ ¯
f / C. ¯
4.4 Homotopy Sheaves with Transfers 4.4.1 Homotopy Invariance Definition 4.4.1. Let S be a scheme. A presheaf F on SmS is said to be homotopy invariant if for all smooth S-schemes X, the morphism induced by the canonical projection F (X) → F (A1X ) is an isomorphism. When S is regular, we denote the category of sheaves (resp. presheaves) with transfers over S which are homotopy invariant by HNStr (resp. HPtr S ). Such sheaves (resp. presheaves) will simply be called homotopy sheaves (resp. homotopy presheaves). The following lemma relates homotopy presheaves to correspondences up to homotopy. Lemma 4.4.2. Let F be a presheaf with transfers over a regular scheme S. The following conditions are equivalent: 1) F is homotopy invariant.
Correspondences and Transfers
199
2) For all smooth S-schemes X, considering s0 : X → A1X (resp. s1 : X → A1X ) the zero (resp. unit) section of A1X , s∗0 = s∗1 . cor 3) F can be factored through the canonical morphism Smcor S → πSmS . Proof For a smooth S-scheme X, we denote by pX : A1X → X (resp. µX : A1X ×X A1X → A1X ) the canonical projection (resp. multiplication) of the ringed X-scheme A1X . The lemma now follows easily from the relations pX ◦ s0 = pX ◦ s1 = 1X and the fact that µX defines a homotopy from s0 to s1 . 4.4.3. In particular, a homotopy presheaf (resp. homotopy sheaf) is nothing but a presheaf on πSmcor S (resp. a presheaf whose restriction to SmS is a tr Nisnevich sheaf). As a corollary, the forgetful functor HPtr S → PS admits a ˆ left adjoint h0 (.) constructed as follows. Let F be a presheaf with transfers, s∗0 −s∗1 ˆ 0 (F ) (X) as the cokernel of the morphism F (A1 ) − and define h → F (X). X −−− ˆ 0 (F ) is homotopy invariant and has the The preceding lemma implies that h adjunction property. Consider now a sheaf F with transfers. We denote by (1) ˆ 0 (F ) (cf. Corollary h0 F the sheaf with transfers associated to the presheaf h 4.2.7). In general, this sheaf is not homotopy invariant - unless S is the spectrum of a perfect field (see 4.4.14). For a natural integer n, we denote by (n) (1) h0 the n-th composition power of h0 . We deduce a sequence of morphisms (1)
(n)
F → h0 F → · · · → h0 F → . . . and define (n)
h0 (F ) = lim h0 F, −→ n∈N
where the limit is taken in the category of sheaves with transfers. Proposition 4.4.4. Let S be a regular scheme and F a sheaf with transfers over S. Then the sheaf with transfers h0 (F ) defined above is homotopy invariant. Moreover, the functor h0 : NStr → HNStr is left adjoint to the obvious forgetful functor. Proof Let X be a smooth S-scheme, s0 and s1 the zero and unit sections of A1X /X. According to the preceding lemma, we have to show that s∗0 = s∗1 on h0 (F )(A1X ). Let x be an element of (n)
h0 (F )(A1X ) = lim h0 F (A1X ). −→ n∈N
(n)
By definition, it is represented by a section xn in h0 F (A1X ) for an integer
200
F. D´eglise
n ∈ N. The transition morphism of level n in the above inductive limit can be factored out as (n) a /ˆ b / (n+1) (n) h0 h0 F h0 F h0 F. ˆ 0 h(n) F is homotopy invariant. From what we saw before, the sheaf h 0 Thus s∗0 (axn ) = s∗1 (axn ). As a is a natural transformation, we deduce that as∗0 (xn ) = as∗1 (xn ), thus bas∗0 (xn ) = bas∗1 (xn ) and s∗0 (x) = s∗1 (x). 4.4.2 Fibers Along Function Fields Let us fix a field k. To simplify notation, we write Smk (resp. πSmcor k ) in stead of SmSpec(k) (resp. πSmcor ). Spec(k) —Open Immersions. The following proposition is analogous to [Voe00a, Cor. 4.19]; its proof uses the same arguments. Proposition 4.4.5. Let F be a presheaf over πSmcor k and let G be one of the following presheaves over Smk : 1) the Zariski sheaf FZar over Smk associated to F , ˇ ˇ 0 F over Smk associated with F 2) the 0-th Cech cohomology presheaf H for the Nisnevich topology. Then for any smooth k-scheme X, any dense open subscheme U of X, the restriction morphism G(X) → G(U ) is a monomorphism. Proof Consider a ∈ G(X) such that a|U = 0. We shall show that a = 0. We may assume that there exists an element b ∈ F (X) such that a is the image of b by the canonical morphism F (X) → G(X). Indeed, there exists a Nisnevich covering (in the first case, even a Zariski covering) of X such that a|W can be lifted along the morphism F (W ) → G(W ). The open scheme W ×X U of W is still dense and we have a|W ×X U = 0 in G(W ×X U ), hence we can replace X by W and make the above assumption. Moreover, in both cases there exists by hypothesis a Nisnevich covering p W − → U such that b|W = 0. As W is a Nisnevich covering of U , there exist a dense open subscheme U0 of U and an open subscheme W0 of W such that p induces an isomorphism between W0 and U0 . Thus, b|U0 = 0. Applying Corollary 4.3.22, we find a Zariski cover W 0 of X and a finite kcorrespondence α : W 0 → U0 such that the diagram W 0 commutes α qq xqqq /X U0
Correspondences and Transfers
201
up to homotopy. Applying F to this diagram, we thus obtain that b|W 0 = 0 in F (W 0 ) which implies a = 0. Corollary 4.4.6. Let F be a homotopy sheaf over k. Consider a smooth k-scheme X and a dense open subscheme U of X. Then the restriction morphism F (X) → F (U ) is a monomorphism. —Generic Points. Let X be a smooth S-scheme, and x be a generic point of X. The local ring OX,x of X in x is a field. Thus it is henselian. If we let Vx (X) be the category of open neighbourhoods of X, and define the localisation of ˜ X in x as the pro-object Xx = lim U . Its limit is Spec (OX,x ). As ←− U ∈Vx (X)
OhX,x ,
OX,x = we have a canonical isomorphism F (Xx ) = F (Xxh ). Note that OX,x is a separable field extension of k of finite type. We call such an extension a function field. We let Ek be the category of function fields with arrows the k-algebra morphisms. When E/k is a function field, we put Msm (E/k) = A ⊂ E | Spec (A) ∈ Smk , Frac(A) = E as an ordered set, the order coming from inclusion. This set is in fact non empty and right filtering. We define the pro-scheme (E) = A∈M
˜ lim ← − sm
Spec (A) .
(E/k)op
Thus, according to our general conventions, for any presheaf F over Smk , F (E) =
lim −→
F (Spec (A)).
A∈Msm (E/k)
Moreover, for any A ∈ Msm (E/k), if x denotes the generic point of X = Spec (A), we have canonical isomorphism F (E) = F (Xxh ) = F (Xx ) as Spec (E) is the limit of all the pro-schemes (E), Xxh and Xx . In particular, the morphism F 7→ F (E) from Nisnevich sheaves to abelian groups is a fiber functor. The following proposition due to Voevodsky shows the fiber functors defined above form a conservative family of ”fiber functors” for homotopy sheaves. Proposition 4.4.7. Let F , G be homotopy sheaves over k, and η : F → G be a morphism of sheaves with transfers. If for any field E/k in Ek the induced morphism ηE : F (E) → G(E) is a monomorphism (resp. an isomorphism), then η is a monomorphism (resp. an isomorphism). Proof Indeed, it is sufficient to apply the next lemma to the morphism η.
202
F. D´eglise
Lemma 4.4.8. Let F , G be presheaves over πSmcor k and η : F → G be a natural transformation. The following conditions are equivalent: 1) The morphism ηZar : FZar → GZar between the associated Zariski sheaves over Smk is a monomorphism (resp. isomorphism). 2) For all extension E/k in Ek , ηE : F (E) → G(E) is a monomorphism (resp. isomorphism). Proof Clearly, (1) implies (2). For the converse we consider N , the kernel of η in the category of presheaves with transfers. It is homotopy invariant. Let X be a smooth irreducible kscheme with residue field E. Clearly, we have a canonical isomorphism N (E) = lim NZar (U ) −→ U ⊂X
where the limit runs over the open dense subschemes of X. Then Proposition 4.4.5 implies that the canonical morphism NZar (X) → lim NZar (U ) = N (E) −→ U ⊂X
is a monomorphism. But N (E) is the kernel of ηE : F (E) → G(E), hence N (E) = 0 and N (X) = 0. We now conclude the proof by applying the same reasoning to the cokernel of η.
4.4.3 Associated Homotopy Sheaf ˇ —Example: Cech Cohomology of Curves. Let k be a field and C/k be an algebraic curve. We introduce the following property for the curve C: (N) For all finite extensions L/k, Pic (C ⊗k L) = 0. Remark 4.4.9. If this property is true for C, it is true for any open subscheme of C. If C is affine with function ring A, property (N) is equivalent to the property that for any finite extension L/k, the ring A ⊗k L is factorial; cf. [EGA4, 21.7.6 and 21.7.7.]. Note this property implies that for any closed subscheme Z of C nowhere dense, Pic (C ×k Z) = 0. We deduce from that fact the following proposition which is in fact a generalisation of [Voe00a, 5.4]: Proposition 4.4.10. Let k be a field. Consider C/k, a smooth affine curve satisfying property (N), and F a presheaf over πSmcor k . Then for all integers
Correspondences and Transfers
203
ˇ n ≥ 0, the n-th Cech cohomology group of C with coefficients in F for the Nisnevich topology is F (C) if n = 0 n ˇ H (C; F ) 0 otherwise Proof First we remark that for any Nisnevich covering W → C there exists a distinguished square l
U ×X V h
U
j
/V f /X
with U and V affine such that the covering U t V → C is a refinement of W → C. Indeed, we may assume that W is affine. As W/C is a Nisnevich covering, there exists a dense open subset U of X such that W ×C U → U admits a section. As this morphism is ´etale, we have W ×C U = U t U 0 . Put Z = (C − U )red . Then W ×C Z is a finite set of closed points of W . As W ×C Z → Z is a Nisnevich covering, any point of Z has a preimage in W ×C Z which is isomorphic to it ; that it W ×C Z = Z t Z 0 . If we now put V = W − Z 0 , then V is affine as W is regular and Z 0 is a finite set of points; we have obtained our distinguished square. Consider now a distinguished square as above. As C/k satisfies property (N), Theorem 4.3.24 implies that the complex j ∗ +f ∗
h∗ −l∗
0 → F (C) −−−−→ F (U ) ⊕ F (V ) −−−−→ F (U ×X V ) → 0 ˇ is contractible. This implies that the Cech cohomology group associated with the covering U t V /C is F (C) in degree 0 and 0 in other degrees; this concludes the proof of the proposition by the remark at the beginning of the proof. Corollary 4.4.11. Let C/k be a smooth curve satisfying property (N) and F be a presheaf over πSmcor k . Then for all integers n > 0, we have Hn (C; FNis ) = 0. Proof Indeed, the Nisnevich cohomology of C/k vanishes in dimension ˇ strictly greater than 1, and the Cech cohomology coincides with the usual cohomology in degree 1. Remark 4.4.12. It is not only sufficient, but also necessary that C/k satisfy property (N). Let us assume C/k satisfies H1 (C; FNis ) = 0 for every presheaf F over πSmcor k . Let Gm be the sheaf over Smk represented by Gm . It has
204
F. D´eglise
a canonical structure of a sheaf with transfers: let X and Y be smooth kschemes, and α be a finite k-correspondence from X to Y . We assume X is integral and α is an integral closed subscheme of X ×k Y . Let κ(X) and κ(Z) be the respective function fields of X and Z. Then κ(Z)/κ(X) is a finite extension, as Z → X is finite surjective. Let Nκ(Z)/κ(X) be the associated norm morphism. Then we construct α∗ by the commutative diagram: α∗
/ OZ (Z)× _ _ _ _ _/, OX (X)× _ _ Nk(Z)/k(X) / k(X)× . k(Z)×
OY (Y )×
The dotted arrow exists as OX (X) is integral. These transfers are compatible with the composition of finite correspondences using the property of the norm homomorphism.† Let L/k be a finite extension, j : Spec (L) → Spec (k) the canonical morphism. Then the sheaf with transfers j∗ j ∗ Gm is still homotopy invariant and we have H 1 (C; j∗ j ∗ Gm ) = Pic (C ⊗k L). ˇ —Example: the 0-th Cech Cohomology Presheaf. Recall from Lemma 4.2.6 that for any presheaf F with transfers over k, ˇ the 0-th Cech cohomology presheaf associated to F in the Nisnevich topology has a canonical structure of a presheaf with transfers. We denote this 0 F . Recall that for any smooth k-scheme X, ˇ tr presheaf with transfers by H 0 F) = H ˇ tr ˇ 0 (X; F ). The following proposition is the very point where Γ(X; H our proof of the technical results concerning homotopy sheaves differs from that of [Voe00a] (especially 4.26 and 5.5). Proposition 4.4.13. Let k be field and F be a presheaf over πSmcor k . Then ˇ 0 F is homotopy invariant. the presheaf H tr Proof Let X/k be a smooth scheme. If s : X → A1X is the 0-section, we have 0 F (A1 ) → H 0 F (X) is a monomorphism. We ˇ tr ˇ tr to prove in fact that s∗ : H X may assume X is irreducible. Applying Proposition 4.4.5, we find that for ˇ 0 F (X) → H ˇ 0 F (U ) any nonempty open subscheme U of X the morphism H tr tr is a monomorphism. Thus in the commutative diagram below ˇ 0 F (A1 ) H tr X s∗
0 F (X) ˇ tr H
ˇ 0 F (A1 ) H / lim U −→ tr U ⊂X
/ lim −→
σ
ˇ 0 F (U ) H tr
U ⊂X
† It is also a consequence of [D´ eg05, 6.5 and 6.6] applied to Gm = A0 (.; K∗M )1 .
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205
where U runs over the nonempty open subschemes of X, the horizontal arrows are injective and we only have to prove that σ is injective. Denote by E the function field of X, and let τ : Spec (E) → Spec (k) be the canonical tr morphism. We let τˆ∗ : Ptr k → PE be the base change functor for presheaves with transfers (cf. (4.3)). Let sE : Spec (E) → A1E be the 0-section. Then from Proposition 4.2.18 and the remark that follows about functoriality, we deduce that the morphism ˇ 0 F (A1U ) → lim H ˇ 0 F (U ) σ : lim H −→ tr −→ tr U ⊂X
U ⊂X
is isomorphic to ˇ 0 F (A1 ) → τˆ∗ H ˇ 0 F (Spec (E)). s∗E : τˆ∗ H tr E tr ˇ0 = H ˇ 0 τˆ∗ . To conclude Let us recall that by Lemma 4.2.21 we have τˆ∗ H tr tr that σ is an isomorphism, we apply Proposition 4.4.10 to the curve A1E /E and to the homotopy presheaf τˆ∗ F over E. Corollary 4.4.14. Let k be a field. For any presheaf over πSmcor k , the tr sheaf FNis is homotopy invariant. In particular, the functor atr : Pk → Nktr tr of Corollary 4.2.7 induces an exact functor aHtr : HPtr k → HNk which is left adjoint to the inclusion functor HNktr ,→ HPtr k. Corollary 4.4.15. The category HNktr is a Grothendieck abelian category which admits arbitrary limits. The inclusion functor HNktr ,→ Nktr is exact. In the case where the base is a field k, use the same notation as introduced (1) just before Proposition 4.4.4. The corollary above implies that h0 = h0 . The generators of HNktr are the elements of the (essentially small) family h0 L[X] X∈Sm . Note finally that we also get the analogue of the results of k [Voe00a] for presheaves with transfers in the Zariski topology: Corollary 4.4.16. Let F be a homotopy presheaf over a field k. Then the canonical morphism FZar → FNis is an isomorphism. Proof By the preceding results, we know FNis is a homotopy invariant sheaf over k. Thus the result follows from Lemma 4.4.8, applied to the morphism of homotopy presheaves with transfers F → FNis .
4.5 Homotopy Invariance of Cohomology We aim to prove the following theorem:
206
F. D´eglise
Theorem 4.5.1 (Voevodsky). Let k be a perfect field and F be a homotopy sheaf over k. Then the Nisnevich cohomology presheaf H ∗ (.; F ) is homotopy invariant over Smk .
4.5.1 Lower Grading Definition 4.5.2. Let S be a regular scheme and F be a homotopy presheaf with transfers over S. We associate to F the homotopy presheaf with transfers F−1 over S such that for all smooth S-schemes X, F−1 (X) = coker F (A1 × X) → F (Gm × X) . Using the homotopy invariance of F , with this definition, we always have a split short exact sequence j∗
0 → F (A1 × X) −→ F (Gm × X) → F−1 (X) → 0; s
1 a canonical retraction of j ∗ is induced by the morphism A1 × X → X −→ Gm × X, given by projection followed by the unit section of Gm × X/X. Using this canonical splitting, we may assume that F−1 (X) ⊂ F (Gm × X). As a consequence, we deduce that if F is a homotopy sheaf then F−1 is a homotopy sheaf.
4.5.2 Local Purity —Relative Closed Pairs. We first introduce the analogue of Def. 4.3.6 over a base. Definition 4.5.3. Let S be a scheme. A closed pair over S is a pair (X, Z) such that X is a smooth S-scheme and Z is a closed subscheme of X. We will say that (X, Z) is smooth (resp. has codimension n) if Z is smooth (resp. Z is of pure codimension n in X). A morphism of closed pair (Y, T ) → (X, Z) is a pair of morphisms (f, g) which fits into the commutative diagram of schemes T g
Z
/Y f
/ X.
which is cartesian on the corresponding topological spaces. We will say (f, g) is cartesian (resp. excisive) if the preceding square is cartesian in the category of schemes (resp. gred : Tred → Zred is an isomorphism).
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Remark 4.5.4. Given a closed immersion i : Z → X into a smooth S-scheme, we will usually identify the schemes Z with the closed subscheme i(Z) of X when no confusion can arise. 4.5.5. The following method gives a general process to construct excisive morphisms. Let S be a scheme. Consider two closed pairs (X, Z) and (X 0 , Z) over S such that X and X 0 are ´etale over S. Let ∆ be the diagonal of Z over S. It is a closed subscheme of Z ×S Z and thus we identify it as a closed subscheme of X ×S X 0 . Similarly, ∆ is a closed subset of X ×S Z and Z ×S X 0 . Lemma 4.5.6. We adopt the hypotheses and notation as stated above. Define the set Ω = X ×S X 0 − [(X ×S Z − ∆) ∪ (Z ×S X 0 − ∆)], and consider p
q
the canonical projections X o Ω / X 0 . Then Ω is an open subscheme of X ×S X 0 and contains ∆ as a closed subscheme. Thus, identifying ∆ with Z, (Ω, Z) is a closed pair over S such that Ω/S is ´etale. Moreover, the projections p, q induce cartesian excisive morphisms (X, Z) o
p
(Ω, Z)
q
/ (X 0 , Z).
Proof We only have to prove that Ω is open in X ×S X 0 . Consider the closed immersion ι : ∆ → X ×S Z. Identifying ∆ with Z, ι is a section of the ´etale morphism f ×S Z : X ×S Z → Z. In particular, ι is an open immersion and X ×S Z − ∆ is a closed subscheme of X ×S X 0 . By symmetry, we get the conclusion. Remark 4.5.7. The reader can check that the preceding construction is functorial with respect to cartesian ´etale morphisms (Y, T ) → (X, Z) and (Y 0 , T ) → (X 0 , Z) of closed pairs over S. Let X be a S-scheme and s0 : X → AnX be the 0-section. We will often consider the closed pair (AnX , s0 (X)) which we will always denote by (AnX , X). Definition 4.5.8. Let S be a scheme and (X, Z) a closed pair over S. A parametrisation of (X, Z) over S is a cartesian ´etale morphism n (f, g) : (X, Z) → (Ac+n S , AS ) for a pair of integers (n, c). n Suppose given a parametrisation (X, Z) → (Ac+n S , AS ). Then (X, Z) is smooth of codimension c. Moreover, X has pure dimension n over S. In particular, the integers (n, c) are uniquely determined by (X, Z). Conversely, when the closed pair (X, Z) is smooth, for any point s of Z there always exist an open neighbourhood U of s in X and a parametrisation of (U, Z ∩U ) over S. We will loosely speak of a local parametrisation of (X, Z) at s.
208
F. D´eglise
4.5.9. The following process is the geometric base for the local purity theorem. Suppose given a closed pair (X, Z) and a parametrisation n (u, v) : (X, Z) → (Ac+n S , AS ). We associate a commutative square Z
s0
/ Ac Z
1×S v / Ac+n S
X
to this parametrisation, where s0 is the 0-section of AcZ . This gives two closed pairs (X, Z) and (AcZ , Z) over An+c S . From the preceding lemma, we obtain cartesian excisive morphisms (X, Z) → (Ω, Z) ← (AcZ , Z). (f,g)
Remark 4.5.10. Consider in addition a cartesian ´etale morphism (Y, T ) −−−→ (X, Z). Then we associate to the induced parametrisation of (Y, T ) over S a closed pair (Π, T ) which fits into the commutative diagram of closed pairs over S (Y, T ) o (f,g)
(X, Z) o
/ (Ac , T )
(Π, T )
(Ω, Z)
/
T
(1×S g,g) c (AZ , Z).
We note that this process allows us to deduce the following structure theorem for points in the Nisnevich topology: Corollary 4.5.11. Let S be a scheme and (X, Z) a smooth pair over S. Then for any point s of Z, there exists an isomorphism Xsh ' (AcZ )hs of pro-schemes over S which is the identity on Zsh . The integer c is the codimension of Z in X at s. Proof Indeed, as we can find a local parametrisation of (X, Z) at s, the preceding construction gives an open neighbourhood U of s in X and excisive morphisms (U, Z ∩ U ) → (Ω, Z ∩ U ) ← (AcZ∩U , Z ∩ U ). This implies that Ω is a Nisnevich neighbourhood of s in U (resp. AcZ∩U ), hence in X (resp. AcZ ). This concludes the proof. —The Case of Homotopy Presheaves.
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Let (X, Z) be a pair over a regular scheme S. For any point s ∈ Z, we have a canonical isomorphism Zsh =
˜ lim ←− h
V ×X Z.
V ∈Vs (X)
It is natural to consider the pro-object Xsh − Zsh =
˜ lim ←− h
(V − V ×X Z).
V ∈Vs (X)
We thus have canonical morphisms of pro-objects (pro-immersions): ι
Xsh − Zsh − → Xsh ← Zsh . For any presheaf over SmS , we consider the induced morphism ι∗ : F (Xsh ) → F (Xsh − Zsh ), and denote by F (Xsh − Zsh )/F (Xsh ) the cokernel of ι∗ . Note this is a little abuse of notation, as ι∗ is not necessarily a monomorphism. Proposition 4.5.12. Let S be a regular excellent scheme, (X, Z) be a smooth closed pair over S and F be a homotopy presheaf over S. Let s be a point of X such that Z is of codimension 1 in X at s. Then any local parametrisation of (X, Z) at s induces a canonical isomorphism F (Xsh − Zsh )/F (Xsh ) ' F−1 (Zsh ). Proof Indeed, Corollary 4.5.11 shwos that a local parametrisation of (X, Z) at s induces a canonical isomorphism Xsh → (A1Z )hs that is the identity on Zsh . Thus we are reduced to the case of the closed pair (A1Z , Z). Let V be a Nisnevich neighbourhood of s in Z. Then A1V is a Nisnevich neighbourhood of s in A1Z . Thus we get a canonical morphism lim −→ h
(∗)
F (A1V − V )/F (V ) −−→
V ∈Vs (Z)
lim −→
F (W − WZ )/F (W ).
1 W ∈Vh s (AZ )
Lemma 4.5.13. Let S be a regular excellent scheme and Z a smooth Sscheme. For every point s in Z, the canonical morphism described above (∗) F−1 A1Z h −−→ F (A1Z )hs − Zsh /F (A1Z )hs s
is an isomorphism.
210
F. D´eglise
Let Z = Spec OhZ,s be the limit of Zsh , and τ : Z → S the canonical morphism. Note that Z is regular and noetherian. As filtered inductive limits are exact, using Proposition 4.2.18 we obtain a canonical isomorphism (ˆ τ ∗ F )−1 (A1Z ) ' F−1 A1Z h . s
Moreover, we can write F (A1Z )hs − Zsh /F (A1Z )hs =
lim −→
lim −→
F (W − WZ )/F (W ).
1 h V ∈Vh s (Z) W ∈Vs (AV )
This expression, together with Proposition 4.2.18, gives us a canonical isomorphism τ ∗ F (A1Z ) ' F (A1Z )hs − Zsh /F (A1Z )hs . τˆ∗ F (A1Z )hs − Z)/ˆ Thus we are reduced to the case where S = Z is a local henselian scheme with closed point s. Indeed, the two isomorphisms just constructed are compatible with the morphism (∗) (cf. the remark after Proposition 4.2.18). Note also that Z is still a regular excellent scheme (cf. [EGA4, 18.6.10 g and 18.7.6]). We consider the category I of ´etale morphisms V − → A1S such that V is affine and g −1 (S) → S is an isomorphism, with arrows the A1S f → A1S morphisms. Then I is a final subcategory of Vhs A1S . Indeed, let V − be a Nisnevich neighbourhood of s in A1S . As S is henselian, the morphism g : VS → S induced by f admits a section. Thus there exists an open subscheme V 0 of V such that V 0 ∩ VS = S. As we can always reduce V 0 in a f
neighbourhood of s, we can assume V 0 is affine, that is, V 0 ∈ I. Let V − → A1S be an object in I. To conclude the proof of the lemma, we will prove that the morphism F (A1S − S)/F (A1S ) → F (V − VS )/F (V ) induced by f is an isomorphism. Zariski’s main theorem implies that there exist an S-scheme V¯ and morphisms V f
A1S
k
j
/ V¯ f¯
/ P1 S
such that f¯ is finite and k is an open immersion. Replacing V¯ by the reduced closure of V in V¯ , we can assume that V is dense in V¯ . As S is excellent and V is normal, we can assume furthermore that V¯ is normal, replacing it by its normalisation. We claim that V¯ /S is a good compactification of (V, S). Let W = f¯(V¯ − V ) as a reduced closed subscheme of P1S , and Ws ⊂ P1s be
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211
the special fiber of W . Then Ws is a finite set, as it is nowhere dense in P1s : we can find a regular function h on P1s such that D(h) ∩ (Ws ∪ {0}) = ∅. Let l be an extension of h to P1S . As the projection V¯ → S is proper, we necessarily have W ∩ D(l) = ∅. Thus, W ∪ {0S } is contained in the affine open subset V (h) = P1S− D(h). Then (V¯ − V ) ∪ VS is contained in the affine open subset f¯−1 D(h) as required. Finally, when we apply Theorem 4.3.24 to the square V − VS g
A1S
−S
l
/V
j
f / A1 S
we find that the complex j ∗ +f ∗
(h∗ ,−l∗ )
0 → F (A1S ) −−−−→ F (A1S − S) ⊕ F (V ) −−−−−→ F (V − VS ) → 0 is split exact. This implies the desired result by an easy diagram chase. —The Case of Sheaves. For any scheme X, we let XNis be the site of ´etale X-schemes with the ˜ Nis be the topos associated to XNis . We also Nisnevich topology, and X ˜ Nis the category of abelian sheaves on XNis . When f : Y → X denote by Z.X is any morphism of schemes we have, following [SGA4, exp. IV], a pair of adjoint functors ˜ Nis (f∗ , f ∗ ) : Y˜Nis → X with f ∗ exact. When we restrict our attention to the category of abelian sheaves, the functor f∗ can be classically derived on the right and induces a functor ˜ Nis ). Rf∗ : Db (Z.Y˜Nis ) → Db (Z.X Recall that for any q ∈ N, and any (abelian) Nisnevich sheaf FY on Y , Rq f∗ FY is the Nisnevich sheaf associated to the presheaf U/X 7→ H q (Y ×X U ; FY ). This implies that f∗ is exact whenever f is finite as a finite scheme over a local henselian scheme is a disjoint union of local henselian schemes. Let S be a regular scheme and F be a homotopy sheaf over S. For any smooth S-scheme X, we will denote by FX the restriction of F to XNis . Remark 4.5.14. 1) For any U in XNis , we obviously have H n (U ; FX ) = H n (X; F ). 2) When f : Y → X is a smooth morphism, we have f ∗ FX = FY . More generally, let (Xi )i∈I be a pro-object of ´etale X-schemes affine over
212
F. D´eglise
Z, and let X be its limit. We can consider X• = (Xi )i∈I as a proobject of smooth affine S-schemes. Let τ : X → S be the canonical morphism. We have defined the sheaf with transfers τ ∗ F over X in § 4.2.5. Recall that Proposition 4.2.19 implies that it is homotopy invariant. Let now τX : X → X be the canonical morphism. Then, as ∗ F = (τ ∗ F ) . another application of Proposition 4.2.19, we obtain τX X X Note that the coherence of the Nisnevich topos together with [SGA4, VI], implies that H n (X• ; F ) = H n (X; τ ∗ F ). Let (X, Z) be a smooth closed pair over S of codimension 1, j : X − Z → X and i : Z → X the canonical immersions. Applying Corollary 4.4.6, the adjunction morphism FX → j∗ j ∗ FX is a monomorphism. Let C be ˜ Nis . Then the adjunction morphism C → i∗ i∗ C is an its cokernel in Z.X isomorphism as for any point s ∈ X − Z, C(Xsh ) = 0. Definition 4.5.15. Using the above notation, we define F(X,Z) to be the Nisnevich sheaf on ZNis equal to i∗ C. Thus, by the above construction, we have a canonical exact sequence of sheaves on XNis 0 → FX → j∗ FU → i∗ F(X,Z) → 0.
(4.4)
For any cartesian morphism (f, g) : (Y, T ) → (X, Z) there is an induced morphism F(X,Z) → g∗ F(Y,T ) which makes the above exact sequence functorial. Lemma 4.5.16. Let F be a homotopy sheaf over a regular scheme S, and (X, Z) be a smooth codimension 1 closed pair over S. Then for any s ∈ Z, we have a canonical isomorphism F(X,Z) (Zsh ) = F (Xsh − Zsh )/F (Xsh ) using the notations preceding Proposition 4.5.12. Proof By definition, F(X,Z) (Zsh ) = (i∗ FX )(Xsh ). Hence the result follows by taking fibers along Xsh in the exact sequence (4.4).
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Lemma 4.5.17. Let F be a homotopy sheaf over a regular scheme S, and (f, g) : (Y, T ) → (X, Z) be an excisive morphism of smooth codimension 1 closed pairs over S. Then the canonical morphism F(X,Z) → g∗ F(Y,T ) is an isomorphism. Proof Let s be a point of Z. It suffices to check the assertion by evaluating the sheaves on Zsh . As (f, g) is excisive, Y is a Nisnevich neighbourhood of s in X. Thus f induces an isomorphism Yth → Xsh , with t being the point of Y such that g(t) = s. The result now follows from the preceding lemma. Let F be a homotopy sheaf over S and Z be a smooth S-scheme. For any ´etale Z-scheme V , let IV be the category with objects the pairs (U, η) such that U is an ´etale A1Z -scheme and η : V → UZ is a Z-morphism. The arrows in IV are the X-morphisms in U compatible with η. Then, by contruction of the pullback on sheaves over small Nisnevich sites, F(A1 ,Z) is the sheaf Z associated to the presheaf G : V /Z 7→
lim −→
F (U − UZ )/F (U ).
(U,η)∈IV
There is an obvious morphism F−1 |Z → G of presheaves over ZNis , which induces a morphism of sheaves over ZNis (A1 ,Z) : F−1,Z → F(A1 ,Z) . Z
Z
Lemma 4.5.18. The morphism (A1 ,Z) : F−1 |Z → F(A1 ,Z) described above Z Z is an isomorphism. Proof It suffices to check the assertion on the fibres. Using the computation of Lemma 4.5.16 and the homotopy invariance of F−1 , this is just Lemma 4.5.13. The two previous lemmas imply the main result of this section: Corollary 4.5.19. Let (X, Z) be a closed pair over S and ρ : (X, Z) → p q n (An+1 → (Ω, Z) ← − (A1Z , Z) be S , AS ) a parametrisation over S. Let (X, Z) − the morphisms constructed in 4.5.9. Let T be a smooth S-scheme. For any smooth S-scheme Y , we put Y 0 = Y ×S T ; this defines an endomorphism of smooth S-schemes. Then, all the morphisms in the sequence (p0 )∗
(A1
(q 0 )∗
0 0 ,Z )
Z F(X 0 ,Z 0 ) ←−−− F(Ω0 ,Z 0 ) −−−→ F(A1 0 ,Z 0 ) −−− −−→ F−1 |Z 0 Z
are isomorphisms.
214
F. D´eglise
We thus have associated to the parametrisation ρ an isomorphisms of sheaves over ZNis , ρ,T : F(X×k T,Z×k T ) → F−1 |Z×k T . This isomorphism is obviously functorial in the smooth S-scheme T , using the naturality of φT with respect to T . Remark 4.5.20. This isomorphism is functorial in ρ in a suitable sense, but we will not use this functoriality. Note moreover that by using deformation to the normal cone, we can show at this point that ρ does not depend on the choice of ρ. This could be used to construct such an isomorphism for any smooth closed pair of codimension 1 without requiring the existence of a global parametrisation.
4.5.3 Localisation Long Exact Sequences Let S be a regular scheme and let (X0 , Z0 ) be a smooth closed pair over S of n codimension 1 and ρ : (X0 , Z0 ) → (An+1 S , AS ) a parametrisation over S. Let T be a smooth S-scheme, and put (X, Z) = (X0 ×S T, Z0 ×S T ). Let j : U → X and i : Z ← X be the canonical closed embeddings. The isomorphism constructed in Corollary 4.5.19 induces a canonical exact sequence of sheaves on XNis 0 → FX → j∗ FU → i∗ F−1 |Z → 0. Recall that i∗ is exact. Thus, for any ´etale X0 -scheme V0 , taking cohomology on V = V0 ×S T we get a localisation long exact sequence, · · · →H n−1 (V ; FX ) → H n−1 (V ; j∗ FU ) → H n−1 (VZ ; F−1 |Z ) → H n (V ; FX ) → H n (V ; j∗ FU ) → . . .
(4.5)
This sequence is functorial in V0 with respect to ´etale X0 -morphisms and in T with respect to any S-morphisms. Remark 4.5.21. We could also have considered the closed pair (V0 , V0 ×X Z) n and the induced parametrisation (V0 , V0 ×X Z) → (An+1 S , AS ). By the very construction, the sequence obtained is exactly the above sequence. To conclude, we note that if we know the vanishing of Rm j∗ for m > 0, this exact sequence as the form j∗
· · · → H n−1 (V ; F ) −→ H n−1 (V −VZ ; F ) → H n−1 (VZ ; F−1 ) → H n (V ; FX ) → . . . which is in fact the localisation exact sequence associated to (V, VZ ).
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215
4.5.4 The Proof of Theorem 4.5.1 Let now S be the spectrum of a perfect field k. The proof of Voevodsky proceeds by induction on the dimension. He inductively proves both the homotopy invariance of H n (.; F ) and the existence of the localisation long exact sequence (for smooth divisors) in dimension less than n; according to what we have seen above, this amounts to proving the vanishing of Rm j∗ for m < n, and for any open immersion j of the complement of a smooth divisor. Reduction step 1.— Let n be a positive integer. Suppose that for any smooth k-scheme X, any point s ∈ X, and any integer 0 < i ≤ n, H i (A1X h ; F ) = 0. Then for any s
smooth k-scheme X and any integer 0 < i ≤ n, H i (A1X ; F ) = H i (X; F ). Indeed, we let π : A1X → X be the canonical projection and apply the Leray spectral sequence to π and to FA1 : X
E2p,q
p
q
= H (X; R π∗ FA1 ) ⇒ H p+q (A1X ; FA1 ). X
X
Now it suffices to note that the hypothesis implies Rq π∗ FA1 = X Indeed, for any point s ∈ X, Rq π∗ FA1 (Xsh ) = H q (A1X h ; F ). X s
0 if 0 < q ≤ n.
Reduction step 2.— The following step is exactly the point where we need the base to be a perfect field. Let n be a positive integer. For a closed pair (X, Z) over k, we consider the property P (X, Z)
H n (A1X h ; F ) s
the induced morphism → H n (A1X h −Z h ; F ) s s is a monomorphism.
If the property P (X, Z) holds for any smooth closed pair (X, Z) of codimension 1, then for any smooth k-scheme X, and any point s ∈ X, H n (A1X h ; F ) = s 0. We start by showing that our assumptions imply that for any closed pair (X, Z) over k property P (X, Z) holds. For this, we use the following lemma of Voevodsky (cf. [Voe00a, Lem. 4.31]) Lemma 4.5.22. Let X be a smooth k-scheme over a perfect field k and Z a nowhere dense closed subscheme of X. Then for every point s ∈ X, there exists an open neighbourhood U of x in X and an increasing sequence of closed subschemes ∅ = Y0 ⊂ · · · ⊂ Yr of U for r > 0 such that 1) for any 1 ≤ i ≤ r, Yi − Yi−1 is a smooth divisor of U − Yi−1 . 2) Z ∩ U ⊂ Yr .
216
F. D´eglise
Proof We assume that X is connected and use induction on the dimension n ≥ 1 of X. The case n = 1 is trivial, as k is perfect. Let n ≥ 2 and X be a smooth n-dimensional scheme. Necessarily, there exist an open subscheme U0 of X and a morphism p : U0 → Y to a smooth k-scheme Y such that p is smooth of relative dimension 1. Let Zsing be the singular locus of Z. As k is perfect, dim(Zsing ) < dim(Z) < n. Let T be the reduced closure of p(Zsing ) in Y . We thus have dim(T ) ≤ n − 2. As Y has pure dimension n − 1, T is nowhere dense in Y . By the induction hypothesis applied to Y and T in a neighbourhood of p(s), there is a neighbourhood V of p(s) in Y and an increasing sequence Y00 ⊂ · · · ⊂ Yr0 of closed subschemes of V satisfying conditions 1 and 2 for T and V . Put U = p−1 (V ), Yi = p−1 (Yi0 ) for 0 ≤ i ≤ r and Yr+1 = Yr ∪ (Z ∩ U ). Then the sequence Y0 ⊂ · · · ⊂ Yr+1 of closed subschemes of U satisfies conditions (1) and (2) for Z and U . With this lemma, we easily obtain P (X, Z) for any pair (X, Z), as P (X, T ) ⇒ P (X, Z) if Z ⊂ T ⊂ X and P (X, Z) is a local property on X. Fix now a smooth k-scheme X and a point s ∈ X. Let E be the quotient field † ˜ X h − Z h of ´etale X-schemes has the of OhX,s . The pro-object (E) = lim s ←− s Z⊂X
scheme Spec (E) as limit. The property P (X, Z) for any Z implies that the canonical morphism H n (A1X h ; F ) → H n (A1(E) ; F ) s
is a monomorphism. Let τ : Spec (E) → Spec (k) be the canonical morphism. Then Remark 4.5.14 implies H n (A1(E) ; F ) = H n (A1E ; τ ∗ F ). Thus finally, this reduction step follows from Corollary 4.4.11. We are now ready to prove the following assertions by induction on n ≥ 1: i) For any smooth closed pair (X, Z) of codimension 1, j : X − Z → X the open immersion, we have Rm j∗ (FX−Z ) = 0 for all 0 < m < n. ii) For any smooth closed pair (X, Z) of codimension 1 with a given parametrisation ρ, j : X − Z → X the open immersion, and V an ´etale X-scheme, the localisation exact sequence (4.5) induces an exact sequence j∗
H n−1 (V − VZ ; F ) → H n−1 (VZ ; F−1 ) → H n (V ; F ) −→ H n (V − VZ ; F ) functorial in X with respect to ´etale morphisms (the parametrisation of an ´etale X-scheme being the parametrisation induced by ρ). iii) H n (.; F ) is homotopy invariant. † The extension field E/k, though of finite transcendence degree, is not necessarily of finite type.
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217
Suppose that n = 1. Let (X, Z) be a smooth pair of codimension 1 and ρ a parametrisation of (X, Z). Let T = Spec (k) or A1k , (X 0 , Z 0 ) = (X ×k T, Z ×k T ), U 0 = X 0 − Z 0 and j : U 0 → X 0 the canonical immersion. The localisation exact sequence (4.5) associated with these data is j∗
ν
0 → F (X 0 − Z 0 ) −→ F (X 0 ) → F−1 (X 0 ) → H 1 (X 0 ; F ) − → H 1 (X 0 ; j∗ FX 0 −Z 0 ). Applying the Leray spectral sequence for j and FX 0 −Z 0 , we obtain a canonical b
morphism H 1 (X 0 ; j∗ FX 0 −Z 0 ) − → H 1 (X 0 − Z 0 ; FX 0 −Z 0 ) which is a monomorphism and such that j ∗ = b ◦ ν. This implies that Ker(ν) = Ker(j ∗ ), hence we obtain the sequence of property (ii) for (X 0 , Z 0 ). Functoriality in X with respect to ´etale morphisms now follows from the functoriality of the sequence (4.5). Let s be a point of Z. We now consider the limit of this exact sequence by replacing X by an arbitrary Nisnevich neighbourhood of s in X. The functoriality in T of the sequence (4.5) implies that the following diagram, in which the lines are exact, is commutative: F (Xsh − Zsh ) ∼
F (A1X h −Z h ) s
s
/ F (Z h ) −1 s
/0
∼
/ F−1 (A1 h ) Zs
/ H 1 (A1 h ; F ) X
(1)
s
/ H 1 (A1
Xsh −Zsh
; F ).
Thus (1) is a monomorphism. As any smooth closed pair (X, Z) locally admits a parametrisation, we are done by reduction steps (2) and (1). We now prove the result for n > 1 using the induction hypothesis. To prove (i), we consider s ∈ X and show that for any 0 < q < n, the fiber of Rq j∗ FU at Xsh is zero. By the induction hypothesis and Corollary 4.2.10, H = H m (.; F ) is a homotopy presheaf over k. Then Proposition 4.5.12 implies that we have an isomorphism Rq j∗ FU (Xsh ) = H(Xsh − Zsh ) = H(Xsh − Zsh )/H(Xsh ) ' H−1 (Zsh ). ˜ Z h − T h . The canonical Let E be the quotient field of OZsh . Put (E) = lim s ←− s T ⊂Z
morphism (E) → Zsh is a pro-immersion. Thus, applying Corollary 4.6, the induced morphism H−1 (Zsh ) → H−1 (E) is a monomorphism. Indeed, Zsh is a point and, though E is not necessarily of finite type over k, it is the filtering union of its sub-k-extensions E 0 of finite type. Thus F 7→ F (E) is still a fiber functor for the Nisnevich topology on Smk . Let τ : Spec (E) → Spec (k) be the canonical morphism. We obtain finally the following inclusion H−1 (E) ⊂ H(Gm × (E)) = H q (Gm × (E); F ) = H q (Gm,E ; τ ∗ F )
218
F. D´eglise
using remark 4.5.14. Since the latter group is zero by Corollary 4.4.11, we are ready for (i). For (ii) it is now sufficient to apply the same reasoning than in the case n = 1. Indeed property (i) and the Leray spectral sequence for j give the b
edge monomorphism H n (X; R0 j∗ FX ) − → H n (X − Z; FX−Z ). For (iii) now, we consider a smooth closed pair (X, Z) of codimension 1 and a point s ∈ Z. It admits a parametrisation in a neighbourhood V of s, which induces a parametrisation of (A1V , A1V ∩Z ). This parametrisation being fixed, we can consider the exact sequence of property (ii) for any ´etale V -scheme. If we take the colimit of these sequences with respect to the Nisnevich neighbourhoods of s in V , we obtain the following exact sequence H n−1 (A1Z h ; F−1 ) → H n (A1X h ; F ) → H n (A1X h −Z h ; F ). s
s
s
s
This concludes the proof using the induction hypothesis, as we can now use again reduction steps (2) and (1).
References ´glise: Premotives and the six functors formalism. [CD] Cisinski, D.-C and F. De ´glise, F.: Transferts sur les groupes de chow `a coefficients, Mathema[D´eg05] De tische Zeischrift 338(1) (2005) 41–46. ´ ements de g´eom´etrie alg´ebrique, ´: El´ [EGA2] Grothendieck, A. and J. Dieudonne II, Publ. Math. de l’IHES (1961). ´ ements de g´eom´etrie alg´ebrique, ´: El´ [EGA3] Grothendieck, A. and J. Dieudonne III, Publ. Math. de l’IHES (1961-1963). ´ ements de g´eom´etrie alg´ebrique, ´: El´ [EGA4] Grothendieck, A. and J. Dieudonne III, Publ. Math. de l’IHES (1966). [FSV00] Friedlander, E.M., A. Suslin and V. Voevodsky: Cycles, Transfers and Motivic homology theories Annals of Mathematics Studies 143 Princeton Univ. Press (2000). [Ful98] Fulton, W: Intersection theory. Springer, second edition, (1998). [Har77] Hartshorne, R.: Algebraic geometry, Graduate Texts in Mathematics 52 Springer-Verlag, New York, (1977). [Mil80] Milne, J.S.: Etale cohomology, Princeton Math. studies 33 (1980). [MV99] F. Morel and V. Voevodsky: A1 -homotopy theory of schemes, Inst. ´ Hautes Etudes Sci. Publ. Math., 90 (2001) 45–143. [Ray70] Raynaud, M.: Anneaux locaux hens´eliens, Lecture Notes in Mathematics 169, Springer-Verlag (1970). [Ser58] Serre, J.P.: Alg`ebre locale , multiplicit´es, Springer, third edition (1957/58). [SGA4] Artin, M., A. Grothendieck and J.L. Verdier: Th´eorie des Topos et ´ Cohomologie Etale des Sch´emas 269,270,305 Springer Verlag (1972–73). [SV96] Suslin, A. and V. Voevodsky: Singular homology of abstract algebraic varieties, Invent. Math., 123(1) (1996) 61–94. [SV00] Suslin, A. and V. Voevodsky: Relative cycles and Chow sheaves. In
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Cycles, transfers, and motivic homology theories Ann. of Math. Stud. 143, Princeton Univ. Press Princeton, NJ, (2000) 10–86. [Voe00a] Voevodsky, V.: Cohomological theory of presheaves with transfers. In Cycles, transfers, and motivic homology theories Ann. of Math. Stud. 143, Princeton Univ. Press Princeton, NJ, (2000) 87–137. [Voe00b] Voevodsky, V.: Triangulated categories of motives over a field. In Cycles, transfers, and motivic homology theories Ann. of Math. Stud. 143, Princeton Univ. Press Princeton, NJ, (2000) 188–238 [Wal96] Walker, Mark E.: Motivic complex and the K-theory of automorphisms, PhD thesis, University of Illinois (1996).
5 Algebraic Cycles and Singularities of Normal Functions Mark Green Institute for Pure and Applied Mathematics, UCLA
Phillip Griffiths Institute for Advanced Study
To our friend, Jacob Murre, on his 75th birthday
Abstract Given the data (X, L, ζ) where X is a smooth 2n-dimensional algebraic variety, L → X is a very ample line bundle and ζ ∈ Hgn (X)prim is a primitive Hodge class, we shall define an analytic invariant νζ ∈ Γ(S, J˜e ) and algebro-geometric invariant δνζ ∈ Γ S, Hen,n−1 ⊗ Ω1S (log D) ∇ where S is a blow-up of PH 0 (OX (L)) and D ⊂ S is the quasi-local normal crossing discriminant locus (see below for definitions). We will also define the singular loci sing νζ and sing δνζ and show that, for L 0, as subvarieties of S sing νζ = sing δνζ and that in a precise sense these loci define the algebraic cycles W on X with the property that hζ, [W ]i = 6 0. The Hodge conjecture (HC) is then equivalent to sing νζ = sing δνζ 6= ∅ 220
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for L 0. In an informal sense we may say that if the HC is true, then there is a systematic geometric procedure for producing the equations of algebraic cycles from Hodge classes. For L 0 an arbitrary class — not one that is rational — ζ ∈ H n (ΩnX )prim may be localized along the locus of singularities of the universal family X → S. The HC is then equivalent to the condition that the integrality of the residues of δνζ along the descriminant locus D give the test that ζ ∈ H 2n (X, Q), which is an explicit form of the absolute Hodge condition. The effective Hodge conjecture (EHC) is the statement that there is an explicit k0 such that there is Xs ∈ |Lk0 | and a subvariety W ⊂ Xs with hζ, [W ]i = 6 0. Heuristic reasons show that in general k0 must be bounded below by an expression whose dominant term is (−1)n ζ 2 (which is positive). The other quantities on which k0 depends and which are independent of ζ are discussed below. The polarizing forms on the intermediate Jacobians define line bundles, including a Poincar´e line bundle P that may be pulled back to νζ∗ (P ) by a normal function ζ. Restricting to one dimensional families with only one ordinary node, the Chern class of νζ∗ (P ) evaluates to ζ 2 . This again suggests the central role of ζ 2 in the study of algebraic cycles. This is an extended research announcement of joint work in progress. The complete details of some of the results have yet to be written out. It is an expanded version of the talk given by the second author at the conference in Leiden in honor of Jacob Murre. We would like to especially thank Mark de Cataldo, Luca Migliorini, Gregory Pearlstein, and Patrick Brosnan for their interest in and comments on this work.
5.1 Introduction and Historical Perspective 5.1.1 Introduction and Statement of Results We shall use the notations X = smooth projective variety p
Z (X) = group of codimension-p algebraic cycles X = {Z = ni Zi : Zi ⊂ X} i
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where Zi is an irreducible codimension-p subvariety, and Z p (X) −−−−→ Hgp (X) = H 2p (X, Z) ∩ H p,p (X) ∪
∪
Z
−−−−→ [Z]
is the mapping given by taking the fundamental class. Hodge’s original conjecture (HC): This map is surjective. It is known that the HC is • True when p = 1 (Lefschetz [30], c. 1924) • False in any currently understood sense for torsion when p ≥ 2 (AtiyahHirzebruch [2] and Koll´ar (see section 5.4.1 below)) • False in any currently understood sense for X K¨ahler, p ≥ 2 (Voisin [37]) The phrase “in any currently understood sense” means this: Atiyah and Hirzebruch showed that for p ≥ 2 there is a smooth variety X and a torsion class in H 2p (X, Z), which being torsion is automatically of Hodge type (p, p), and which is not the fundamental class of an algebraic cycle. Koll´ar showed that there is an algebraic class " # X mi Zi ∈ H 2p (X, Z) i
where mi ∈ Q but we cannot choose mi ∈ Z. Finally, Voisin [37] showed that there is a complex 4-torus T and 0 6= ζ ∈ Hg2 (X) where T has no geometry — i.e., no subvarieties or coherent sheaves — other than those coming from points of T . Conclusion: Any general construction of codimension p cycles for p ≥ 2 must wipe out torsion and must use the assumption that X is an algebraic variety. With the exception of section 5.4.1, in what follows everything is modulo torsion. By standard techniques the HC is reduced to the case dim X = 2n,
p = n,
primitive Hodge classes
where we are given a very ample line bundle L → X with c1 (L) = λ and where the primitive cohomology (with Q coefficients) is as usual defined by λ
H 2n (X)prim = ker{H 2n (X) −−−−→ H 2n+2 (X)}.
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If s ∈ H 0 (OX (L)) and the variety Xs given by {s = 0} is assumed to be smooth then H 2n (X)prim = ker{H 2n (X) → H 2n (Xs )} which by Poincar´e duality is ∼ = ker{H2n (X) → H2n−2 (Xs )} . 0 (O (L)), where the tilde means that we have blown We set S = PH^ X 0 PH (OX (L)) up so that the discriminant locus
D = {s : Xs singular} ⊂ S has quasi-local normal crossings (definition below). We also set S ∗ = S\D so that for s ∈ S ∗ the hypersurface Xs is smooth with intermediate Jacobian J(Xs ), and we set [ J(Xs ) J= s∈S ∗
ˇ n /R2n−1 Z ∼ J = OS ∗ (J) = F = Fn \H2n−1 /Rπ2n−1 Z . π
Here we recall that ˇ 2n−1 (Xs , C)/H2n−1 (Xs , Z) J(Xs ) = F n H ∼ = F n H 2n−1 (Xs , C)\H 2n−1 (Xs , C)/H 2n−1 (Xs , Z) . We consider the picture X∗ ⊂ X yπ yπ S∗
⊂ S
where X ⊂ X × S is the smooth variety given by X = {(x, s) : x ∈ Xs } . In this picture we set H2n−1 = OS ∗ ⊗ Rπ2n−1 C with the Hodge filtration ≥p Fp ∼ = Rπ2n−1 ΩX∗ /S ∗
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satisfying ∇Fp ⊂ Fp−1 ⊗ Ω1S ∗ where ∇ is the Gauss-Manin connection. We set H2n−1−p,p = F2n−1−p /F2n−p , and the cohomology sheaf of the complex ∇
∇
H2n−p,p−1 −→ H2n−1−p,p ⊗ Ω1S ∗ −→ H2n−2−p,p+1 ⊗ Ω2S ∗ (5.1) will be denoted by H2n−1−p,p ⊗ Ω1S ∗ ∇ . Assuming for the moment that we are in the local crossing case, and the unipotency of the local monodromy operators Ti around the branches si = 0 at a point s0 ∈ S, where in a suitable local coordinate system s1 , . . . , sN D = {s1 · · · sk = 0} , it is well-known ([34]) that there are canonical extensions He2n−1 and Fep of H2n−1 and Fp with ∇Fep ⊂ Fep−1 ⊗ Ω1S (log D). We put He2n−1−p,p = Fe2n−1−p /Fe2n−p leading a complex extending (5.1) o n K• =: He2n−p+•,p−1−• ⊗ Ω•S (log D), ∇ (5.2) H k (K • ) =: He2n−p+k,p−1−k ⊗ ΩkS (log D . ∇
A general reference to background material in variation of Hodge structure is [24]. We will use an extension ([31]) of the above to the situation that we will S term quasi-local normal crossings. This means that locally D = i∈I Di is a union of smooth divisors Di = (si ) with the following properties: T (i) On any slice transverse to i∈I Di = DI , any subset of q 5 codim DI of the functions si form part of a local coordinate system in Si , and (ii) most importantly, the local monodromy operators Ti around si = 0 are assumed to commute and are unipotent. We will define • an extension J˜e of J and the space of extended normal functions (ENF) ν ∈ Γ(S, J˜e ) • an infinitesimal invariant δν ∈ Γ
Hen,n−1 ⊗ Ω1S (log D) ∇
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• the singular sets sing ν,
sing δν ⊂ S
The main results concerning sing ν and sing δν are Theorem 5.1.1. There is an isomorphism Hgn (X)prim −−−−→ Γ(S, J˜e )/J(X) ∪ ζ Theorem 5.1.2.
∪ −−−−→
νζ
i) Assume the HC in dimension < 2n. Then
sing νζ = {s ∈ D : hζ, [W ]i = 6 0 where W n ⊂ Xs is a subvariety} . ii) In general sing νζ = {s ∈ D : ζs 6= 0 in IH2n−2 (Xs )} . Corollary. HC ⇔ sing νζ 6= 0 for L 0. Theorem 5.1.3. For L 0 i) ζ 6= 0 mod torsion ⇒ δνζ 6= 0 ii) sing νζ = sing δνζ . Corollary. HC ⇔ sing δνζ 6= 0 for L 0. In (ii), IH(Xs ) refers to intersection homology, general references for which are [16], [17]. The definitions of sing ν, sing δν are geometric and understanding their properties makes extensive use of the theory of degenerations of VHS over arbitrary base spaces developed in recent years [9], [10], [26]. We note that for ζ a torsion class, hζ, [W ]i = 0 for all W as above, and also δνζ = 0. Thus, in the geometry underlying Theorems 5.1.2 and 5.1.3 torsion is indeed “wiped out”, as is necessary. By the basic setting, the results stated require that we be in a projective algebraic — not just a K¨ahler — setting. We remark that our definition of sing νζ should be taken as provisional. Taking S = |L| (not blown up) we feel that the definition is probably the correct one when the singular Xs are at most nodal, but it may well need modification in the most general case.
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The above results will be explained in sections 5.2 and 5.3. In section 5.4 we will explain how all of H n (ΩnX )prim may be localized along the locus of singularities of the Xs , and how when this is done the HC is equivalent to being able to express the condition that the complex class ζ ∈ H n (ΩnX )prim actually be a class in H 2n (X, Q) in terms of the rationality of the residues of δνζ , where δνζ may be defined even when νζ cannot be. Finally, in section 5.5 we will begin the discussion of line bundles over the family of intermediate Jacobians arising from the “polarizing forms” on the primitive cohomology groups. These “polarizations” are bilinear integral valued forms but need not be positive definite (see e.g. [20]) and hence the theory is completely standard. For this reason in section 5.5.1 we include a brief treatment of complex tori equipped with such a polarization (see also [25]). Our results here are very preliminary. They consist of an initial definition of these line bundles and a first computation of their Chern classes. Especially noteworthy is the formula for the “universal” theta line bundle M ∗ ∗ ∗ ∗ 0 c1 νζ+ζ 0 (M ) − c1 νζ (M ) − c1 νζ 0 (M ) + c1 (ν0 (M )) = ζ · ζ , where the LHS is reminiscent of the relation ˙ 0 ) − (a) − (a0 ) + (e) ∼ 0 (a+a ˙ is the group law, (b) is the 0-cycle associated on an elliptic curve E, where + to a point b ∈ E, e is the origin and ∼ is linear equivalence (see Theorem (7) in section 5.5.2).
5.1.2 Historical Perspective In reverse historical order the proofs of HC for p = 1 are (i) λ ∈ Hg1 (X) gives a line bundle Lλ → X (K¨ahler fact) Kodaira-Spencer (ii) L → X gives a divisor (GAGA-requires that λ X be projective) For p ≥ 2 the first step seems to fail in any reasonable form. In fact, as noted above, Voisin has given an example of a 4-dimensional complex torus X with Hg2 (X) 6= 0 but where there are no coherent sheaves or subvarieties other than those arising from points. Lefschetz-Poincar´e: For n = 1 we take a Lefschetz pencil |Xs |s∈P1 to have ˜ is the blow-up of X along the base locus the classic picture, where X
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Xs0
˜ X ←X ↓π P1
X s0 =
s
s0
A primitive algebraic cycle Z on X gives Zs = Z · Xs ∈ Div0 (Xs ) ν(Zs ) ∈ J(Xs ) Z → νZ
∈ Γ(P1 , Je )
where we have 0 → Rπ1 Z → Rπ1 OX˜ −→Je → 0 J(Xs0 ) = H 1 OXs0 /H 1
fibre of ˜s , Z ∼ X = Rπ1 OX˜ / Rπ1 Z s0 . 0 at s0
(By moving Z in a rational equivalence we may assume that its support misses the nodes on the singular fibres.) Poincar´e’s definition of a normal function was a section of Je . Equivalently, setting P1∗ = P1 -{s0 : Xs0 has a node}, J = Je |P1∗ , he formulated a normal function as a section of J with the properties — over ∆∗ it lifts to a section of Rπ1 OX˜ ∆ (i.e. no monodromy) — it extends across s0 to (Rπ1 OX˜ )s0 (moderate growth). Here, ∆ ⊂ P1 is a disc with origin s0 and ∆∗ = ∆\{s0 }.
˜∆ X
∆ s0
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Geometrically we think of
Z
Z where we choose
solid arc
ω rather than
dotted arc
ω for the abelian sums.
Ruled out is a picture (which is not a Lefschetz pencil)
q δ1
p δ2
Here, any path γ joining p to q has monodromy, while we may choose a path γ˜ with ∂˜ γ = 2(p − q) that has no monodromy.
q
γ1 ∂γ1 = p − q γ ∈ H1 (Xs , Z)
p γ = closed loop around the hole
Proof With T = T1 the monodromy operator we have: ∂γ1 = p − q ⇒ (T − I)(2γ1 − γ) = 0 in H1 (Xs , Z). (T − I)γ1 = δ1 (T − I)γ = 2δ1
Moral: For any family
1-dim base
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with a one dimensional base and Z with deg Zs = 0, for some non-zero m∈Z mνZ gives a normal function. As will be seen below this is a consequence of the local invariant cycle theorem. Proof of HC: 0 → Rπ1 Z → Rπ1 OX˜ → Je → 0 gives δ 0 → Pic◦ (X) → Γ(P1 , Je ) → H 1 (Rπ1 Z) → H 1 Rπ1 OX˜ ok
ok
H 2 (X, Z)prim → H 2 (OX ) There are then two steps: (1) ζ ∈ Hg1 (X)prim ∼ = ker{H 1 (Rπ1 Z) → H 1 (Rπ1 OX˜ )} ⇒ ζ = δνζ (2) νζ arises from an algebraic cycle Z (Jacobi inversion with dependence on parameters)
Extensions of (1): dim X = 2n, L → X very ample. The first was the general Lefschetz pencil case (Bloch-Griffiths unpublished notes from 1972), where for a section ν with lifting ν˜ as in the following diagram H2n−1 −→ Fn \H2n−1 /Rπ2n−1 Z ∪ ν˜
∪ −→
ν
we have to add the condition ∇˜ ν ∈ Fn−1 . The next was the definitive extension by Zucker [38] and El Zein-Zucker (cf. [14] and the references cited therein) to a general one parameter family of generically smooth hypersurface sections
˜ X ↓ S s0
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with the assumption on ζ ∈ Hgn (X)prim that we should have Poincar´e’s first condition ˜∆, Z ζ = 0 in H 2n X Their result is now generally referred to as the Theorem on Normal Functions. Now we discuss the Clemens-Schmid exact sequence (cf. Chapter VI in [24]). It implicitely uses the Monodromy Theorem which states that the eigenvalues of T are all roots of unity and so T is quasi-unipotent, i.e. in the decomposition T = Ts Tu in semi-simple and unipotent parts Ts is of finite order k; after base changing via z 7→ z k the monodromy operator T becomes unipotent. We may and do assume that this is the case and put N := log(T ) =
X
(−1)k+1
k≥1
(T − I)k , k
the left-hand side of which is a finite sum with Q-coefficients. This explains we need Q-coefficients in the sequence N ˜∆, ∂ X ˜∆ ˜ ∆ → H p (Xs ) −→ Hp X → Hp X H p (Xs ) → ok ˜∆) H4n+2−p (X ok H4n+2−p (Xs0 )
ok p H (X
s0 )
ok (Rπp Z)s0
ker N = invariant cycles ker N ⊥ = vanishing cycles
With the additional assumption (T − I)2 = 0 ⇒ G = ker(T − I)⊥ /im (T − I) is a finite group we have a N´eron model J¯e with an exact sequence 0 → O(Je ) → O(J¯e ) → G → 0 and Clemens [12] and M. Saito [33] showed that (1) extends using J¯e (G = Z2 in the above example). Issues. Due to the failure in general of Jacobi inversion the above method, at least as it has been applied, fails in general to lead to the construction of cycles (cf. [33]). Among the issues that have arisen in this study are: — the need dim S arbitrary to see non-torsion phenomena and to have δνξ non-trivial — the assumption (T − I)2 = 0 is too restrictive.
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Of these the first may be the more significant, since the second is satisfied when the singularities are nodal and as discussed below, these seem to be sufficient to capture much of the geometry.† However, it is only when the base is higher dimensional that the full richness of the theory of degenerations of Hodge structures and the use of arguments requiring L 0 and all of H 0 (OX (L)) can be brought to bear on the problem. It is also only when L 0 and the full H 0 (OX (L)) is used that the infinitesimal invariant δν captures the information in ν. Example 5.1.4. X = Q ⊂ P3 , L = OX (2, 2), g(Xs ) = 1 and Z = L1 − L2 where the Li are lines from the two rulings on Q. We then have the following picture
|OX (2, 1)|
δ1
|OX (0, 1)|
δ2 δ1 → 0 s1 s2 = 0 in |OX (2, 2)|
δ2 → 0
With ν an extended normal function as defined below we have — ν(s1 , s2 ) ≡ n1 log s1 + n2 log s2 modulo (periods and holomorphic terms) — ν extends to Je ⇔ n1 = n2 — J˜e,s0 /Je,s0 ∼ (ν → n1 − n2 ) =Z — νZ (s1 , s2 ) ≡ 2 log s1 − 2 log s2 modulo (periods and holomorphic terms) | {z } | {z } integrate over
—–
integrate over
----
Here J˜e ⊃ Je is the sheaf of extended normal functions. † Although from the physicists work on mirror symmetry we see that the “most singular” degenerations may also be very useful.
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5.2 Extended Normal Functions and their Singularities 5.2.1 Geometric Motivation Given W n ⊂ X 2n , by a rational equivalence and working modulo torsion and complete intersection cycles, we may assume that W is smooth, and then for L 0 there will be s0 ∈ S such that W ⊂ Xs0 ; we may even assume that Xs0 is nodal (cf. section 5.4.1 below). If hζ, [W ]i = 6 0
(⇒ s0 ∈ D and [W ]prim 6= 0)
then ζ does not satisfy the analogue of Poincar´e’s first condition
˜∆ X
˜∆) ζ = 0 in H 2n (X
s0
∆
This suggests studying the behaviour of νζ (s), defined initially over S ∗ = {s ∈ S : Xs smooth}, as s → s0 . Such a study was attempted in [22] and [23], but this was inconclusive as the understanding of degenerations of Hodge structures over higher dimensional base spaces was not yet in place.
5.2.2 Definition of Extended Normal Functions (ENF) n Near s0 ∈ D where we have quasi-local normal crossings, for ω ∈ Fe,s we 0 have
hν, ωi (s) = P (log s1 , . . . , log sk ) + {meromorphic functionf (s)} for some polynomial P . By definition, moderate growth is the condition that f (s) be holomorphic; we assume this analogue of Poincar´e’s second condi˜ n ; then modulo homomorphic tion. In U∗ choose a (multi-valued) lift ν˜ to F e functions Z h(Ti − I)˜ ν , ωi (s) ≡ ω(s), δi,s ∈ H2n−1 (Xs ) δi,s
where (Ti − I)˜ ν is the change in ν˜ by analytic continuation around the puncture in the disk |si | < 1, sj = constant for j 6= i. The condition that ν
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can be extended to Je is δi,s = (Ti − I)λs
λs ∈ H2n−1 (Xs , Z) ,
(5.3)
and then mν extends to Je if, and only if, (5.3) holds for mδi,s . Definition 5.2.1. An ENF is given by the sections ν of J → S ∗ that near a point of the descriminant locus have moderate growth and satisfy mδi,s = (Ti − I)λi,s
for some integer m.
(5.4)
Thinking of Je = Fen \He2n−1 /Rπ2n−1 Z this is equivalent to X m˜ ν≡ e(Ni log si )λi,s mod He2n−1 i
where ν˜ is a lift of ν to H2n−1 over the punctured polycylinder U∗ and e(Ni log si )λi,s is a multi-valued section of H2n−1 over U∗ . Notation: J˜e is the sheaf of ENF’s. Theorem 5.2.2. νζ gives an ENF. When the base has dimension one this condition to be an ENF is equivalent to mν ∈ Je,s0
for some integer m.
The proof of Theorem 5.2.2 uses the full strength of the Clemens-Schmid exact sequence to show that (5.3) holds. Note. We are indebted to the authors of [15] for pointing out to use the close relationship between our notion of an ENF and M. Saito’s concept of an extended normal function [33]. Briefly, over S ∗ a normal function may be thought of as arising from a variation of mixed Hodge structure (VMHS). Along the discriminant locus D = S\S ∗ the condition of admissibility for a VMHS assumes a simple form for 2-step adjacent mixed Hodge structures; i.e., those for which the weight filtration has only two non-trivial adjacent terms. This is the case for normal functions and, the condition (2) above is essentially equivalent to admissibility as explained in the preprint [15]. 5.2.3 Singularities of ENF’s By definition there is over S an exact sheaf sequence 0 → Je → J˜e → G → 0 . Definition 5.2.3. sing ν is given by the support of the image of ν in H 0 (S, G).
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From the works of Cattani-Kaplan-Schmid ([10]) one is led to consider the complex given by V Bp
= H 2n−1 (Xs , Q), s ∈ U∗ M = Ni1 · · · Nip V , i1 <···
and with a Koszul-type boundary operator. Theorem.
i) There is an injective map Gs0 ⊗ Q → H 1 (B • ) , and
ii) There is an isomorphism H 1 (B ◦ ) ≈ IH 1 (Rπ2n−1 Q) . In the local normal crossing case the second isomorphism is based on the work of Cattani-Kaplan-Schmid [9] and Beilinson-Bernstein-Deligne-Gabber [3]. A proof that works also in the quasi-local normal crossing case has been shown to us by Mark de Cataldo and Luca Migliorini using their theory developed in [13]. The “purity” result of Gabber implies that the weights of Gs0 ⊗ Q are non-positive. This theorem will follow from Theorem 3 in section 4.3.2 below. Theorem 5.2.4.
i) Assuming the HC in dimension < 2n,
sing νζ = {s0 ∈ D : hζ, [W ]i = 6 0 where W n ⊂ Xs0 } . ii) In general sing νζ = {s0 ∈ D : ζs0 6= 0 in IH2n−2 (Xs0 )} . Corollary. HC ⇔ sing νζ 6= 0 for L 0. Example 5.2.5. Perhaps the simplest non-trivial example that illustrates how the singularities of a normal function are captured by the locus where H 1 (B • ) 6= 0 in the dual variety is given by a smooth cubic surface X ⊂ P3 . The dual has a stratification ˇ3 ⊃ X ˇ ⊃X ˇ1 ⊃ X ˇ2 P
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ˇ1 = X ˇ sing , X ˇ 2 = (X ˇ 1 )sing and where the pictures are as follows: where X
ˇ1 s∈X \X
Xs = (i)
Xs =
(ii)
Xs =
ˇ2 ˇ1 \ X s∈X
ˇ2 s∈X
Xs =
As will be seen from the general result quoted in the next section (
H 1 (Bs• )Q ∼ =Q 2 • ∼Q⊕Q H (B )Q = s
ˇ1 − X ˇ 2 of type (i) s∈X ˇ2 . s∈X
(5.5)
ˇ1 − X ˇ 2 of type (ii), since there the We need not consider the locus s ∈ X local monodromy is finite and we are working modulo torsion. We will think of X as the blow up of 6 points P1 , . . . , P6 ∈ P2 that are in general position with respect to lines and conics. The mapping X → P3 is given by the cubics in P3 that pass through P1 , . . . , P6 — thus H 0 (OX (1)) ∼ = H 0 (OP2 (3)(−P1 − · · · − P6 )) and we take the line bundle L on X to be OX (1). The 27 lines on X are given classically by E = blow up of Pi i Fij = image of the line through Pi and Pj G = image of the conic through P , . . . Pˇ , . . . , P . i 1 i 6 The table of intersection numbers is straightforward to write down. A piece
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of it is Ej
Fjk
Gj
Ei
−δij
δij + δ1k
1 − δij
E2
−δ2j
δ2j + δ2k
1 − δ2j
δ2j + δ2k
−δ2j
G2 1 − δ2j
ˇ the corresponding line in the For a line Λ ⊂ X we will denote by Λ⊥ ⊂ X dual projective space, so that with the obvious notation [ [ [ ˇ 1 (i) = X Ei⊥ Fij⊥ G1i . i
ij
i
For purposes of illustration to get started we consider the two Hodge classes ( ζ = [E1 − E2 ] E1 · E2 = 0 ζ 0 = [E1 − G2 ]
E1 · G2 6= 1.
The singular loci of νζ , νζ 0 are unions of the Ei⊥ , Fij⊥ , G⊥ j . We evidently have Λ⊥ ⊂ sing νζ
⇔ E1 · Λ 6= E2 · Λ
Λ⊥ ⊂ sing νζ 0
⇔ E1 · Λ 6= G2 · Λ .
From the above table we have ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ sing νζ = E1 ∪ E2 ∪ F13 ∪ · · · ∪ F16 ∪ F23 ∪ · · · ∪ F26 ∪ G1 ∪ G2 ⊥ ∪ · · · ∪ F ⊥F ⊥ ∪ · · · sing νζ 0 = E1⊥ ∪ Eζ⊥ ∪ · · · ∪ E6⊥ ∪ F13 16 23 ⊥ ∪ G⊥ ∪ · · · ∪ G⊥ · · · ∪ F26 2 6 which have degrees 12 and 18 respectively. In particular, ζ and ζ 0 are distinguished by their singular sets. To formalize this we let in general D ⊃ D1 ⊃ · · · ⊃ DN be the stratification of the discriminant locus, and we set \ D = Di I i∈I
DI0 = non-singular part of DI D0 = irreducible (= connected) components of D0 . I,λ I,λ Then 0 H 1 (Bs• )Q has constant dimension for s ∈ DI,λ .
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0 Thus we may think of H 1 (Bs• )Q as a local system VI,λ on DI,λ and, as described above, a Hodge class ζ induces a section
(sing νζ )I,λ ∈ H 0 (DI,λ , VI,λ ) . We think of this as a map ζ → sing νζ =
X
(sing νζ )I,λ DI,λ
(5.6)
I,λ
which assigns to each Hodge class the formal {VI,λ }-valued cycle as above. The HC is equivalent to the assertion that the mapping (5.6) is injective for L 0 (just how ample L must be will be discussed in section 5.4.1 below). Returning to the cubic surface, it is easy to see that in this case there ˇ 2 . Moreover, for each line is no new information in the components of X ⊥ ˇ 1 we may, by (5.5), canonically identify H 0 (Λ⊥ , VΛ⊥ ) component Λ of X with Q. When this is done, we have X sing νζ = (ζ · Λ)Λ⊥ Λ
where Λ runs over the lines on X and we have only summed over the ˇ Since any primitive Hodge class is codimension one components of X. uniquely specified by its intersection numbers with the lines, we see that for L = OX (1) the map (5.6) is injective. 5.2.4 Nodal Hypersurface Sections
†
. As s → s0 we have vanishing cycles δλ → pλ ∈ ∆s0 . The following numerical invariants reflect the topology, algebraic geometry and Hodge theory associated to the degeneration Xs → Xs0 : ρ(i) = dim {space of relations among δλ ’s} ρ(ii) = dim {image of (H2n (Xs0 ) → H2n (X)prim )} failure of pλ to impose independent ρ((iii) = dim conditions on H 0 (KX ⊗ Ln ) = h1 I∆s0 ⊗ KX ⊗ Ln , L 0 ˜ s ) − hn,n−1 (Xs ) − # double points ρ(iv) = hn,n−1 (X 0 n o ˜ s )/im Hgn−1 (X) ρ(v) = dim Hgn−1 (X 0 ρ(vi) = dim H 1 (B • ) † This section is based in part on correspondence with Herb Clemens and Richard Thomas; cf. [35]
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Theorem. ρ(i) = ρ(ii) = ρ(iii) = ρ(iv) = ρ(v) = ρ(vi). Given a smooth codimension-n subvariety W ⊂ X, for L 0 there exist nodal hypersurfaces Xs0 passing through W . Generically all of the nodes pλ will be on W (Bertini) and there is a Chern class formula for the quantities (a) h0 (IW (L)) (b) number of nodes of Xs0 . Theorem. For Xs0 general among hypersurfaces containing W , the subva˜ ] ∈ H 2n−2 (X ˜s ) riety W is uniquely determined by the fundamental class [W 0 ˜ in the canonical desingularization X ˜ s of Xs . of the proper transform W 0 0 A consequence of this result is that, for L 0, a component of the Hodgetheoretically defined variety sing νζ is equal to the locus {s0 ∈ D : there exists a unique W ⊂ Xs0 with hζ, [W ]i = 6 0} . It is in this precise sense that a Hodge class gives the equations of the dual algebraic cycles. Theorem (Clemens). For L 0 the monodromy action on the nodes pλ ∈ Xs0 , where W is fixed and Xs0 ⊃ W varies, is doubly transitive. A consequence is that for L 0 and Xs0 a general nodal hypersurface containing W ρ(i) = 1; in fact, the generating relation is X
±δλ = 0
λ
where the ± reflects a choice of orientation. From ρ(i) = ρ(vi) in the theorem above we conclude that dim IH 1 Rπ2n−1 Q = 1 where the intersection homology of the local system Rπ2n−1 Q is taken over a neighborhood of s0 . It is easy to check that if Xs0 has nodes p1 , . . . , pm that impose independent conditions on the linear system |Xs |, so that locally D = D1 ∪ · · · ∪ Dm
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where δλ → 0 along Dλ , then H 1 (Bs•0 ) ∼ = {relations along the δλ } . This is the case in the above theorem of Clemens. However, this is a very special circumstance, as illustrated by the following Example. Let X ⊂ P3 be a smooth quartic surface containing a line Λ but otherwise general. The planes containing Λ give a line ˇ3 . ˇ ⊂P Λ⊥ ⊂ X For s0 a general point of Λ⊥ , the picture of Xs as s → s0 is p1 p2 p3
δ2 δ1
δ3
ˇ locally looks like where δi → pi as s → s0 . A transverse plane slice of X
δ2 = 0
δ1 = 0
δ3 = 0
s0
ˇ is the union of three smooth hypersurfaces X ˇi This means that locally X that intersect pairwise transversely along a smooth curve, and where δi = 0 ˇ i . This is a situation where one has quasi-local normal crossings; to on X ˇ along Λ⊥ . obtain the local normal crossing picture we must blow up X In this case the condition (∗) in section 5.1.1 is satisfied. For the complex B • we have α
β
V −−−−→ N1 V ⊕ N2 V ⊕ N3 V −−−−→ N1 N2 V ⊕ N1 N3 V ⊕ N2 N3 V. Since there is one relation among the δi we have dim ker α = 4 ⇒ dim coker α = 1 ⇒ H 1 (B • ) ∼ = Q
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since β = 0. For H ⊂ X a general plane section and ζ = [4Λ − H] ∈ Hg1 (X)prim , one may check that in a neighborhood of s0 as above sing νζ = Λ⊥ . Now suppose we blow up Λ⊥ so as to obtain
δ1 = 0
δ2 = 0
δ3 = 0 E
s0
where along the exceptional divisor E we have δ1 = δ2 = δ3 = 0. Then, say, around s0 we have ( T 1 = T s1 (around δ1 = 0) T2 = Tδ1 + Tδ2 + Tδ3 = 0 (around E), where in the second relation attention must be paid to signs. The complex B • is now α
V −→ N1 V ⊕ N2 V . With suitable choice of bases and signs we will have N1 γ1 = δ1 (
N2 γ1 = 2δ1 + δ2 N2 γ2 = δ1 + 2δ2
and all other Ni γj = 0. Then ( α(γ1 ) = (δ1 , 2δ1 + δ2 ) α(γ2 ) = (0, δ1 + 2δ2 ) so that coker α = H 1 (B • ) has generator (δ1 , 0) and thus is of dimension one.
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Note: We had originally thought that under blowing up the singularities of νζ might disappear, and are grateful to Gregory Pearlstein and Patrick Brosnan for pointing out to us that this is not the case. This example is typical in that when we have the situation W ⊂ Xs0 ∈ |L| ˇ we will have as above, then in a neighborhood U in PH 0 (OX (L)) of s0 ∈ X that ˇ ∩ U = D1 ∩ · · · ∩ Dm X where the conditions in (∗) in 4.1.1 are generally satisfied. More precisely, if we consider the universal local deformation space (Kuraniski space) M for Xs0 , then in many circumstances it will be the case that the nodes pi may be independently smoothed. Denoting by Mi ⊂ M the hypersurface where the node pi remains, Ms0 = ∪ Mi forms a normal crossing divisor in M. If i
U ⊂ |L| is a neighborhood of s0 and we set Di = U ∩ Mi , then we may again generally expect that U meets each MI , so that the quasi-normal condition in 4.1.1 will be satisfied. The general result is: Let I = J ∪ K be a set of nodes with J ∩ K = ∅ and the nodes in each of J, K independent. Then H 1 (B • ) ∼ = {relations among the nodes in I} . 5.3 Infinitesimal Invariant and its Singularities 5.3.1 Definition of the Infinitesimal Invariant Recalling (5.2) we have: Definition. δν = [∇˜ ν ] ∈ H 0 Hen−1,n ⊗ Ω1S (log D) . ∇
The basic properties of δν (cf. [18] and [36]) are as follows. i) δν = 0 if, and only if, ν lifts to a locally constant section ν˜ of H2n−1 . This requires L 0, which also implies that ii) the vanishing cycles are of finite index in H 2n−1 (Xs , Z)/H 2n−1 (X, Z) . From this an argument using the Picard-Lefschetz formula gives iii) ν˜ locally constant ⇒ ν is torsion in J(Xs )/J(X). It then follows that
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iv) νζ torsion ⇒ ζ is torsion. This finally implies the mapping ζ → δνζ is injective modulo torsion, which is (ii) in Theorem 5.1.3.
5.3.2 Singularities of δν Theorem 5.3.1.
i) There is a canonical map Hen−1,n ⊗ ΩpS (log D) ∇,s0 → H p (B • )
which then allows us to define sing δνζ = image of δνζ ∈ H 1 (B • ) , and with this definition we have for L 0 sing νζ = sing δνζ . ii) For L 0 ζ 6= 0 mod torsion ⇒ δνζ 6= 0 . Corollary. HC ⇔ sing δνζ 6= 0 for L 0. Whereas νζ is an analytic invariant, δνζ is an algebro-geometric invariant; by (ii) the information in ζ is, for L 0, captured by δνζ . To sketch the basic idea of the proof, if D has quasi-local normal crossings there is a map Ress0 ΩpS (log D) → ⊕ CI I
where I = {i1 < · · · < ip } and CI is the constant sheaf supported on DI . This gives Ress0 ΩpS (log D) ⊗ He → ⊕(He,s0 )I I
where (He,s0 )I = He,s0 ⊗C CI . If ν˜ is a local multi-valued lifting on ν, then by definition of an ENF, for some integer m we have m(Ti − I)˜ ν ∈ Im (Ti − I)H 2n−1 (Xs0 , Z) where Ti − I is “analytic continuation around Di ”. This relation translates into mRes s0 (∇˜ ν ) ∈ ⊕(Ti − I)HZ,e,s0 . i
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so that from the complex HZ,e;s0 → ⊕(Ti − I)HZ,e;s0 → ⊕ (Ti − I)(Tj − I)HZ,e;s0
(∗)
i<j
we obtain mRes s0 (∇˜ ν ) ∈ H 1 (∗) . Now ν˜ is well-defined up to ν˜ → ν˜ + f + λ
(5.7)
P where f ∈ Fen and λ ∈ HZ,e;s0 . The contribution of λ is (Ti − I)λ which gives a coboundary which This disposes of the ambiguity λ in (5.7). The ambiguity f disappears because ∇f = 0 in the complex (5.2). Next, we need to replace (Ti − I) by Ni = (Ti − I)Ai where Ai is an invertible matrix defined over Q and where all the Ai commute among each other. So over Q the new complex has the same cohomology as the complex (∗). At this stage we have, over Q, essentially described the definition of the map sing ν → H 1 (Bs00 )Q . Next we define the subcomplex df1 dfk • N1 He , . . . , Nk He ⊂ Ω•S (log D) ⊗ He , ΩS f1 fk where the differential on the subcomplex is given by ! X dfi ∧ Ni . fi i
There is then a map of complexes He −−−−→ Res y s0
Ω1S −−−−→ Ω2S Res y s0
He,s0 −−−−→ ⊕ Ni He,s0 −−−−→ i
dfk df1 f1 N1 He , . . . , kk Nk He
Res y s0 ⊕ Ni Nj He,s0
i<j
under which δν maps to sing ν in H 1 (Bs•0 )Q . This is the construction of Theorem 3.
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5.4 Issues and Deeper Structures 5.4.1 The Effective Hodge Conjecture (EHC) Let X be a smooth variety, of any dimension for the moment, L → X a very ample line bundle with Chern class λ = c1 (L), and ζ ∈ Hgp (X) . Since torsion consideration will be important in this section we shall use Z coefficients. The (HC) is equivalent to the statement (1) There exist integers k0 , m0 such that for m = m0 k0 ζ + m(λ)p = [Z]
(5.8)
where Z is an effective, integral algebraic cycle. Indeed, if k0 ζ = [Z 0 − Z 00 ] where Z 0 , Z 00 are effective cycles, then writing X Z 00 = ni Zi , ni ∈ Z , where the Zi are irreducible of codimension p, by passing hypersurfaces of high degree through each Zi we will have Zi + Wi = H1 ∩ · · · ∩ Hp for a subvariety Wi , which then gives [−Zi ] = lλp + [Wi ] for some integer l, from which (1) follows. We note that if (1) holds for m = m0 , then it also holds for m = m0 . The (EHC), in various forms to be discussed below, asks for (1) with estimates on k0 , m0 . We are grateful to the referee for pointing out to us the following result of Koll´ar, which illustrates the care that must be taken in consideration of the torsion coefficient k0 . We consider the space M of smooth hypersurfaces X ⊂ Pn+1 ,
n=3
of degree d = n + 1. Then Hgn−1 (X) ∼ = H 2n−2 (X, Z) ∼ =Z. Denoting by Mk ⊂ M
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the subvariety of X’s containing a curve of degree k, Koll´ar [29, 32] showed that if (k, d) = 1 and d is divisible by a prime power p3 where p > n, then Mk is a proper subvariety; and Soul´e-Voisin [32, § 2] further remark that S (k,d)=1 Mk is dense in M. Of course, for any integer l Mdl = M . One conclusion is that the torsion coefficient k0 in (5.8) could have subtle dependence on X. Now we return to the main situation in this paper where dim X = 2n and we are considering a primitive Hodge class ζ. If ζ = [Z] for an integral algebraic cycle Z, then it follows from results of Kleiman [28] that we will have (n − 1)!ζ + mλm = [W ],
m = m0 ,
(5.9)
where W is a smooth, codimension n subvariety. In fact, we may take W to be the degeneracy classes of general sections σ1 , . . . , σr−1 of a very ample rank r vector bundle F → X. This implies that the normal bundle NW/X is ample, and by a result of Fulton-Lazarsfeld (Annals of Math. 118 (1983), 35–60) cn (NW/X ) > 0 .
(5.10)
When n is odd, so that −ζ 2 > 0 by the Hodge-Riemann bilinear relation, (5.9) and (5.10) give r 2 1 2n −ζ m0 = , n ≥ 2. (n − 1)! λ2n
(5.11)
This suggests that any estimate on m0 in (5.8) must involve ζ 2 . In fact, for n = 1, where the above relation does not make sense, it follows from known results that The (EHC) (5.8) holds for k0 = 1 and where we may take m0 = −ζ 2 + C(ζc1 (X), λc1 (X), c21 (X), c2 (X))
(5.12)
where C is a universal linear combination of the constants listed. Moreover, we shall give an heuristic argument that there exist divisors in surfaces X for which a lower bound (5.11) holds, up to constants.
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The last statement follows by considering the case of quartic surfaces X ⊂ P3 . Letting H be a hyperplane class, we first note that there is no uniform m0 such that for every X and ζ ∈ Hg1 (X)prim ζ + m0 H is effective.
(5.13)
In fact, by considering X for which N S(X) is generated by H plus a noncomplete intersection curve of degree d we see that There is no uniform m such that for every X there are curves W1 , . . . , Wρ in X that span N S(X) and have degree 5 m. Thus there is no uniform m for which (5.13) holds for all X. Next, to see why the lower bound should hold we consider the following statement. Any estimate on m0 in (5.13) must in general involve the lengths of the shortest spanning vectors in Hg1 (X, Z)prim . (5.14) The heuristic argument for (5.14) is based on the following quite plausible (and possibly known) statements: Let Λ = H 2 (X, Z) and Λ0 = {ζ ∈ Λ : ζ · H = 0}. Let P ⊂ Λ be the vectors that are not divisible by any n ∈ Z, n 6= ±1 (these are primitive in a different sense of the term). Then i) there exists ζn ∈ P ∩ Λ, with ζn2 → −∞ ii) there exists a polarized Hodge structure Hn on ΛC with ζn ∈ Hg1 (Hn ) iii) in (ii) we may arrange that the Picard number ρ(Hn ) = 2 (iv) there exists a (possibly singular) Xn ⊂ P3 such that Hn is a direct ˜ n of X. summand of the Hodge structure on a desingularization X ˜ n with Let λ = [H] and Z be an irreducible curve on X a, b ∈ Z .
[Z] = aλ + bζn
Then deg Z = a. By adjunction, since Z is irreducible the arithmetic genus π(Z) satisfies Z ·Z 0 5 π(Z) = +1 2 while Z · Z = 4a2 + b2 ζn2 which implies p a=
−ζn2 . 2
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Returning to (5.11) which only gives information for odd n, for the first even case n = 2 for purposes of illustration we assume that there exists a rank-two vector bundle E → X with c2 (E) = ζ c1 (E) = aλ . Setting as usual E(m) = E⊗Lm , let m0 be such that there is σ ∈ H 0 (OX (E(m0 ))) ∨ ∼ with (σ) = W where NW/X = E(m0 )|W is ample. Then we claim that p m20 + bm0 > c ζ 2
(5.15)
where b, c are constants depending only on X, E and L. Proof By another result due to Fulton-Lazarsfeld (loc. cit.), since NW/X is ample we have c1 (NW/X )2 > c2 (NW/X ) , from which(5.15) follows. This again suggests the possibility of there being, in general, a lower bound on m0 for which (5.8) holds in terms of |ζ|2 . This possibility is reinforced by the following considerations: Let M be a quasi-projective algebraic variety parametrizing a family of smooth projective X’s with reference variety X0 ∈ M. For example, M could be a moduli space if such exists. Letting U be a sufficiently small neighborhood of X0 and ζ ∈ Hgn (Xc )prim , the locus Uζ = U ∩ {X ∈ U : ζ ∈ Hgn (X)prim } of nearby points where ζ remains a Hodge class is an analytic variety. By a theorem of Cattani-Deligne-Kaplan [8] it is part of an algebraic subvariety Mζ ⊂ M . We shall write points of Mζ as (X, ζ) to signify that there is a Hodge class ζ extending the one defined over Uζ , where we may have to go to a finite covering to make ζ single-valued. For each k, m with m > 0, k 6= 0 we consider the subvarieties Mk,m = {(X, ζ) ∈ Mζ : kζ + mλn = [Z]} where Z is an effective algebraic cycle.
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Assuming the (HC) we have [
Mk,m = Mζ .
k,m
It follows that the LHS is a finite union; thus Mζ =
N [
Mki ,mi .
i=1
Letting k0 , m0 be multiples of all the ki , mi respectively, and using that for a positive integer a Mk,m ⊂ Mak,am we have that Mk0 ,m0 = Mζ
(5.16)
from which we conclude: If the (HC) is true, then (5.8) holds for a uniform k0 , m0 when (X, ζ) varies in an algebraic family. Now suppose we now let the Hodge class ζ vary. Then on the one hand, for each positive constant c we shall give an heuristic argument that [ Mζ = Mc is an algebraic subvariety of M. (5.17) |ζ|2 5c
On the other hand, typically [
Mζ is dense in M .
(5.18)
ζ
Letting kζ , mζ be integers such that (5.16) holds with k0 = kζ , m0 = mζ we will then have kζ , mζ are bounded if |ζ|2 < c (5.19) kζ , mζ are not bounded for all ζ . This again suggests the possibility of a lower bound on mζ in terms of |ζ|2 . A proof of (5.17) follows from [8]. Here we give a slightly different way of proceeding, anticipating some possible consequences of the recent work [27]. Heuristic argument for (5.17): Let D be the classifying space for polarized Hodge structures of the same type as H 2n (X0 , Z)prim /mod torsion. Thus we are given a lattice with integral non-degenerate quadratic form (HZ , Q) and
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D consists of all Hodge-type filtrations {F p } on HC satisfying the HodgeRiemann linear relations. There is a period mapping ϕ : M → Γ\D where Γ = Aut(HZ , Q). By the work of Kato-Usui [27], it is reasonable to expect that there will be a partial compactification (Γ\D)Σ such that ϕ extends to ϕ : M → (Γ\D)Σ
(5.20)
for a suitable compactification of M of M. Here, Σ stands for a set of fans that arise in the work of Kato-Usui (loc. cit.) Now let HZprim be the lattice vectors ζ that are primitive in the arithmetic sense; i.e., ζ is only divisible by ±1 in HZ . Then it is known that There are only finitely many Γ orbits in HZprim with fixed Q(ζ, ζ). (5.21) For each ζ ∈ HZprim we let Dζ = {F 0 } ∈ D : ζ ∈ F n is a Hodge class . Then, by (5.21), Dζ projects to a closed analytic subvariety (Γ\D)ζ ⊂ Γ\D . Analysis similar to that in Cattani-Deligne-Kaplan (loc. cit.) suggests that (Γ\D)ζ extends to a closed log-subvariety (Γ\D)ζ ⊂ (Γ\D)Σ . Then Mζ = ϕ −1 ((Γ\D)ζ ) will be an algebraic subvariety (which, as noted above, we know by CattaniDeligne-Kaplan) and essentially because of (5.21), there are only finitely many such Mζ ’s with |ζ|2 5 c. Summary. i) There is heuristic evidence that any bounds on k0 , m0 such that (5.8) holds will depend on |ζ|2 , together with quantities a|ζ|+ b, where a, b are constants independent of ζ. ii) For n = 1 we may take k0 = 1 and there is an upper bound (5.12) on m0 . For a general surface X, this bound is sharp. In a subsequent work, we shall show that obtaining an estimate on codim(sing νζ ) requires that we let X vary in its moduli space and consider the NoetherLefschetz loci. Heuristic reasoning then suggest the following formulation of an effective (HC)
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(EHC): There is a relation (5.8) where the constants k0 , m0 depend on |ζ 2 |, aζ + b, and universal characters constructed from H ∗ (X) and H ∗ (M) that do not depend on ζ.
5.4.2 Localization of Primitive Cohomology along the Singular Locus The central issue is that δνζ — but not νζ — can be defined for any class ζ ∈ H n,n (X)prim ⊂ H n (ΩnX ) ,
(5.22)
and then as a consequence of Theorem 3 ζ ∈ Hgn (X)⊥ ∩ H n,n (X)prim ⇒
sing (δνζ ) = 0
for L 0 .
Thus, any existence result involving δνζ must involve the condition that ζ be an integral class, or equivalently that νζ exist. Roughly speaking the residues of δνζ must be integral in order to be able to “integrate” and enable us to define R νζ = “ δνζ ”. This brings us to the Question 5.4.1. Given ζ as in (5.53), how can we tell if ζ ∈ H 2n (X, Q) — i.e. Z ζ∈Q, Γ ∈ H2n (X, Z) ? Γ
It turns out that there is a very nice geometric structure underlying this question. It is based on two principles ˇ the dual variety of X and by H → X ˇ the 1st Principle: Denoting by X hyperplane bundle, the group H n (ΩnX )prim may be expressed globally along the singular locus ˇ ∆⊂X ×X by the failure collectively of the ∆s = ∆ ∩ Xs × {s}, s ∈ D, to impose independent conditions on |KX ⊗ Ln ⊗ H n |. Here we are thinking algebraically with H n (ΩnX ) being defined in the Zariski topology.
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2nd Principle: For p ∈ ∆s there is a map KX,p ⊗ Lnp → C , well-defined up to sign and given by ω ω → ±Res p n , s
ω ∈ KX,p ⊗ Lnp ,
where ω ˜ is any extension of ω to a neighborhood and s ∈ H 0 (OX (L)) defines Xs . Remark. This leads to an integral structure expressed by (5.27) below. The injection (5.22) also arises from a canonical section 2 η ∈ H 0 O∆ KX ⊗ L2n ⊗ H 2n and we may think of the image of Zζ in (5.17) as being √ Z η ⊂ O∆ (KX ⊗ Ln ⊗ H n ) . The section η is constructed as follows: At points p ∈ ∆, the universal section s = quadratic + (higher order terms) and the quadratic terms give a canonical symmetric map ∗ TX,p → TX,p ⊗ Lp ⊗ Hp
which by exterior algebra induces (recalling that dim X = 2n) ∗ 2n Λ2n TX,p → Λ2n TX,p ⊗ L2n p ⊗ Hp
and then we obtain ∗ η(p) ∈ Λ2n TX,p
2
2n ⊗ L2n p ⊗ Hp
with the property that η(p) 6= 0 ⇔ p is a node. Globalizing over X × S, this map gives an injection of sheaves √ Z η → O∆ (KX ⊗ Ln ⊗ H n ) .
(5.23)
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Combining these two principles leads to a commutative diagram for L 0 H n (ΩnX )prim ∼ y=
H 0 (O∆ (KX ⊗ Ln ⊗ H n )) ≈ −→ H 1 (I∆ ⊗ KX ⊗ Ln ⊗ H n ) H 0 (OX×S (KX ⊗ Ln ⊗ H n )) S
(5.24)
Λ The horizontal isomorphism is the standard long exact cohomology sequence arising from 0 → I∆ → OX×S → O∆ → 0 and using L 0, and Λ is the subgroup arising from (5.23) and the numerator in (5.24) under the horizontal isomorphism there. The vertical isomorphism is more interesting. It uses the Koszul complex associated to ds ∈ H 0 (OX×S (Σ∗ ⊗ L)) where the prolongation bundle Σ arises from 0 → Ω1X×S → OX×S (Σ) → OX×S → 0 with extension class c1 (L ⊗ H), and the vanishing theorems necessary to have the isomorphism require L 0 to ensure Castelnuovo-Mumford type of regularity. This isomorphism is constructed purely algebraically. The Leray spectral sequence applied to the universal family X ⊂X ×S . &π X S lead to a spectral sequence which, when combined with (5.24), gives a diagram α H 1 (I∆ ⊗ (KX ⊗ Ln ⊗ H n )) −→ H 0 Rπ1 I∆ ⊗ (KX ⊗ Ln ⊗ H n ) ko
H n (Ωn )prim S Hgn (X)prim Theorem. Combining (5.23) and (5.24) we have Hgn (X) → image of Λ in H 0 Rπ1 I∆ ⊗ KX ⊗ Ln ⊗ H n ; i.e., Hodge classes have integral residues.
(5.25)
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Then it may be shown that (this is essentially ρ(iii) = ρ(v) in section 5.2.4 above) HC ⇒ α is injective on Hgn (X)prim .
(5.26)
From (5.24) a consequence is that a class ζ ∈ H n (ΩnX )prim is integral ⇔ the residues of ζ are integral (5.27) The spectral sequence argument also gives H 1 Rπ0 I∆ ⊗ (KX ⊗ Ln ⊗ H n ) = 0 for L 0 ⇒ α is injective ⇒ HC.
(5.28)
The statements (5.26) and (5.28) give precise meaning to the general principle: The HC may be reduced to (in fact, is equivalent to) a statement about the global geometry of ∆⊂X ×S .
(5.29)
We thus have: HC ⇔ geometric property of (5.29) when L 0. Above we have discussed the question: Can we a priori estimate how positive L must be? The condition L 0 in this section requires sufficient positivity to have vanishing of cohomology plus Castelnuovo-Mumford regularity. Above, we gave a heuristic argument to the effect that for each ζ the condition L 0 must also involve ζ 2 . ˇ k the dual variety to the image of Discussion: Denote by X ˇ 0 (OX (Lk )) . X → PH One may ask the question ˇ k,sing of X ˇ k for k 0? What are the properties of the singular set X Although we shall not try to make it precise, one may imagine two types ˇ 0 (OX (L)) of singularities: (i) Ones that are present for a general X ⊂ PH having the same numerical characters as X; in particular, they should be invariant as X varies in moduli. (In this regard, one may assume that L → X is already sufficiently ample so as to have those vanishing theorems that will ˇ k can be computed from the numerical characters of X1 ). ensure that dim X (ii) Ones that are only present for special X. What our study shows is that:
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If the HC is true, then non-generic singularities of type (ii) are necessarily present when Hgn (X)prim 6= 0. One may of course ask if singularities of type (ii) are caused by anything other than Hodge classes?
5.4.3 Remarks on Absolute Hodge Theory Recent works [19] on Hodge-theoretic invariants of algebraic cycles have shown that in codimension = 2 arithmetic aspects of the geometry — specifically the spread of both X and of cycles in X — must be taken into account and this might be a consideration for an effective HC. In considering spreads a central issue is that one does not know that A Hodge class is an absolute Hodge class.
(5.30)
That is, for X defined over a field k of characteristic zero, a class in H n (ΩnX(k)/k ) (sheaf cohomology computed algebraically in the Zariski topology) that is a Hodge class for one embedding k ⊂ C using H n ΩnX(k)/k ⊗ C ∼ = H n ΩnX(C) (GAGA) is a Hodge class for any embedding of k in C (here we also assume ¯ We shall refer to the statement (5.30) as absolute Hodge (AH). that k = k). We close by remarking that the above geometric story works over any algebraically closed field of characteristic zero — in particular one has the diagram (5.24) and integrality conditions on H n (ΩnX(k)/k ) given by the image of Λ in H 0 (Rπ1 I∆ ⊗ KX ⊗ Ln ⊗ H n ). For any embedding k ⊂ C such that the (well-defined) map Hgn (X)prim → Λ is injective (which is implied by the HC), one has a direct geometric “test” n n for when a class in H ΩX(k)/k is in H 2n (X(C), Q). Remark. We shall give a very heuristic argument to suggest that AH ⇒ HC .
(5.31)
The reasoning is as follows. i) Assuming AH, the statement of HC is purely algebraic; ii) When p = 1 the HC is true, and although the existing proofs both
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use transcendental arguments, by model theory there will be a purely algebraic proof of the algebraic statement sing νζ 6= ∅ for L 0 ,
(5.32)
which is equivalent to the HC; iii) Finally (and most “heuristically”), because the geometric picture of the structure of sing νζ is “uniform” for all n — in contrast, for example, to Jacobi inversion — any purely algebraic proof of (5.32) that works for n = 1 will work for all n.
5.5 The Poincar´ e Line Bundle Given a Hodge class ζ ∈ Hgn (X)prim there is an associated analytic invariant νζ ∈ H 0 (S, Jf E ) and its singular locus sing νζ ⊂ D . Although the local behaviour of νζ and subsequent local structure of sing νζ can perhaps be understood, the direct study of the global behaviour of νζ and of sing νζ — e.g., is sing νζ 6= ∅ for L 0 — seems of course to be more difficult. In this section we will begin the study of potentially important global invariants of νζ obtained by pulling back canonical line bundles (or rather line bundle stacks) that arise from the polarizations on the intermediate Jacobians J(Xs ). We shall do this only in the simplest non-trivial case and there we shall find, among other things, that ∗ (Poincar´e line bundle) = ζ 2 . c1 νζ×ζ This is perhaps significant since as we have given in section 5.4.1 an heuristic argument to the effect that any lower bound estimate required for an EHC will involve ζ 2 .
5.5.1 Polarized Complex Tori and the Associated Poincar´ e Line Bundle The material in this section is rather standard; see for instance [25, Ch. 2]. We shall use the notations • V is a complex vector space of dimension b, • Λ ⊂ V is a lattice of rank 2b. • T = V /Λ is the associated complex torus.
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We then have, setting ΛC = Λ ⊗Z C, ΛC = V ⊕ V¯ where the conjugation is relative to the real structure on ΛC . There are canonical identifications (i)
H p,q (T ) ∼ = Λp V ∗ ⊗ Λq V¯ ∗
(ii)
(Λp V ∗ ⊗ Λq V¯ ∗ )∗ ∼ = Λb−p V ∗ ⊗ Λb−q V¯ ∗
(5.33)
where (ii) is given by Z ϕ⊗ψ →
ϕ∧ψ . T
Definition. A polarization on T is given by a non-degenerate, alternating bilinear form Q:Λ⊗Λ→Z which, when extended to ΛC , satisfies Q(V, V ) = 0 .
(5.34)
The polarization is principal in case det Q = ±1. We shall see that a polarization gives a holomorphic line bundle M →T , well-defined up to translation, and satisfying c1 (M )b [T ] 6= 0 . We shall also see that Q defines an Hermitian metric in M whose Chern form is expressed as √ −1 X hi¯j dv i ∧ d¯ vj c1 (M ) = 2π where v1 , . . . , vb ∈ V ∗ gives a basis and ( ¯¯ hi¯j = h ji det khi¯j k = 6 0. Thus the Hermitian matrix khi¯j k is non-degenerate but, in contrast to the usual terminology, we do not require that it be positive or negative definite.
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We shall give two constructions of M . For the first, assuming as we shall always do that the polarization is principal, we may choose a Q-symplectic basis x1 , . . . , xb ; y 1 , . . . , y b for Λ∗Z so that X Q= dxi ∧ dy i . i
Thinking of Λ as H1 (T, Z) and with the canonical identification H 2 (T, Λ) = Hom(Λ2 H1 (T, Z), Z) we have Q ∈ H 2 (T, Z) . By (5.34), when expressed in terms of dv 1 , . . . , dv b , d¯ v 1 , . . . , d¯ v b we have that √ −1 X vj Q= hi¯j dv i ∧ d¯ 2π where the matrix khi¯j k is Hermitian and non-singular. Thus Q ∈ Hg1 (T ) , and since T is a compact K¨ahler manifold it is well-known that there exists a holomorphic line bundle M → T with a Hermitian metric and with Q = c1 (M ) being the resulting Chern form. It is also well-known that the subgroup Pic0 (T ) of line bundles with trivial Chern class has a canonical identification Pic0 (T ) ∼ = V¯ ∗ /Λ∗
(5.35)
and that the action of T on Pic0 (T ) by translation is the natural linear algebra one using the above identification. Thus, M is uniquely determined by c1 (M ) up to translation. Before doing that we want to recall the construction of the Poincar´e line bundle P → T × Pic0 (T ) . For this we have the canonical identification H1 (T × Pic0 (T ), Z) ∼ = Λ ⊗ Λ∗ and from this the canonical inclusion Hom(Λ∗ , Λ∗ ) ⊂ H 2 (T × Pic0 (T ), Z) .
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The class T in H 2 (T ×Pic0 (T ), Z) corresponding to the identity in Hom(Λ∗ , Λ∗ ) is easily seen to lie in Hg1 (T × Pic0 (T )) and therefore defines a class of line bundles P → T × Pic0 (T ) . We may uniquely specify P by requiring that both P | T × {e◦ } P | {e} × Pic0 (T ) are trivial, where e, e◦ are the respective origins in T, Pic0 (T ). The map Pic0 (T ) → Pic0 (T ) given by a◦ → P | T × {a◦ },
a◦ ∈ Pic0 (T )
is the identity. The above construction gives what is usually called the Poincar´e line bundle. However, for the purposes of this work we assume given a principal polarization Q in T and will canonically define a line bundle PQ → T × T
(5.36)
which will induce an isomorphism T ∼ = Pic0 (T ) by a → PQ | T × {a},
a∈T
and via this isomorphism the identification P ∼ = PQ . Definition. Denoting by µ:T ×T →T the group law, the Poincar´e line bundle (5.36) is defined by PQ = µ∗ M ⊗ p∗1 M ∗ ⊗ p∗2 M ∗ ⊗ Me
(5.37)
where the pi : T × T → T are the coordinate projections of M → T in any line bundle with c1 (M ) = Q and Me is the fibre of M over the identity.
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Proof that (5.37) is well defined: Denote by PM the RHS of (5.37) . Since PM ⊗R = PM ⊗ PR we have to show that for a line bundle R → T PR = 0 if c1 (R) = 0 .
(5.38)
In fact we will show that PR is canonically trivial. We will check that c1 (R) = 0 ⇒ c1 (PR ) = 0 .
(5.39)
Assuming this we have PR ∈ Pic0 (T × T ) ∼ = Pic0 (T ) ⊕ Pic0 (T ) . Then, by definition, for a, a0 ∈ T (PR )(a,e) = Re∗ (PR )(e,a0 ) = Re∗ so that the two “coordinates” of PR are zero, hence PR is trivial. To make the trivialization canonical we need to show independence of scaling, and this is the role of the Me factor. To verify (5.39), in general we may choose coordinates xi , y i ∈ Λ∗Z so that any line bundle R has c1 (R) represented by X λi dxi ∧ dy i . i
Using coordinates (xi , y i , x0 j , y 0 j ) on ΛR ⊕ ΛR and using that ˙ 0 , y 0 )) = (xi + x0 i , y i + y 0 i ) µi ((x, y)+(x ˙ is the group law on T , it follows that c1 (PR ) is represented by where + X i i λi (dxi ∧ dy 0 + dx0 ∧ dy i ) . i
In particular, if the λi = 0 then (5.39) follows. Remark. For later use we note for Q as above X i i c1 (PQ ) = dxi ∧ dy 0 + dx0 ∧ dy i . i
In particular c1 (PQ )2b [T × T ] = 2b .
(5.40)
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5.5.2 Topological Properties of the Poincar´ e Line Bundle in Smooth Families We let X and X ⊂X ×S p y S be as above. For simplicity of exposition we assume that for a general point s ∈ S all of H 2n−1 (Xs ) is primitive, so that the intermediate Jacobian J(Xs ) = F n H 2n−1 (Xs )\H 2n−1 (Xs )/H 2n−1 (Xs , Z) ∼ ˇ 2n−1 (Xs )/H2n−1 (Xs , Z) = F nH has a principal polarization as discussed in the preceding section. In this section we will assume the existence of a smooth curve B ⊂ S such that all the Xs , s ∈ B, are smooth. This is a very rare circumstance, but one that will help to prepare the way for the treatments below of the case when B is a general curve in S. We set XB = p−1 (B) and denote by JB → B the smooth analytic fibre space of complex tori with fibres J(Xs ), s ∈ B. Then (1) There exists a complex line bundle stack MB → JB whose restriction to each fibre gives a line bundle, defined up to translation by a point of finite order, and whose Chern class is the polarizing form. The meaning of the term “stack” in the present context will be explained below. (2) There exists a complex line bundle PB → JB ×B JB whose restriction to each fibre of JB ×B JB → B is the Poincar´e line bundle. The point is that in each case the Chern classes c1 (MB ), c1 (PB ) ∈ H 2 (B, Q) may be defined. We let D be the classifying space for polarized complex tori and T→D
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the universal family of these tori. Then there is a line bundle M → T ×D T y
(5.43)
D such that over each T × T ∼ = T × Pic0 (T ) the line bundle M restricts to be the family of all principally polarized line bundles whose Chern class is the polarizing form, as explained in the preceding section. We denote by Me the restriction of M to T ×D {e} ∼ = T. ∼ Let ΓZ = Sp(b, Z) be the arithmetic group associated to the above situation. Then D/ΓZ is an analytic variety whose points are in one-to-one correspondence with equivalence classes of principally polarized complex tori. As usual in the theory of stacks, there is no universal family of complex tori over all of D/ΓZ , although the quotient Cb × D/Z2b × ΓZ exists as a family of complex tori over the automorphism-free ones. Given XB → B as above, letting Γ ⊂ ΓZ be the monodromy group we have the picture JB y
(5.44) ϕ
B −→ D/Γ which one thinks of as the family of complex tori that would be induced by pulling back the universal family if the latter existed. Turning to (5.56), there is an action of Z2b × ΓZ on D × Cb × Cb × C that would represent the descent of (5.56) to D/ΓZ were it not for the presence of automorphisms. In addition, it can be shown that for γ ∈ ΓZ , Z2b × {γ} maps Me to Ma(γ) where Ma(γ) | T = M | T × {a(γ)} where a(γ) is a division point in T . (This is well-known phenomenon for principally polarized abelian varieties.) Turning to (5.44), there will be a subgroup Γ0 ⊂ Γ of finite index and such that a(γ) = e for γ ∈ Γ0 . Let ˜ → B be a finite covering such that (5.44) lifts to π:B JB˜ y ϕ ˜ ˜ −→ B D/Γ0 .
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Then Me is invariant under Γ0 and induces a line bundle ϕ˜∗ (Me ) −→ JB˜ y ˜ B that on each fibre induces a line bundle whose Chern class is the polarizing π ˜→ form. If d is the degree of B B, then even though MB is not defined, we may set 1 c1 (MB ) = c1 (π∗ (ϕ˜∗ (Me )) ∈ H 2 (B, Q) . (5.45) d One may check that this is independent of the choice of covering ˜ → B. B The discussion of the Poincar´e line bundle is similar but easier since it is uniquely characterized by the properties discussed in the preceding section. Having defined MB and PB we now consider normal functions νζ viewed as cross-sections of
"
JB " νζ B
and ask:
∗ What is the dependence of c1 (νζ∗ (MB )) and c1 (νζ×ζ 0 (PB )) 0 on ζ, ζ ?
Here, the Chern classes are considered as rational numbers using H 2 (B, Q) ∼ = Q. To discuss this question we set Zζ = νζ (B) − ν0 (B) and define the quantities (i)
∗ ∗ Q1 (ζ, ζ 0 ) = c1 (νζ+ζ 0 (MB )) − c1 (νζ (MB )) − c1 (νζ∗0 (MB )) + c1 (ν0∗ (MB ))
(ii)
∗ ∗ Q2 (ζ, ζ 0 ) = c1 (νζ×ζ 0 (PB )) − c1 (ν0×0 (PB )) 0 ∗ ∗ ∗ 0 Q3 (ζ, ζ ) = p1 [Zζ ] ∪ p2 [Zζ ] ∪ c1 (µ (MB )) .
(iii)
(5.46)
Here in (iii), we are working in the cohomology ring of JB ×B JB and the pi are the projections onto the two coordinate factors. We remark that: The definition of Q1 (ζ, ζ 0 ) is motivated by the property ˙ 0 ) − (a) − (a0 ) + (e) ∼ 0 (a+a
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˙ is the group law on E, (b) is the 0-cycle on an elliptic curve E, where + associated to a point b ∈ E and ∼ is linear equivalence. Theorem 5.5.1. Q1 (ζ, ζ 0 ) = Q2 (ζ, ζ 0 ) = Q3 (ζ, ζ 0 ) = ζζ 0 . The last term is to be taken as the value of the ζ ∪ ζ 0 on [X]. Since for ζ 6= 0 primitive, (−1)n ζ 2 > 0 we have the following curious ∗ (P ), we have for m ∈ Z Corollary 5.5.2. Setting PB,ζ = νζ×ζ B (±)
h0 (PB,mζ ) = m2 (ζ 2 ) + (terms not depending on m) . where ± is the parity of n. Sketch of the proof: The real dimension of JB is 4b + 2, and denoting the Leray filtration by FLp we have that the fundamental class [Zζ ] ∈ FL1 H 2b (JB ) → H 1 (B, R2b−1 ZJB ) where we use Z coefficients throughout and R2b−1 ZJB is the (2b − 1)st direct image of Z under the map JB → B. Here and below the notation means that 1 H 2b (J ) ∼ H 1 (B, R2b−1 Z ). [Zζ ] ∈ FL1 H 2b (JB ) which then maps to GrL B = JB Denoting by JB the sheaf of holomorphic sections of JB → B we have JB =
R2n−1 CXB R2b−1 CJB ∼ = Fn R2n−1 CXB + R2n−1 ZXB Fb R2b−1 CJB + R2b−1 ZJB
where R2n−1 CXB is understood to be OB ⊗ R2n−1 CXB and R2n−1 CXB is R2n−1 C for the projection p : XB → B. Now ζ → νζ ∈ H 0 (B, JB ) is linear in ζ, and we have H 0 (B, JB ) → H 1 (B, R2n−1 ZXB ) ok H 1 (B, R2b−1 ZJB ) where in the top row νζ → ζ ∈ FL1 H 2n−1 (XB ) → H 1 (B, R2n−1 ZXB ) and under the vertical isomorphism ζ → [Zζ ] ∈ FL1 H 2b (JB ) → H 1 (B, R2b−1 ZJB ) .
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Thus this mapping is linear in ζ. For simplicity of notation we set = νζ∗ (MB )
Mζ
∗ = νζ×ζ 0 (PB )
Pζ×ζ 0
∗ ∗ ∗ ∗ ∗ ∗ = νζ×ζ 0 (µ MB ⊗ p1 MB ⊗ p2 MB ⊗ M0 )
where MB,0 → B is the line bundle stack whose fibres are the fibres of MB over the 0-section and µ is the fiberwise addition map. Then we have c1 (Pζ×ζ 0 ) = c1 (Mζ+ζ 0 ) − c1 (Mζ ) − c1 (Mζ 0 ) + c1 (M0 ) .
(5.47)
The first step is to analyze p∗1 [Zζ ] ∪ p∗2 [Zζ 0 ] .
(5.48)
Since cup product is Poincar´e dual to intersection on a smooth manifold, and since p∗1 [Zζ ] is the cycle traced by {νζ (s) × Js − ν0 (s) × Js }s∈S and similarly for ζ 0 , we see that (5.48) is Poincar´e dual to the cycle traced out by {νζ (s) × νζ 0 (s) − νζ (s) × ν0 (s) − ν0 (s) × νζ 0 (s) + ν0 (s) × ν0 (s)}s∈S . Call this cycle Zζ×ζ 0 , so that p∗1 [Zζ ] ∪ p∗2 [Zζ 0 ] = [Zζ×ζ 0 ] . For the second step, since [Z
ζ×ζ 0
∗
] ∪ c1 (µ MB ) =
Z
µ∗ c1 (MB )
Zζ×ζ 0
where the RHS is the sum with signs of the values of µ∗ c1 (MB ) on the four curves in the definition of the cycle Zζ×ζ 0 , we have from (5.47) that p∗1 [Zζ ] ∪ p∗2 [Zζ 0 ] ∪ c1 (µ∗ MB ) = c1 (Pζ×ζ 0 ) .
(5.49)
For the next step, since ζ and ζ 0 are primitive and thus live in FL1 H 2n (XB , Z) where FL is the Leray filtration, they define classes [ζ], [ζ 0 ] ∈ H 1 (B, R2n−1 ZXB ) 1 = F 1 /F 2 . As above, the notation R2n−1 Z 2n−1 Z for the in GrL XB means Rp L L projection p : XB →B. We then have
H 1 (B, R2n−1 ZXB ) ⊗ H 1 (B, R2n−1 ZXB ) → H 2 (B, R4n−2 ZXB ) ∼ = Z , (5.50)
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where the last isomorphism uses R4n−2 ZXB ∼ = Z and H 2 (B, Z) ∼ = Z. It is known that Under the mapping (5.50), [ζ] ⊗ [ζ 0 ] → ζ · ζ 0
(5.51)
Passing to JB → B, we have by definition R2b−1 ZJB ∼ = R2n−1 ZXB and that the image of [Zζ ] in H 2b (JB , Z) ∼ = H 1 (B, R2n−1 ZJB ) corresponds to the image of ζ in Gr1 H 2b (XB , Z) ∼ = H 1 (B, R2n−1 ZXB ). Moreover, since the polarization is principal we have Qb−1 : R1 ZJB ∼ = R2b−1 ZJB . Thus [Zζ ] defines a class [Zζ ]Q ∈ H 1 (B, R1 ZJB ) , and it may be shown from (5.50) that under the pairing H 1 (B, R1 ZJB ) ⊗ H 1 (B, R2b−1 ZJB ) → H 2 (B, R2b ZJB ) ∼ =Z,
(5.52)
where the last isomorphism results from R2b ZJB ∼ = Z and H 2 (B, Z) ∼ = Z, we have in (5.52) [Zζ ]Q ⊗ [Zζ 0 ] maps to ζ · ζ 0 .
(5.53)
For the final step, for a torus T = V | Λ with principal polarization Q ∈ Λ2 Λ∗ we have i) Λ2b Λ∗ ∼ (using [35]) =Z ∗ 2b−1 ∗ ii) Λ ⊗ Λ Λ →Z (cup product) where we have iii) Qb−1 ⊗ identity: Λ∗ ⊗ Λ2b−1 Λ∗ ' Λ2b−1 Λ∗ ⊗ Λ2b−1 Λ∗ , and iv) the diagram H 1 (T, Z) ⊗ H 2b−1 (T, Z)
−→
ok
Z k
Q
H 2b−1 (T, Z) ⊗ H 2b−1 (T, Z) y
−→
H 4b−2 (T × T, Z)
c1 (PQ )
Z k
−→
Z
commutes where the top vertical isomorphism is (iii).
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This gives the conclusion p∗1 [Zζ ] ⊗ p∗2 [Zζ 0 ] ⊗ c1 (µ∗ MB ) → Z computes ζ · ζ 0 , where the LHS is in H 2b−1 (JB ×B JB , Z) ⊗ H 2b−1 (JB ×B JB , Z) ⊗ H 2 (JB ×B , Z) and the mapping is cup product. This completes the sketch of the proof of Theorem 5.5.1.
5.5.3 Generalized Complex Tori and Their Compactifications For the purposes of this work one needs the construction and properties of the Poincar´e line bundle in families in which there are singular fibres. In fact, heuristic reasoning suggests that this line bundle may have some sort of “topological discontinuity” along the locus H 1 (Bs•0 ) 6= 0. What we are able to do here is only to take some first steps in this program. Specifically, for smooth curves B ⊂ S such that the fibres of XB → B have at most one ordinary node as singularities we shall i) construct an analytic fibre space JB → B of complex Lie groups whose fibre over s ∈ B is J(Xs ) when Xs is smooth and is the generalized intermediate Jacobian Je (Xs ) when Xs has a node; and where OB (JB ) = Je as defined in sections 5.2.1, 5.5.2 above, ii) construct a compactification J¯B ⊃ JB y y B
=
B
where J¯B is a smooth compact complex manifold and for Xs0 nodal defn
(J¯B,s0 )sing = J∞,s0 has dimension b − 1 and is smooth,
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(iii) contruct a desingularlization J¯B^ ×B J¯B → J¯B ×B J¯B y y B
=
B;
where (J¯B ×B J¯B )sing =J¯∞ ×B J¯∞ , iii) although we shall not construct the line bundle stack MB → J¯B and Poincar´e line bundle PB → J¯B^ ×B J¯B , we will show that their Chern classes c1 (MB ) ∈ H 2 (J¯B , Q) c1 (PB ) ∈ H 2 (J¯B^ ×B J¯B , Q) can be defined, and iv) finally, we shall show that the arguments in the preceding section can be extended to give the main result Theorem 5.5.1 in this context. Remark. There is a substantial literature on compactification of quasi-abelian varieties and of generalized Jacobians of singular curves, both singly and in families. Although we shall not get into it here, for our study the paper [4] by Lucia Caporaso and its sequel [5] together with [1] are especially relevant. In those papers there is an extensive bibliography to other work on the compactifications referred to above. In addition the papers [7] and [6] have been useful in that they directly relate Hodge theory to compactifications. We now realize our program outlined above. i) We begin by recalling the construction for a family of elliptic curves. The question is local over a disc ∆ = {s : |s| < 1}, where Xs is smooth for s 6= 0 and Xs has a node p. It is well-known that the normalized period matrix of Xs is log s 1, + a(s) , 2πi where a(s) is a holomorphic function at s = 0. It represents an inessential peturbation term and for simplicity of exposition will be assumed
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to be zero. The period lattice of Xs , s 6= 0, thus has generators
1 i log 2π |s|
1+
arg s 2π
We let Z2 act on C × ∆ by arg s , s) e1 · (z, s) = (z + 1 + 2π ( ) ! i 1 z + 2π log |s| , s 6= 0 e1 · (z, s) = ,s . z, s=0 The quotient by this action is J∆ → ∆. To see that it is an analytic fibre space of complex Lie groups, we first restrict to the axis Im s = 0 and factor out the action of e1 by setting w = e2πiz ∈ C∗ . Then e2 acts on C∗ × ∆ by e2 · (w, s) =
|s| · w s ∈ 0 w s=0
.
By similar but more complicated expressions one may extend this to all s, and when this is done the resulting action is visibly properly discontinuous and exhibits J∆ → ∆ as an analytic fibre space of complex Lie groups. For a curve of genus g the normalized period matrix is (Ig , Z(s)) where Z(s) ∈ Hg , the Siegel generalized upper-half-plane, is given by ! log s Tb(s) + a(s) 2πi Z(s) = (5.54) ˜ b(s) Z(s) ˜ where a(s) is holomorphic at s = 0, and b(s) ∈ Cg−1 and Z(s) ∈ Hg−1 are holomorphic at s = 0. The above discussion extends to define an
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analytic fibre space J∆ → ∆ of complex Lie groups. The fibre J∆,s over s 6= 0 is the Jacobian J(Xs ), and over s = 0 we have ˜o) → 0 1 → C∗ → J∆,o → J(X
(5.55)
˜ o → Xo is the normalization. The extension class of (5.55) is where X ˜ 0 ), J∆ is a product represented by b(0). Locally over a point of J(X C∗ × U ˜ 0 ) is an open set. This local splitting is as complex where U ⊂ J(X manifolds, not as complex Lie groups, and locally refers to the strong property of holding outside a compact set in the C∗ factor. In general, for Xs ⊂ X 2n as above and for L sufficiently ample so that hn,n−1 (Xs ) 6= 0 for s 6= 0, it is known (cf. [21]) that ˜ the period matrix will have the form (5.33) where now Z(s) represents the period matrix of a family of polarized complex tori with hn,n−1 = hn,n−1 (Xs ) − 1, s 6= 0. Thus the same conclusion — that J∆ → ∆ may be constructed as an analytic fibre space of complex ˜ 0 ) is Lie groups — holds. Moreover, we have (5.55) where now J(X ˜ the intermediate Jacobian of the standard desingularization X0 → X0 obtained by blowing up the node p ∈ X0 . We shall refer to J∆,0 as the generalized intermediate Jacobian of X0 . We may summarize as follows: The analytic fibre space of complex Lie groups JB → B is locally biholomorphic to the product of a smooth fibre space (5.56) and an elliptic curve acquiring a node across a disc. Here, as noted above, locally has the strong meaning of “outside a compact set in the C∗ factor”. ii) Because of (5.56) does not work !! it will suffice to analyze the elliptic curve picture in a way that will extend to the local product situation as described above. Here we may be guided by the geometry. Namely, locally in the analytic topology around a nodal elliptic curve Xs0 there are local coordinates x, y on XB and s on B such that s0 is the origin and the map XB → B is given by
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(5.57)
xy = s
r
q
Then dx dy ≡− x y
mod ds ,
and using the above notation, on C∗ with coordinate w we have dx dw ≡ near w = 0 w x dw ≡ − dy near w = ∞ w y where ≡ denotes congruence modulo holomorphic terms. Then we compactify C∗ by adding one ideal point p with Z q Z r dw dw lim = lim − q→p r→p w w in the above figure. Of course, in this case the compactification of JB,o ∼ = C∗ is just the original elliptic curve Xs0 . But using (5.43) and the above coordinate description enables us to infer the general case from the particular case. Remark. One obvious but slightly subtle point is that we are not saying that a general family XB → B has around a node the local coordinate description (5.57). Rather, for n = 2 that is x21 + · · · + x22n = s . What we are saying is that in the family JB → B, the “C∗ direction” has the coordinate description (5.57).
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iii) We now turn to the study of the singularities of J¯B ×B J¯B . Again, locally in the sense explained above the situation is a product of the elliptic curve picture with some parameters. Around a point on a smooth fibre, respectively a node on a singular fibre, the map J¯B → B is (x, y) −→ s = y (smooth case) (x, y) −→ s = xy (nodal case) . From this it follows directly that J¯B ×B J¯B
sing
⊆ J¯∞ ×B J¯∞
(5.58)
where J¯B,∞ ⊂ J¯B is the set of singular points on fibres. Moreover, in coordinates (x, y, x0 , y 0 , s) ∈ C5 such that J¯B,∞ ×B J¯B,∞ is locally given by f = xy − s = 0 , (5.59) f 0 = x0 y 0 − s = 0 from df ∧ df 0 = 0 ⇔ x = y = x0 = y 0 = 0 we see that we have equality in (5.58). Moreover, for the Jacobian of (f, f 0 ) we have that rank(J(f, f 0 )) = 1 along J¯B,∞ . Finally, (5.59) gives xy = x0 y 0 which is a quadric cone in C4 and has a canonical desingularization. Remark 5.5.3. For later reference we note that a) the 0-section of JB ⊂ J¯B is a smooth section not meeting J¯∞ ; b) for the group law µ : JB ×B JB → JB we have that µ−1 (0) = W , and in J¯B ×B J¯B we have for the closure W = J¯∞ ×B J¯∞ . The model here is ∗
C = (P1 , {0, ∞}) =
p
µ : C∗ × C∗ → C∗ is multiplication ⇒ (p, p0 ) = µ−1 (1) ∩ (P 1 , {0, ∞})\C∗ × (P1 , {0, ∞})\C∗
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5.5.4 Topological Properties of the Poincar´ e Line Bundle in some Families with Singular Fibres The objective of this section is to show that, using J¯B → B, the argument sketched above for the proof of Theorem 5.5.1 may be extended to the case in which there are singular fibres as in the preceding section. First we shall explain why JB → B is not the right object. We give three reasons. i) Although JB is a smooth manifold it is non-compact; in particular, it does not have a fundamental class and Poincar´e duality does not hold (both of which were used in the proof of Theorem 5.5.1. ii) The local invariant cycle theorem does not hold for JB → B, whereas it does hold for J¯B → B. Thus, for Xs0 having a simple node and s close to s0 with (as usual) T representing monodromy, we have (R2b−1 ZJ¯B )s0 ∼ = ker{T − I : H 2b−1 (Js , Z) → H 2b−1 (Js , Z)}
(5.60)
but (R2b−1 ZJB )s0 6= ker{T − I : H 2b−1 (Js , Z) → H 2b−1 (Js , Z)} . (5.61) Note: This is related to the fact that for ∆ a disc around s0 and with J¯∆ = p¯−1 (∆) J¯∆ retracts onto J¯s 0
while this fails to be the case for J∆ . iii) Relatedly, the Leray spectral sequence for p¯ : J¯B → B degenerates at E2 while this fails to be the case for JB → B. Example. Let dim Js = 1 so that J¯B → B is an elliptic surface whose singular fibres J¯si are all nodal elliptic curves while Jsi = J¯si \{pi } ∼ = C∗ .
(5.62)
Then (Rq ZJ¯B )s ∼ = (Rq ZJB )s for all points s ∈ B and all q, except that (R3 ZJ¯B )si = 0,
(R3 ZJB )si ∼ =Z.
(5.63)
This follows from localizing (5.62) over a disc ∆i around si . Then Z∼ = H 2 (B, R2 ZJ¯B ) ∼ = H 2 (B, R2 ZJB )
(5.64)
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but whereas the Leray spectral sequence for J¯B → B degenerates at E2 and the first isomorphism in (5.64) gives H 4 (J¯B , Z) ∼ =Z, the Leray spectral sequence for JB → B has by (5.63) d
2 H 0 (B, R3 ZJB ) −→ H 2 (B, R2 ZJB )
ok ⊕Z i
so that the right term in (5.64) is killed by d2 . Before begining the argument we remark that we are not claiming that the line bundle stack MB → J¯B and Poincar´e line bundle PB → J¯B^ ×B J¯B exist, although this may well be true. What we shall use is that what would be images of their Chern classes ( c1 (MB ) ∈ H 0 (B, R2 ZJ¯B ) c1 (PB ) ∈ H 0 (B, R2 Z ¯ ^¯ ) JB ×B JB
do exist, and their pullbacks under νζ and νζ×ζ 0 are all that is really required for the argument. Thus we are able to proceed pretending that MB and PB exist as in the case treated in section 5.5.2. We think that the issue of defining M and P over the family of all J(Xs ), s ∈ S is a very attractive and probably important geometric problem. Referring to the proof of Theorem 5.5.1 in section 5.5.2, we note that both νζ and Zζ avoid the singularities in the fibres of XB → B and J¯B → B, respectively. Moreover, the argument that [Zζ ] → H 1 (B, R2b−1 ZJ¯B ) is defined and is linear in ζ carries over verbatim. The next step, which uses Poincar´e duality on J¯B and J¯B^ ×B J¯B , also carries over to give p∗ [Zζ ] ∪ p∗2 [Zζ 0 ] = [Zζ×ζ 0 ] p∗1 [Zζ ] ∪ p∗2 [Zζ 0 ] ∪ c1 (µ∗ MB ) = c1 (P ) as before. Additionally, (5.50) and the discussion just under remain as stated there, with J¯B replacing JB .
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Next comes the main somewhat subtle point; namely, that Qb−1 : R1 ZJ¯B ∼ = R2b−1 ZJ¯B
(5.65)
continues to hold. Essentially this is because of (ii) above. Namely, we have Qb−1 : H 1 (J¯B,s , Z) ∼ = H 2b−1 (J¯B,s , Z)
(5.66)
for s near si and where we have set J¯B,s = p¯−1 (s). Moreover since TQ = Q
(5.67)
(R1 ZJ¯B )si ∼ = ker{T − I : H q (J¯B,s , Z) → H 1 (J¯B , Z)}
(5.68)
and
we may infer (5.65) from (5.66)–(5.68). The final step is essentially the same as before, where over si we replace Λ by (R2b−1 ZJ¯B )si
∼ = RHS of (5.68) ∼ = (R1 ZJ¯B )si
of (5.65), and then the pairing (R1 ZJ¯B )si ⊗ (R2b−1 ZJ¯B )si → Z follows from the fact that the compact analytic variety J¯B,si has a fundamental class. Note: The condition to be able to fill in a family of intermediate Jacobian {Js }s∈∆∗ with a compactification J0 of the generalized intermediate Jacobian over the origin is probably very special to the case n = 1. Namely, first recall that for s 6= 0 H 1 (Js , Z) ∼ (5.69) = H2n−1 (Xs , Z) . Suppose that we can compactify the family π
J∆ −−−−→ ∆ where π −1 (s) = Js to have J∆ → ∆ . It is reasonable to expect that the total space J∆ will be a K¨ahler manifold,
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and Clemens [11] has shown that in this situation the local monodromy theorem holds, so that after after passing to a finite covering the monodromy T : H 1 (Jη , Z) → H 1 (Jη , Z)
(η 6= 0)
will satisfy (T − I)2 = 0 .
(5.70)
But by (5.70) all we can expect in general is (T − I)2n = 0 . In other words, (5.70) (which is satisfied in the model case) is perhaps a necessary condition to be able to compactify J0 in a family. More plausible is that J∆ → ∆ will have a partial compactification of the sort appearing in the work of Kato-Usui [27].
5.6 Conclusions The theory discussed above is, we feel, only part of what could be a rather beautiful story of the geometry associated to a Hodge class ζ ∈ Hgn (X)prim through its normal function νζ ∈ H 0 (S, JE ) where S is either PH 0 (X, Lk ), or is a suitable blowup of that space If one wants to use the theory to construct algebraic cycles, i.e. to show that sing νζ 6= ∅ , then the following four assumptions must enter: i) ζ is an integral class in H 2n (X, Z) ii) ζ is of Hodge type (n, n) iii) a) k = k0 (ζ) b) where the ζ-dependence of k0 is at least |ζ 2 |; and iii) all of H 0 (X, Lk ) is used. In our work above, there are two main approaches to studying the geometry associated to ζ A) the “capturing” of ζ along the singular locus ∆ ⊂ X (cf. section 5.4.2); and B) the (as yet only partially defined) line bundles νζ∗ (M ) and νζ×ζ 0 (P ).
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In A) we have used the assumptions (iii), (iiia), (iv) in order to have the necessary vanishing theorems so as to have the isomorphism H n (ΩnX )prim ∼ = H 1 (I∆ ⊗ KX ⊗ Ln ⊗ H n )
(5.71)
with the resulting conclusion HC ⇔ Hgn (X)prim ,→ H 0 (Rp1 I∆ ⊗ K ⊗ Ln ⊗ H n )
(5.72)
where p : ∆ → D is the projection. We refer to section 5.4.2 for a discussion of how the assumption (i) should enter, and in fact will enter if the HC is true. We remark that, based on the heuristics discussed in section 5.4.1, one may reasonably expect that the stronger assumption (iiib) must be used. In this regard, the condition (iiia) needed to have (5.33) is locally uniform in the moduli space of X, whereas the stronger assumption (iiib) cannot have this local uniformity. In B) we have used from the very outset the assumptions (i) and (ii), ∗ (P )). However, and moreover the quantity ζ 2 appears naturally in c1 (νζ×ζ the assumptions (iiib), (iv) have as yet to appear, even heuristically, in the ∗ geometry of νζ∗ (M ) and νζ×ζ 0 (P ). In closing we would like to suggest three examples whose understanding would, we feel, shed light on the question of existence of singularities of νζ . These are all examples in the case n = 1 of curves on an algebraic surface, where of course the HC is known. However, one should ignore this and seek to analyze sing νζ in the context of this paper. Example 5.6.1. (i) X = P1 × P2 , L = OX (2, 2) and ζ is the class of L1 − L2 where the Li are lines from different rulings of X realized as a quadric in P. (ii) X is a general smooth quartic surface in P3 containing a line Λ, L = OX (1) and ζ = [H − 4Λ] where H is a hyperplane. (iii) X is a general smooth surface of degree d = 4 in P3 containing a twisted curve C, L = OX (1) and ζ = [H − dC]. In example 5.6.1.1 the general fibre Xs is an elliptic curve where degenerations are well understood, although in this case the base space is 8 dimensional and the non-torsion phenomena in our extened N´eron-type model J˜E is what is of interest. In example 5.6.1.2 we have the situation where the nodes do not impose independent conditions on |L|, which must then be blown up so that the discriminant locus D has local normal crossings. This example has the
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advantage that dim S = 3 so that the analysis of, e.g., the “singularities” of the Poincar´e line bundle should be easier to do directly. Example 5.6.1.3 exhibits the phenomenon that νζ has no singularities on PH 0 (OX (L)); one must pass to L2 to have sing νζ 6= 0. This will of course be the general case. Of course these examples could be extended, e.g. to smooth hypersurfaces in P5 where in example 5.6.1.3 the condition is to contain a Veronese surface. As explained in section 5.4.3, we see no a priori reason why the geometric picture as regards sing νζ should be significantly different from the n = 1 case, although analyzing the geometry will of course be technically much more involved.
References [1] Alexeev, V.: Compactified Jacobians and the Torelli map, Publ. RIMS Kyoto Univ. 40 (2004), 1241–1265. [2] Atiyah, and F. Hirzebruch: Analytic cycles on complex manifolds, Topology 1 (1962), 25–45. [3] Beilinson, A., J. Bernstein and P. Deligne: Faisceaux pervers, Ast´erisque 100 (1981). [4] Caporaso, L.: A compactification of the universal Picard variety over the moduli space of stable curves, Jour. AMS 7 (1994), 589–560. [5] Caporaso, L.: N´eron models over moduli of stable curves, to appear. [6] Cattani, E.: Mixed Hodge structures, compactifications and monodromy weight filtration, Chapter IV in Topics in Transcendental Algebraic Geometry, Annals of Math. Studies 106 (1984), Princeton Univ. Press, Princeton, NJ. [7] Carlson, J., E. Cattani and A. Kaplan: Mixed Hodge structures and compactifications of Siegel’s space, in Journ´ees de g´eometrie alg´ebrique d’Angers, Sijthoff & Hoordhoff, 1980, pp. 1–43. [8] Cattani,E. P. Deligne and A. Kaplan: On the locus of Hodge classes, J. Amer. Math. Soc. 8 (1995), 483–506. [9] Cattani, E. A. Kaplan and W. Schmid: Degeneration of Hodge structures, Ann. of Math. 123 (1986), 457–535. [10] Cattani, E. A. Kaplan and W. Schmid: L2 and intersection cohomologies for a polarizable variation of Hodge structure, Invent. Math. 87 (1987), 217– 252. [11] Clemens, C.H.: Degeneration of K¨ahler manifolds, Duke Math. J. 44 (1977). [12] Clemens, C.H.: The N´eron model for families of intermediate Jacobians acquiring “algebraic” singularities, Publ. Math. I.H.E.S. 58 (1983), 5–18. [13] de Cataldo M. and L. Migliorini: The Hodge theory of algebraic maps, arXiv:math.AG/0306030v2 May 2004. [14] El. Zein, F. and S. Zucker: Extendability of normal functions associated to algebraic cycles, in Topics in Transcendental Algebraic Geometry, Ann. of Math. Studies 106 (1984), 269–288, Princeton Univ. Press, Princeton, N.J. [15] Fang, H., Z. Nie and J. Pearlstein: Note on singular extended normal function, preprint.
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[16] Goresky, M. and R. MacPherson: Intersection homology theory, Topology 19 (1980), 135–162. [17] Goresky, M. and R. MacPherson: Intersection homology II, Invent. Math. 72 (1983), 77–129. [18] Green, M.: Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Diff. Geom. 29 (1989), 545–555. [19] Green M. and P. Griffiths: Hodge theoretic invariants for algebraic cycles, IMRN 9 (2003), 477–510. [20] Griffiths, P.: Periods of integrals on algebraic manifolds I, II Am.J.Math. 90 (1968) 568–626, 805–865. [21] Griffiths, P.: Periods of rational integrals II, Annals of Math. (2)90 (1969), 498–541 [22] Griffiths, P.: A theorem concerning the differential equations satisfied by normal functions associated to algebraic cycles, Amer. J. Math. 101 ( 1979) no. 1, 94–131. [23] Griffiths, P.: Infinitesimal variations of Hodge structure. III. Determinantal varieties and the infinitesimal invariant of normal functions, Compsitio Math. 50 (1983) no. 2-3, 267–324. [24] Griffiths, P.: Topics in Transcendental Algebraic Geometry, Annals of Math. Studies 106 (1984), Princeton Univ. Press, Princeton, NJ. [25] Lange, H. and Ch. Birkenhage: Complex Abelian Varieties, Springer Verlag 1992. [26] Kashiwara, M.: The asymptotic behavior of a variation of polarized Hodge structure, Publ. R.I.M.S. Kyoto Univ. 21 (1985), 853–875. [27] Kato, K. and S. Usui: Classifying spaces of degenerating polarized Hodge structures, preprint. [28] Kleiman, S.: Geometry on Grassmannians and applications to splitting bundles and smoothing cycles, Publ. Math. IHES 36 (1969), 281–297. ´r, J: Lemma on p. 134 in Classification of algebraic varieties, Lext. Notes [29] Kolla in Math. 1515 (1990). [30] Lefschetz, S.: L’analysis situs et la g´eometrie alg´ebrique, Paris, GauthiersVillars (1924). ´methi and J. Steenbrink: Extending Hodge bundles for abelian varieties, [31] Ne Ann. of Math. 143 (1996), 131–148. ´, C. and C. Voisin: Torsion cohomology classes and algebraic cycles on [32] Soule complex projective manifolds, preprint 2005. [33] Saito, M.: Admissible normal functions, J. Alg. Geom. 5 (1996), 235–276. [34] Steenbrink, J.: Limits of Hodge structures, Inv. Math. 31 (1975–76), 229– 257. [35] Thomas, R.P.: Nodes and the Hodge conjecture, J. Alg. Geom. 14 (2005), 177–185. [36] Voisin, C.: Une remarque sur l’invariant infinit´esimal des functions normales, C. R. Acad. Sci. Paris S´er I 307 (1988), 157–160. [37] Voisin, C.: A counterexample to the Hodge conjecture extended to K¨ahler varieties, IMRN 20 (2002), 1057–1075. [38] Zucker, S.: Generalized intermediate jacobians and the theorem on normal functions, Inv. Math. 33, (1976), 185–222.
6 Zero Cycles on Singular Varieties Amalendu Krishna V. Srinivas
[email protected]
[email protected]
In this article, we give an overview of some recent progress in the study of 0-cycles on singular varieties. We also discuss proofs of some results, which are basically corollaries of our results or methods, but which have not been made explicit earlier. These statements are refinements of the earlier published results, and are of interest in the theory of projective modules. We have made an attempt here to make our work more accessible to those interested in such applications.
6.1 Quick review of the smooth case We begin by recalling some standard things about 0-cycles on smooth varieties. Let X be an irreducible, non-singular algebraic variety of dimension n over an algebraically closed field k. Recall that the Chow group of 0-cycles of X is defined by CHn (X) =
Free abelian group on (closed) points of X . h(f )C | C ⊂ X is an irreducible curve, f ∈ k(C)∗ i
Here (f )C = (zeroes of f ) − (poles of f ) where the zeroes and poles of the rational function f are counted with multiplicity. If X is projective (or even proper ) over k, let X X deg CHn (X)deg 0 = ker(CHn (X)−−→ Z), quad deg( ni xi ) = ni . i
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Then one has a surjective Abel-Jacobi mapping n AJX : CHn (X)deg 0 → Alb(X) = Albanese variety of X.
Here if k = C, then one has Alb(X) =
H 2n−1 (X, C) Ω(X)∗ ∼ = n 2n−1 H1 (X, Z) F H (X, C) + H 2n−1 (X, Z)
where Ω(X)∗ = dual C-vector space to the space Ω(X) of holomorphic 1-forms on X n
F H
2n−1
(X, C) = n-th level of the Hodge filtration on H 2n−1 (X, C),
and the two descriptions of Alb(X) are equivalent by Hodge theory and Poincar´e duality. The map H1 (X, Z) → Ω(X)∗ is given by integration, Z γ 7→ (ω 7→ ω), γ
and the Abel-Jacobi map has a related description X X Z yi ω) (mod periods). (xi ) − (yi ) 7→ (ω 7→ i
i
xi
Let S m (X) denote the m-th symmetric product of X, S m (X) = X × · · · × X/Sm = parameter space for 0-cycles
P
i ni (xi )
withni > 0,
X
ni = m
i
= space of effective 0-cycles of degree m. There are natural maps γm : S m (X) × S m (X) → CHn (X)deg 0 ,
γm (A, B) = [A] − [B] ∈ CHn (X).
Then Im γm ⊂ Im γm+1 ,
[
Im γm = CHn (X)deg 0 .
m≥1
Definition. We say CHn (X)deg 0 is infinite dimensional if none of the γm is surjective; if some γm is surjective, we say CHn (X)deg 0 is finite dimensional. The above definition is due to Mumford.
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Definition. If G is (the group of closed points of) an algebraic group, a group homomorphism f : CHn (X)deg 0 → G is called regular if f ◦ γm : S m (X) → G is a morphism of varieties for each m (it suffices to check this for m = 1). The Abel-Jacobi map n AJX : CHn (X)deg 0 → Alb(X)
is an example of a regular homomorphism. We summarize some standard results on 0-cycles in the smooth case, in the following result. Theorem 6.1.1. Let X be an irreducible smooth proper variety of dimension n over k. (a) The Abel-Jacobi map is universal among regular homomorphisms f : CHn (X)deg 0 → G to algebraic groups (i.e. any such f factors uniquely as n for some homomorphism g : Alb(X) → G of algebraic groups). f = g ◦ AJX (b) (Roitman) Let k = k be uncountable. Then n is an isomorphism. CHn (X)deg 0 is finite dimensional ⇔ AJX n is always an isomorphism on torsion subgroups. (c) (Roitman) AJX (d) (Mumford, Roitman) If k = C,and X supports a non-zero holomorphic q-form, then for any closed subvariety Y ⊂ X of dimension < q, the 0-cycles supported on Y do not span CHn (X) ⊗ Q. (e) (Bloch) Let k be uncountable. If q q Het (X, Q` ) 6= N 1 Het (X, Q` ),
then for any closed subvariety Y ⊂ X of dimension < q, the 0-cycles supported on Y do not span CHn (X) ⊗ Q. Two well-known open problems on 0-cycles are as follows (we do not state the questions in the most general form, since they are unknown even in the special cases considered). Conjecture (Bloch Conjecture). Let X, Y be smooth proper surfaces over C, and α ∈ CH2 (X × Y ) a correspondence. Then α∗ : H 0 (X, Ω2X ) → H 0 (Y, Ω2Y ) is an isomorphism m 2 → ker AJ 2 is an isomorphism. α∗ : ker AJX Y
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In particular, if X is a smooth projective surface, then H 0 (X, Ω2X ) = 0 ⇔ 2 is injective (this is the special case when X = Y and α = 0). AJX Conjecture (Bloch-Beilinson Conjecture). Let X be a smooth projective surface over Q, the field of algebraic numbers. Then 2 AJX : CH2 (X)deg 0 → Alb(X)
is an isomorphism. In particular, if V = Spec A is any affine smooth surface over Q, then CH2 (V ) = 0. The Bloch Conjecture is known in a few non-trivial situations. However the Bloch-Beilinson conjecture is at present unknown for any projective smooth surface X/Q supporting a non-zero regular 2-form. The affine version of the conjecture also unknown, for an open subset V of any such surface X.
6.2 The singular case This section is a quick review of key definitions, and results on this topic, before our joint work [1]. A more detailed discussion, with appropriate references, may be found in an earlier survey article [17]. Let X be a reduced variety of dimension n over an algebraically closed field k, and let Xreg denote the open subset of smooth codimension n points of X. Let Xsing = X − Xreg be the singular locus (thus points lying on any lower dimensional component of X are considered singular). Levine and Weibel have defined CHn (X) = D
Free abelian group on points of Xreg (f )C | C ⊂ X is a “Cartier curve”, f ∈ O∗C∩Xsing
E.
Here a Cartier curve C is a reduced (purely 1-dimensional) curve in X such that C ∩ Xsing is finite, and C is a local complete intersection in X at points of C ∩ Xsing . The expression f ∈ O∗C∩Xsing means that f is a rational function on C, non-zero on each component, and invertible at points of C ∩ Xsing . Note that if C is a Cartier curve, there is a natural map K0 (C) → K0 (X), inducing Pic C → F n K0 (X), such that F n K0 (X) is the subgroup of K0 (X) generated by the images of Pic C for all Cartier curves C in X. Theorem 6.2.1. (Levine)
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1) CHn (X) ⊗ Q ∼ = F n K0 (X) ⊗ Q, where F n K0 (X) is the subgroup of the Grothendieck group K0 (X) of vector bundles on X generated by the classes of points of Xreg . 2) If n = dim X = 2, then CH2 (X) ∼ = H 2 (X, K2,X ) ∼ = {α ∈ K0 (X) | rank (α) = 0, det (α) = 0}. 3) If codimX (Xsing ) ≥ 2, then CHn (X) =
Free abelian group on points of Xreg . h(f )C | C ⊂ X irred. curve, C ∩ Xsing = ∅, f ∈ k(C)∗ i
This gives some evidence as to why the Levine-Weibel definition is the “correct” one, in dealing with the “Chow cohomology” group of zero cycles and K0 (X), the Grothendieck group of vector bundles (as opposed to the “Chow homology”, which is the Chow group defined in Fulton’s book [8], which is naturally associated to the Grothendieck group G0 (X) of coherent sheaves on X). In the case when X is a quasi-projective variety with isolated singularities, it is known from work of Collino, and Pedrini and Weibel [16], that CHn (X) ∼ = H n (X, Kn,X ) if n = dim X is arbitrary. However, in unpublished work of M. Levine and V. Srinivas, it is shown that in general, the isomorphism fails when n > 2. There are results of several people (Collino, Levine, Weibel, Srinivas, ...) in the 80’s extending most of Theorem 6.1.1 to the case of normal projective varieties X. However: a) no analogue of Bloch’s result has been proved (or even plausibly conjectured, to our knowledge) b) the Roitman torsion theorem was not known then for p-primary torsion cycles in characteristic p > 0 (we return to this point later). One other interesting result from this period was the following, due independently to Levine and Srinivas. Theorem 6.2.2. Let X be a normal quasi-projective surface with quotient singularities, and f : Y → X a resolution of singularities. Then f ∗ : CH2 (X) → CH2 (Y ) is an isomorphism. There have been some further developments in the subject in the last few years, which we now describe.
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Let X be reduced and projective of dimension n over an algebraically closed field k. Then there is a degree homomorphism degX : CHn (X) → Zt ,
t = no. of n-dimensional irred. components of X.
The next result was obtained by Jishnu Biswas and Srinivas [5], generalizing earlier work of Barbieri, Pedrini and Weibel for surfaces [2] (this work represented the first step in the resurgence of activity on this topic). Theorem 6.2.3. Let k = C. There is an Abel-Jacobi map n AJX : CHn (X)deg 0 → J n (X)
with J n (X) =
H 2n−1 (X, C) F n H 2n−1 (X, C) + H 2n−1 (X, Z) (a semi-abelian C-variety.)
This is a surjective regular homomorphism, and is an isomorphism on torsion subgroups. Another set of results, of Esnault, Srinivas and Viehweg, is summarized in the following theorem (see [7]). Theorem 6.2.4. Let X be projective of dimension n over k = k. 1) There exists a smooth connected commutative algebraic group An (X) defined over k, and a surjective regular homomorphism ϕ : CHn (X)deg 0 → An (X) which is universal among regular homomorphisms from CHn (X)deg 0 to algebraic groups over k. 2) If k is uncountable, then CHn (X) is finite dimensional ⇔ ϕ is an isomorphism. 3) If K is an algebraically closed extension field of k, then An (XK ) ∼ = n A (X) ×k K. 4) If k = C, then n
A (X) = =
n−1 H2n−1 (X, OX → · · · → ΩX/C )
H 2n−1 (X, Z) n−2 coker (H n (X, ΩX/C ) → H n (X, Ωn−1 X/C ))
H 2n−1 (X, Z)
.
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5) (Lie (An (X))∗ := Ω(X) ⊂ {closed, regular (algebraic) 1-forms on Xreg ; the integration map H1 (Xreg , Z) → Ω(X)∗ = Lie An (X), Z γ 7→ (ω 7→ ω), γ
factors through 2n−1 (X H1 (Xreg , Z) ∼ H 2n−1 (X, Z) reg , Z) = H −→ c → → = π1 (An (X)), torsion torsion torsion and the map ϕ : CHn (X)deg 0 → An (X) is given by X X Z yi ω). (xi ) − (yi ) 7→ (ω 7→ i
i
xi
The Mumford-Roitman theorem (Theorem 6.1.1(d)) has the following analogue, proved in detail in [17]. n−q Theorem 6.2.5. Let X/C be projective of dimension n with H n (X, ΩX/C ) 6= 0. Then for any closed Y ⊂ X with dim Y < q, the 0-cycles supported on Y ∩ Xreg do not span CHn (X) ⊗ Q.
6.3 Some recent results for normal surfaces In this section, we give an overview of some more recent results on 0-cycles on normal surfaces. These are based on the paper [1]. Theorem 6.3.1. Let X be a normal projective surface over C, and π : Y → X a resolution of singularities. Then π ∗ : CH2 (X) → CH2 (Y is an isomorphism m dim H 2 (X, OX )
= dim H 2 (Y, OY ) m
every holomorphic 2-form on Xreg extends to a holomorphic 2-form on Y . The above result is in the spirit of the Bloch conjecture. Srinivas’ 1982 Chicago thesis contained a proof that CH2 (X) ∼ = CH2 (Y ) ⇒ H 2 (X, OX ) ∼ = H 2 (Y, OY ) in the situation of the theorem. The converse implication ⇐ is in fact valid
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for any projective surface X over an algebraically closed field of characteristic 0 (this is easily deduced from the complex case). Corollary 6.3.2. Let X be a quasi-projective surface over an algebraically closed field of characteristic 0, with only rational singularities. Then for a resolution π : Y → X, the map π ∗ : CH2 (X) → CH2 (Y ) is an isomorphism. This is a substantial improvement on Theorem 6.2.2. Example. Let X ⊂ An+1 be the affine cone over a smooth projective curve C C ⊂ PnC . Then CH2 (X) = 0 ⇔ H 1 (C, OC (1)) = 0, i.e., the embedding C ⊂ PnC is by a “non-special” linear system, in the sense of classical algebraic geometry. Theorem 6.3.1 has the following analogue in characteristic p. Theorem 6.3.3. Let X be a normal projective surface over an uncountable algebraically closed field k of characteristic p > 0, and π : Y → X a resolution of singularities. Then π ∗ : CH2 (X) → CH2 (Y ) is an isomorphism m dimQ`
H´e2t (X, Q` (1)) N S(X) ⊗ Q`
= dimQ`
H´e2t (Y, Q` (1)) . N S(Y ) ⊗ Q`
If X is defined over a subfield of k finitely generated over the prime field, so that there are natural Galois representations on the ´etale cohomology groups in the Theorem, the condition for the isomorphism on Chow groups is equiv2 (X, Q ) is pure of weight 2. alent to asserting that the representation on Het ` Srinivas’ 1982 Chicago thesis contained a proof that, in the situation of the theorem, the isomorphism on Chow groups implies the cohomological condition (the characteristic p result in that thesis is equivalent to this statement, but phrased differently). The next result is in the spirit of the Bloch-Beilinson Conjecture (stated above in Section 6.1). Theorem 6.3.4. Let X be a normal affine surface whose coordinate ring A = ⊕n≥0 An is a finitely generated graded algebra over Q. Then i) CH2 (X) = 0 ii) K0 (X) = Z, and all projective A-modules are free (all algebraic vector bundles on X are trivial) iii) maximal ideals in A of smooth points of X are complete intersections.
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Example. Let A=
Q[x, y, z] , n ≥ 4. + yn + zn)
(xn
Then all algebraic vector bundles on X = Spec A are trivial, all smooth maximal ideals are complete intersections, etc. In contrast, for the corresponding complex surface XC ⊂ C3 , we have CH2 (XC ) 6= 0, and it is in fact infinite dimensional in the sense of MumfordRoitman. The smooth points of XC = Spec A ⊗Q C whose maximal ideals are complete intersections are precisely those which lie on a ruling over a point of the Fermat curve with algebraic number coordinates. This is analogous to the prediction of the Bloch-Beilinson conjecture. n : CHn (X) Theorem 6.3.5. AJX deg 0 → Alb(X) is an isomorphism on torsion (including p-primary torsion in char. p > 0), for any normal projective variety X over an algebraically closed field.
The proof is by using a reduction, due to Levine, to the case of normal surfaces, following a proof of Bloch in the smooth case. The surface case is settled by a new argument, using K-theory, and reducing to the smooth case. This fills the remaining gap in the proof of the Roitman torsion theorem for normal projective varieties. The main new ingredient in the above results is the following K-theoretic result. Let X be a normal quasi-projective surface, π : Y → X a resolution of singularities, with exceptional locus E (taken with reduced structure, say). Let nE ⊂ Y be the subscheme defined by the n-th power of the ideal sheaf of E in Y . One can define relative algebraic K-groups Ki (Y, nE), and a “relative Chow group” F 2 K0 (Y, nE) ⊂ K0 (Y, nE), such that there is a commutative diagram of surjective maps for each n, F 2 K0 (Y, nE)
fMMM MMM MMM M
/ F 2 K (Y ) = CH2 (Y ) 0 6 nnn n n n n nn nnn
F 2 K0 (X) = CH2 (X)
compatible in an obvious way as n varies. Theorem 6.3.6. Let π : Y → X be a resolution of singularities of a normal quasi-projective surface k over a field k, with exceptional set E. Then the maps F 2 K0 (X) → F 2 K0 (Y, nE) and F 2 K0 (Y, (n + 1)E) → F 2 K0 (Y, nE)
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are isomorphism for all large enough n. Hence F 2 K0 (Y, nE). CH2 (X) ∼ = lim ←− n
This result verifies a conjecture of Bloch and Srinivas, made in the latter’s Chicago thesis (1982). The other new ingredient in characteristic 0 is an exact sequence, in the situation where the exceptional locus E has normal crossings, and IE is the ideal sheaf of E in OY : H 1 (Y, IE /IEn ) ⊗k Ω1k/Z → SK1 (nE) → SK1 (E) → 0. For k = Q, one has Ω1k/Z = 0, while if H 2 (X, OX ) ∼ = H 2 (Y, OY ), one can show that H 1 (Y, IE /IEn ) = 0 for all n > 1. In either case we get SK1 (nE) = SK1 (E), so that one obtains CH2 (X) ∼ = F 2 K0 (Y, E). This formula for CH2 (X) is also proved when the ground field k is algebraically closed of characteristic p, which is related to the Roitman torsion theorem in that case.
6.4 Some applications In this section, we explain some consequences of the above results, which are of interest in algebra. The two results considered are for algebras over the algebraic closure Fp of a finite field, and for graded algebras over the field Q of algebraic numbers. Theorem 6.4.1. Let A be a finitely generated algebra of dimension d > 1 over the algebraic closure of a finite field. Then any projective A-module of rank d has a nonzero free direct summand, and any smooth maximal ideal of A of height d is a complete intersection. Proof The reduction of this theorem to our results is basically due to M. P. Murthy. His main theorem in [13] reduces the above theorem to the vanishing of the Chow group CHd (X), where X = Spec A. This vanishing assertion, in turn, is proved in [14], in the cases (i) when X is non-singular of dimension d ≥ 2, and
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(ii) when dim X = d ≥ 3. In both cases, the argument is to show that any 0-cycle on X lies on a smooth affine surface Y ⊂ X which is disjoint from the singular locus of X. This is done using a version of Bertini’s theorem, and the fact that X is affine (in case (ii), this requires a little care). Thus, the remaining case is when d = dim X = 2. If X is normal, then we may view X as an affine open subset of a normal projective surface Z, whose singular locus is contained in X (this follows from resolution of singularities for surfaces over Fp ). Then the natural map CH2 (Z)deg 0 → CH2 (X) is a surjection, with kernel given by the classes of 0-cycles of degree 0 supported on Z − X. Since the ground field is the algebraic closure of a finite field, CH2 (Z)deg 0 is a torsion group, and hence by Theorem 6.3.5, the Abel-Jacobi map AJZ2 : CH2 (Z)deg 0 → Alb(Z) is an isomorphism. Since X is an affine open subset of Z, the complement Z − X supports an ample divisor on Z, and hence the Abel-Jacobi images of 0-cycles supported on Z − X generate Alb(Z). Hence these 0-cycles generate CH2 (Z)deg 0 as well, and so CH2 (X) = 0, as desired. If X = Spec A is an arbitrary reduced affine surface, we may compare ˜ where X ˜ = Spec A˜ is the normalization of X. By CH2 (X) with CH2 (X), 2 ˜ ˜ the above, CH (X) = 0, and so it suffices to show that CH2 (X) → CH2 (X) is injective. There is a natural map CH2 (X) → K0 (X), which is injective (since X is affine, this is a simple corollary of the Murthy-Swan cancellation theorem for projective modules [15]); the image of this map is the subgroup F 2 K0 (X) of K0 (X) generated by the classes of smooth points of X. It ˜ is injective. suffices to prove that F 2 K0 (X) → F 2 K0 (X) ˜ From the Let I ⊂ A be the conductor ideal for the extension A ,→ A. Mayer-Vietoris sequence [12, Theorem 3.3], there is an exact sequence ∂ ˜ →K ˜ ˜ · · · → K1 (A/I) 0 (A) → K0 (A) ⊕ K0 (A/I) → K0 (A/I).
Clearly F 2 K0 (A) is in the kernel of K0 (A) → K0 (A/I). For any commuta˜ ⊕ B ∗ , where ˜ = SK1 (B) tive ring B, there is a natural isomorphism K1 (B) ∗ B is the group of units, and one has also that ˜ ⊂ ∂(SK1 (A/I)). ˜ ker F 2 K0 (A) → K0 (A) ˜ So we are reduced to showing that, in our context, SK1 (A/I) = 0. Now SK1 (B) = SK1 (Bred ) for any commutative ring B, by a result of ˜ red is a reduced, finitely generated algebra over Fp Bass. The ring (A/I) which has dimension ≤ 1. Hence it suffices to prove that SK1 (R) = 0 for any
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finitely generated reduced Fp -algebra R of dimension 1 (the 0-dimensional case is trivial). ∗ The product K0 (R) ⊗ R∗ → K1 (R) induces a pairing Pic (R) ⊗ Fp → SK1 (R). It is shown in [9], Theorem 3, that this pairing is surjective, since Spec R is a reduced affine curve over an algebraically closed field. Since Spec R is an affine curve over Fp , its Picard group is a torsion group. On ∗ the other hand, Fp is a divisible abelian group; hence the above pairing is in fact 0. Hence SK1 (R) = 0. Theorem 6.4.2. Let A = ⊕n≥0 be a finitely generated reduced graded algebra of dimension d > 1 over A0 = Q, the field of algebraic numbers. Then all projective A-modules of rank d are free, and all smooth maximal ideals of height d are complete intersections. Proof As in the preceding result, the main result of [13] reduces us to proving that the Chow group CHd (X) = 0, where X = Spec A. We first make a reduction to the case when d = 2. Since our assertion here is about graded Q-algebras, the reduction used in [14] is not applicable. Instead, we argue as follows. Assume d ≥ 3. It suffices to show that, for a dense Zariski open subset U ⊂ X, any smooth closed point x ∈ U represents 0 in CHd (X). We make the following choice of U . Let Z = ProjA be the corresponding reduced projective Q-variety. There is a natural morphism ϕ : Spec A − {M} → Z, where M is the graded maximal ideal of A. This is a Spec Q[t, t−1 ]-bundle over a Zariski open subset V of Z, such that V ∩ Zreg is dense in Zreg . Let U = ϕ−1 (V ∩ Zreg ) ⊂ Xreg . Choose any point x ∈ U , and let y = ϕ(x) ∈ Zreg . We can find a reduced divisor W ⊂ Z of the form Proj(A/hA), where h is a homogeneous element of positive degree, such that h vanishes at x, and p y ∈ W is a smooth point of codimension d − 2. Let B = (A/hA)red = A/ (h). We may further assume that the nilradical of A/hA is M-primary, i.e., that h does not vanish at any associated point of the scheme Spec A − {M}. Let Y = Spec B. By induction, we may assume x represents 0 in CHd−1 (Y ). This means (from the definition of rational equivalence in this context) that there is a Cartier curve C on Y , and a suitable rational function f on C, so that the divisor of f on C equals the 0-cycle x. By a moving lemma, we may assume C may be chosen to be disjoint from any given codimension 2 set in W which is contained in the singular locus. In particular, we may assume C does not pass through the “vertex” M. Hence Y is a Cartier divisor in X at all points of C. Further, C is assumed to be a local complete intersection in
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Y at all points of C ∩ Ysing . Hence C is in fact a local complete intersection in X at all points of C ∩ Xsing ⊂ C ∩ Ysing . Thus C is a Cartier curve on X as well, and f defines a rational equivalence on X trivializing the point x. We now prove the result in the 2 dimensional case. In the normal 2 dimensional case, this is Theorem 6.3.4. As in the case of Theorem 6.4.1, to get the general case, the Mayer-Vietoris sequence [12] reduces the problem to the vanishing of SK1 (R), where now R is a graded 1-dimensional Q-algebra. We may again assume further that R is reduced. We may compare R with ˜ which is a direct product of copies of the polynomial its normalization R, algebra Q[t] (and hence has SK1 = 0); this is done as follows. Let I be ˜ There is a commutative diagram with exact rows the conductor of R ,→ R. (compare [12, Theorem 6.2]) ˜ ˜ I) → K1 (R) ˜ → K1 (R/I) ˜ K2 (R/I) → K1 (R, ↑ ↑ ↑ ↑ K2 (R/I) → K1 (R, I) → K1 (R) → K1 (R/I) which yields the diagram ˜ ˜ I) → SK1 (R) ˜ →0 K2 (R/I) → SK1 (R, ↑ ↑ ↑ K2 (R/I) → SK1 (R, I) → SK1 (R) → 0 By the excision theorem for K1 for curves (see Geller and Roberts [10], and ˜ I), and so we have an Geller and Weibel [11]), we have SK1 (R, I) ∼ = SK1 (R, ˜ ˜ = 0. induced map K2 (R/I) → SK1 (R), which is a surjection since SK1 (R) ˜ So it suffices to prove K2 (R/I) = 0. ˜ which (as noted Now I is a graded ideal in the graded Q-algebra R, ˜ is a direct prodearlier) is a product of polynomial algebras, and so R/I n uct of truncated polynomial algebras Q[t]/(t ) (for perhaps different values of n). By a particular case of an old result of Bloch [4], we have that K2 (Q[t]/(tn )) ∼ = K2 (Q), since Ω1Q/Z = 0. Finally, we have K2 (Q) = 0, for example as a very particular case of results of Borel [6]. References [1] Krishna, A. and V. Srinivas: Zero-Cycles and K-theory on normal surfaces, Ann. of Math., 156, 155–195 (2002) [2] Barbieri-Viale, L., C. Pedrini and C.A. Weibel: Roitman’s theorem for singular complex projective surfaces, Duke Math. J. 84, 155–190 (1996) [3] Bass, H.: Algebraic K-theory, W. A. Benjamin, Inc., New York–Amsterdam (1968) [4] Bloch, S.: K2 of Artinian Q-algebras with application to algebraic cycles, Comm. Alg. 3, 405–428 (1975)
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[5] Biswas, J. and V. Srinivas: Roitman’s theorem for singular projective varieties, Compositio Math. 119, 213–237 (1999) [6] Borel, S.: Stable real cohomology of arithmetic groups, Ann. Sci. E.N.S. 7, 235–272 (1974) [7] Esnault, H., V. Srinivas and E. Viehweg, E.: The universal regular quotient of the Chow group of points on projective varieties, Invent. Math. 135, 595–664 (1999) [8] Fulton, W.: Intersection theory, Ergeb. Math. Grenzgebiete (3) 2, SpringerVerlag, Berlin (1984) [9] Geller, S. and L. Roberts: The relationship between Picard groups and SK1 of some curves, J. Algebra 55 213–230 (1978) [10] Geller, S. and Roberts, L.: Kahler differentials and excision for curves, J. Pure Appl. Algebra 17 85–112 (1980) [11] Geller, S. and C. A. Weibel: K1 (A, B, I), J. Reine Angew. Math. 342, 12–34 (1983) [12] Milnor, J.: Introduction to Algebraic K-theory, Annals of Math. Studies 72, Princeton (1971) [13] Murthy, M. P.: Zero cycles and projective modules, Ann. of Math. 140 405– 434 (1994) [14] Murthy, M. P., N. Mohan Kumar and A. Roy: A cancellation theorem for projective modules over finitely generated rings, in Algebraic geometry and commutative algebra, Vol. I (in honour of Masayoshi Nagata), Kinokuniya, Tokyo 281–287 (1988) [15] Murthy, M. P. and R. Swan: Vector bundles over affine surfaces, Invent. Math. 36 125–165 (1976) [16] Pedrini, C. and C. A. Weibel: Bloch’s formula for varieties with isolated singularities I, Commun. Alg. 14 1895–1907 (1986) [17] Srinivas, V.: Zero cycles on singular varieties, in The Arithmetic and Geometry of Algebraic Cycles, NATO Science Series C, 548, Kluwer 347–382 (2000)
7 Modular Curves, Modular Surfaces, and Modular Fourfolds Dinakar Ramakrishnan, Department of Mathematics California Institute of Technology, Pasadena, CA 91125.
[email protected]
To Jacob Murre
7.1 Introduction We begin with some general remarks. Let X be a smooth projective variety of dimension n over a field k. For any positive integer p < n, it is of interest to understand, modulo a natural equivalence, the algebraic cycles P Y = j mj Yj lying on X, with each Yj closed and irreducible of codimension P p, together with codimension p + 1 algebraic cycles Zj = i rij Zij lying on Yj , for all j. There is a natural setting in which to study such a chain (X ⊃ Yj ⊃ Zij )ij of cycles, namely when the following hold: (a) Each Zj is, as a divisor on Yj , linearly equivalent to zero, i.e., of the form div(fj ) for a function fj on Yj ; P (b) The formal sum j mj Zj is zero as a codimension p + 1 cycle on X. Those satisfying (a), (b) form a group Zp+1 (X, 1). An easy way to construct elements of this group is to take a codimension p − 1 subvariety W of X, with a pair of (non-zero) functions (ϕ, ψ) on W , and take the formal sum P j (Yj , TYj (ϕ, ψ)), where {Yj } is the finite set of codimension p subvarieties where ϕ or ψ has a zero or a pole, and ! ordj (ψ) ϕ TYj (ϕ, ψ) = (−1)ordj (ϕ)ordj (ψ) |Yj , ψ ordj (ϕ) the Tame symbol of (ϕ, ψ) at Yj , where ordj denotes the order at Yj . It P is a fact that j TYj (ϕ, ψ) is zero as a codimension p + 1 cycle on X. Let CHp+1 (X, 1) denote the quotient of Zp+1 (X, 1) by the subgroup generated by such elements. This is a basic example of Bloch’s higher Chow group 293
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([B]). For any abelian group A, let AQ denote A ⊗ Q. Then it may be worthwhile to point out the isomorphisms 2p+1 CHp+1 (X, 1)Q ' HM (X, Q(p + 1)) ' Grp+1 γ K1 (X) ⊗ Q, ∗ (X, Q(∗∗)) denotes the bigraded motivic cohomology of X, and where HM Grrγ denotes the r-th graded piece defined by the gamma filtration on K∗ (X). There is another way to construct classes in this group, and that is to make use of the product map
CHp (X) ⊗ k ∗ → CHp+1 (X, 1), the image of this being generated by the decomposable classes {(Y, α)}, where each Y is a codimension p subvariety and α a non-zero scalar. It should also be mentioned, for motivation, that when k is a number field, Beilinson predicts that the elements in CHp+1 (X, 1)Q which come from a regular proper model X over Z span a Q-vector space of dimension equal to the order of vanishing at s = p of the associated L-function L(s, H 2p (X)), which we will denote by L(s, X) if there is no confusion. By the expected functional equation, the order of pole at s = p+1 of L(s, X), denoted ran (X), will be the difference of the order of pole of the Gamma factor L∞ (s, X) at s = p and the order of zero of L(s, X) at s = p. The celebrated conjecture of 2p Tate asserts that ran (X) is the dimension of Im CHp (X) → Het (XQ , Q` ) for any prime `. One of the main objects here is to sketch a proof of the Tate conjecture in codimension 2 for Hilbert modular fourfolds, and also deduce the Hodge conjecture under a hypothesis. Going back to CHp+1 (X, 1), the first case of interest is when X is a surface and p = 1. When X is the Jacobian J0 (37) of the modular curve X0 (37), Bloch constructed a non-trivial example β ∈ CH2 (X, 1)Q by using the curve and the fact that J0 (37) is isogenous to a product of two elliptic curves over Q. This was generalized by Beilinson ([Be]; see also [Sch]) to a product of two modular curves by going up to a (ramified) cover X0 (N )×X0 (N ) and by taking {Yj } to be the union of the diagonal ∆ and the curves X0 (N ) × {P } and {Q} × X0 (N ), where P, Q are cusps; the existence of the functions fj came from the Manin-Drinfeld theorem saying that the difference of any two cusps is torsion in the Jacobian. Later the author generalized this ([Ra1, Ra2]) to the case of Hilbert modular surfaces X by using a class of curves on X called the Hirzebruch-Zagier cycles, carefully chosen to have appropriate intersection properties; in general these curves meet in CM points or cusps. The second main goal of this article is to describe briefly the ideas behind an ongoing project of the author involving the construction of (Q-rational)
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classes in CH3 (X, 1)Q for certain modular fourfolds X/Q. We will restrict ourselves to Hilbert modular fourfolds defined by a biquadratic, totally real field F . Very roughly, the basic idea is to use suitable translates Yj of embedded Hilbert modular surfaces (coming from the three quadratic subfields), choose Zj to be made up of (translates of) embedded modular curves which are homologically trivial, hence rationally trivial (as the Yj are simply connected), and also use the fact that the Tate and Hodge conjectures are known (by the author [Ra4]) for codimension 2 cycles on X (for square-free level), as well as the knowledge (loc. cit.) that a basis of Q-rational cycles modulo homological equivalence (in the middle dimension) on X, and on the embedded Hilbert modular surfaces, is given by appropriate Hirzebruch-Zagier cycles and their twists. We will first construct decomposable classes and then indicate some candidates for the indecomposable part. The ultimate goal is to understand this phenomenon for Shimura varieties X, of which modular curves and Hilbert modular varieties are examples. It is known that the most interesting (cuspidal) part of the cohomology of X is in the middle dimension, which leads us to consider such X of dimension n = 2m and take p = m. When there are Shimura subvarieties Y of X of dimension m with H 1 (Y ) = 0, like for Siegel modular varieties, one can hope to construct promising classes in CHm+1 (X, 1)Q . This will be taken up elsewhere. This article is dedicated to Jaap Murre, from whom I have learnt a lot over the years – about algebraic cycles and about the (conjectural) ChowK¨ unneth decompositions, though they exist for simple reasons in the cases considered here. We have a long term collaboration as well on the zero cycles on abelian surfaces. I would also like to acknowledge a helpful conversation I had with Spencer Bloch about CH∗ (X, 1) in the modular setting (see section 7.14). I thank the referee and Mladen Dimitrov for spotting various typos on an earlier version, and for making suggestions for improvement of exposition. Finally, I am pleased to acknowledge the support of the National Science Foundation through the grant DMS-0402044.
7.2 Notation Let X be a smooth projective fourfold over a number field k. Set: VB = H 4 (X(C), Q) Hg 2 (X) = VB ∩ H 2,2 (X(C)) rHg = dimQ Hg 2 (X) Gk = Gal(k/k)
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`: a prime V` = H 4 (XQ , Q` ): Gk -module ralg,k = dim Im(CH2 (X)Q → V` (2)) T`,k = V` (2)Gk r`,k = dimQ` T`,k Sbad := {P | V` 6= V`IP } ∪ {P | `} S: a finite set of places ⊃ Sbad F rP : geometric Frobenius at P, ∀P ∈ / Sbad with norm N (P) Q −1 L(s, X) = det(I − F rP T |V` )| −s P∈S /
T =N (P)
By Deligne’s proof of the Weil conjectures, the inverse roots of F rP on V` are, for P ∈ / S, of absolute value N (P)2 , implying that the L-function L(s, X) converges absolutely in {<(s) > 3}. The boundary point s = 3, where L(s, X) could be divergent, is called the Tate point. The fourfolds of interest to us will admit meromorphic continuation to the whole s-plane and satisfy a functional equation relating s to 5 − s. Put ran,k = −ords=3 L(s, X). Tate’s conjecture is that this analytic rank equals the algebraic rank ralg,k of codimension 2 algebraic cycles on X modulo (`-adic) homological equivalence. It is also expected that these two ranks are the same as the `-adic cycle rank r`,k , and one always has ralg,k ≤ r`,k .
7.3 Hilbert Modular fourfolds Let K be a quartic, Galois, totally real number field with embedding K ,→ R4 given by the archimedean places. Fix a square-free ideal N in the ring OK of integers of K, and write Γ for the congruence subgroup Γ1 (N) ⊂ SL(2, OK ) of level N. Then there is a natural embedding Γ ,→ SL(2, R) × SL(2, R) × SL(2, R) × SL(2, R), γ → (γ σ )σ∈Hom(K,R) . Using this one gets an action of Γ on the four-fold product of the upper half plane H = SL(2, R)/SO(2). The quotient Y = Γ\H4 is a coarse moduli space of polarized abelian fourfolds A with Γ-structure, with End(A) ←- O ⊂ K. It is a quasi-projective variety, with Baily-Borel-Satake compactification Y ∗ , and a smooth toroidal compactification X := Y˜ = Y ∪ Y˜ ∞ , all defined over Q. For simplicity of exposition, we have used here the classical formalism. Later, we will need to work with the adelic version SC1 (N) relative to the
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standard compact open subgroup C1 (N) of G(AK,f ), where AK,f denotes the ring of finite adeles of K; one has C1 (N) ∩ GL(2, F ) = Γ1 (N). Moreover, SC1 (N) (C) = GL(2, K)\(C − R)4 × GL(2, AK,f )/C1 (N), which is finitely connected, and Γ1 (N)\H4 occurs as an ´etale quotient of the connected component. (If the object is to realize Γ1 (N)\H4 as exactly the connected component, one needs to consider instead the Shimura variety associated to the Q-group G with G(Q) = {g ∈ GL(2, K) | det(g) ∈ Q∗ }.) The Shimura variety SC1 (N) is defined over Q, and the same holds for its Baily-Borel-Satake compactification SC∗ 1 (N) . One can also choose a smooth toroidal compactification X = S˜C1 (N) over Q. Again we will use Y , resp. Y˜ ∞ , to denote SC (N) , resp. the boundary, so that X = Y ∪ Y˜ ∞ . 1
7.4 Results on Cycles of codimension 2 on X Let X be the smooth toroidal compactification over Q of a Hilbert modular fourfold of square-free level N as above, relative to a quartic Galois extension K of Q. Theorem 7.4.1 ([Ra4]). i) The Tate classes in V` (2) are algebraic. In fact, r`,k = ralg,k = ran,k . ii) If N is a proper ideal, the Hodge classes in VB (2) are algebraic when they are not pull-backs of classes from the full level, and moreover, they are not all generated by intersections of divisors. The next few sections indicate a proof of this, while at the same time developing the theory and setting the stage for what is to come afterwards. Now let K be biquadratic so that Gal(K/Q) = {1, σ1 , σ2 , σ3 = σ1 σ2 }, with 2 σj = 1 for each j. Let Fj ⊂ K be the real quadratic field obtained as the fixed field of σj . For every g ∈GL+ 2 (K), let YFj ,g denote the closure in X of the image of gH2 , which identifies with the Hilbert modular surface attached to Fj and the congruence subgroup g −1 Γg ∩ SL(2, OFj ). It is the natural analogue of the Hirzebruch-Zagier cycle on a Hilbert modular surface. The proof of 7.4.1 has as a consequence the following: Theorem 7.4.2. Let K be biquadratic with intermediate quadratic fields F1 , F2 , F3 . Define N > 0 by N Z = N ∩ Z, and assume that the modular curve X0 (N ) has genus > 0. Then there exist g1 , g2 ∈ GL+ 2 (K) such that YF1 ,g1 , YF2 ,g2 span a 2-dimensional subspace of ralg,Q . Consequently, dimQ CH2 (X) ≥ 2.
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By the product structure, one gets non-trivial, decomposable classes in CH3 (X, 1). A refinement will be discussed in section 7.11.
7.5 Contribution from the boundary By the decomposition theorem, there is a short exact sequence 0 → IH 4 (Y ∗ ) → H 4 (X) → HY4˜ ∞ (X) → 0, both as Galois modules and as Q-Hodge structures. Here IH ∗ is the GoreskyMacPherson’s middle intersection cohomology. We obtain, for α = {`, k}, an, alg, rα (X) = rα (Y ∗ ) + rα∞ , where rα (Y ∗ ), resp. rα∞ , is the α-rank associated to IH 4 (Y ∗ ), resp. HY4˜ ∞ (X). The cohomology with supports in Y˜ ∞ has a nice description: ∗ HY∗˜ ∞ (X) ' ⊕σ:cusp HD (X), σ
where Dσ is a divisor with normal crossings (DNC) with smooth irreducible components Dσi . If Dσi,j denotes Dσi ∩ Dσj , there is an exact sequence X X 4 H 2 (Dσi )(1) → HD (X)(2) → H 0 (Dσi,j ) σ i
i6=j
Since Dσi is toric, its H 2 is generated by divisors. This implies the following string of equalities for large k: ∞ ∞ ∞ ∞ = rHg . = ran,k = r`,k ralg,k
All but the last equality on the right remain in force for any number field k. Hence the problem reduces to understanding the rα (Y ∗ ) for various α and explicating their relationships with each other.
7.6 The action of Hecke correspondences −1 If g ∈GL+ 2 (K), there are two maps YΓg → YΓ , with Γg = Γ∩g Γg, inducing ∗ an algebraic correspondence Tg , which extends to YΓ . The algebra H of such Hecke correspondences acts semisimply on cohomology. This leads to a H × GQ -equivariant decomposition
IH 4 (Y ∗ ) ' Vres ⊕ Vcusp
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where the submotive Vres is algebraic, and the cuspidal submotive Vcusp is the “interesting” part (see below). To be precise, the residual part Vres is spanned by the intersections of Chern classes of certain universal line bundles Lij , 1 ≤ i, j ≤ 4 occurring at every level N. In the complex realization they are defined by the SL(2, R)4 invariant differential forms ηi ∧ ηj , 1 ≤ i 6= j ≤ 4 on H4 , where for each j, ηj = pr∗j (dzj ∧ dz j ), with prj : H4 → H being the j-th projection. Remark: In our case one knows enough about the Galois modules which occur in H · (X) to be able to get a direct sum decomposition: H · (X) ' IH · (Y ∗ ) ⊕ HY·˜ ∞ (X) IH · (Y ∗ ) can be cut out as a direct summand by an algebraic cycle modulo homological equivalence. (This has recently been done for general Shimura varieties by A. Nair ([N]).) The reason is that the Hecke correspondences act on an inverse limit of a family of toroidal compactifications of Y , though not on any individual one. However, IH · (Y ∗ ) is not immediately a Chow motive, since the algebra of Hecke correspondences modulo rational equivalence is not semisimple. The K¨ unneth components of ∆ in IH 8 (Y ∗ × Y ∗ ) are algebraic, since it is known that IH 1 (Y ∗ ) = IH 3 (Y ∗ ) = 0 and IH 2 (Y ∗ ) is algebraic, being purely of Hodge type (1, 1). There is a further H × GQ -equivariant decomposition: Vcusp = ⊕ϕ V (ϕ)m(ϕ) , where ϕ runs over holomorphic Hilbert modular cusp forms of level N, which have (diagonal) weight 2, and m(ϕ) is a certain multiplicity which is 1 if ϕ is a newform, i.e., not a cusp form of level a proper divisor of N.
7.7 The submotives of rank 16 It is now necessary to understand V (ϕ) for a Hilbert modular newform ϕ of weight 2 and level N. It is easy to see that VB (ϕ) is 16-dimensional, generated over C by the differential forms ϕ(z)dzI ∧ dz J of degree 4 for partitions {1, 2, 3, 4} = I ∪ J. Let π be the cuspidal automorphic representation of GL(2, AK ) of trivial central character associated to ϕ. We will write V (π) instead of V (ϕ). By R.L. Taylor and Blasius-Rogawski, one can associate a 2-dimensional
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irreducible representation W` (π) of GK such that L(s, W` (π)) = L(s, π), i.e., ∀P - N , tr (F rP |W` (π)) = aP (π). By Brylinski-Labesse, as refined by Blasius, tr (F rP |V` (π)) = L(s, πP ; r), ∀P - N . Since V` (π) and W` (π) are semisimple, we get the following isomorphism by Tchebotarev: V` (π)|GK ' ⊗τ ∈Gal(K/Q) W` (π)[τ ] To be precise, V` (π) is the tensor induction ([Cu-R]) of W` (π) from K to Q: G
V` (π) ' ⊗ IndGQK (W` (π)) This identity makes it possible to compute the Tate classes. We need the following result: Theorem 7.7.1 ([Ra3, Ra4]). The L-function of V` (π) admits a meromorphic continuation to the whole s-plane with a functional equation of the form L(s, V` (π)∨ ) = ε(s, V` (π))L(5 − s, V` (π)), where ε(s, V` (π)) is an invertible function on C and the superscript ∨ indicates the dual. From this one also gets the analogous statement about the L-function of X, which is a product of these L(s, V` (π)) with an abelian L-function. 7.8 Strategy for algebraicity When r` (π) 6= 0, we show first that L(s, V` (π)) has a pole at s = 3. But then we also show, using a specific integral representation, that for a suitable quadratic field F ⊂ K, the function L1,F (s) := L(s, ⊗ IndGGFK (W` (π)) ⊗ ν) has ν a simple R pole. What we do then is to construct an algebraic cycle ZF , and prove ν ZF ω 6= 0 for a (2, 2)-form ω by realizing it as ress=2 L1,F (s). Using the previous section, we first prove Proposition For any Dirichlet character χ, r` (π, χ) = ran (π, χ) ≤ 2 r` (π, χ) = 1 iff a twist π ⊗ ν is fixed by an involution τ ∈Gal(K/Q), while r` (π, χ) = 2 iff K is biquadratic and π ⊗ ν is Gal(K/Q)-invariant.
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Now the problem is to construct algebraic cycles in the (π ⊗ν)-eigenspace, which are of infinite order modulo homological equivalence, to account for these poles when π ⊗ ν is fixed by one or more (non-trivial) elements of Gal(K/Q).
7.9 Cycles Let F be a quadratic subfield of K, with corresponding embedding h2 ,→ 2 h4 . Recall that if g ∈ GL+ 2 (K), the image of the translate gh under the projection h4 → Γ\h4 defines a surface ∆g \h2 with closure ZF,g in X. This is an example of a Hirzebruch-Zagier cycle; see [K] for a definition of such cycles for orthogonal Shimura varieties. For any abelian character µ, there is a µ-twisted Hirzebruch-Zagier cycle µ of codimension 2 in X. This is defined (cf. [Ra4]) by composing the ZF,g above construction with a twisting correspondence defined by µ, which sends, for every π, the π-isotypic subspace onto the (π ⊗ µ)-isotypic subspace of the cohomology. Suppose r` (π, χ)(= ran (π, χ)) is > 0. Then there is a quadratic subfield F and a cusp form π1 on GL(2)/F such that π ⊗ ν ' (π1 )K . As one would µ provides the requisite algebraic hope, a twisted Hirzebruch-Zagier cycle ZF,g cycle to get the Tate conjecture. There is a real subtle point here which separates it from the work of Harder, Langlands and Rapoport ([H-L-R]) on the divisors on Hilbert modular surfaces: µ is non-zero, but The period of a (2, 2)-form on X (defined by π) over ZF,g it is the residue of a different L-function, namely L1,F (s), which does not divide L(s, V` (π))! The residues of the two L-functions are presumably related in a non-trivial way, but this is not known. It is an intriguing problem to try to understand this better.
7.10 Hodge classes By hypothesis, we need only consider those π which are of level N 6= OK . We may then fix a prime divisor P of N and consider a quaternion algebra B/K which is ramified only at three infinite places and at P. By the EichlerShimizu-Jacquet-Langlands correspondence, there exists a corresponding cusp form π 0 on B ∗ giving rise to a submotive V (π 0 ) of H 4 (RK/Q (C)) for a
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Shimura curve C = ∆\h defined over K. As Gal(Q/K)-modules, V` (π 0 ) ' V` (π), implying that r`,E (π 0 ) = r`,E (π),
(7.1)
for any number field E ⊃ K. On the Hodge side we need the following result, proved jointly with V.K. Murty, which is really a statement about periods: Theorem ([Mu-Ra2])). As Q-Hodge structures, VB (π 0 ) ' VB (π). The proof compares the coefficients of the Shimura liftings of π and π 0 to forms of weight 3/2, which live on the two-sheeted covering group of GL(2)/K. By Deligne [D-M-O-S], every Hodge class on an abelian variety gives rise to a Tate class. One gets from this the equality over a sufficiently large field E ⊃ K: r`,E (π 0 ) = rHg (π 0 ).
(7.2)
On the other hand, by the Theorem with Murty, one also gets rHg (π 0 ) = rHg (π).
(7.3)
Combining (7.1), (7.2) and (7.3), we see that the Hodge conjecture for VB (π) follows from the Tate conjecture for V` (π) over E.
7.11 Where the cycles come from The method of proof furnishes the following: Theorem 7.11.1. Let F be a quadratic subfield of K. Then a twisted Hirzebruch-Zagier cycle of codimension 2 on X associated to F contributes to V (π) iff a twist of π is a base change from F . When K is biquadratic and a twist of π is base changed from Q, i.e., attached to an elliptic cusp form h, we get ralg,Q (π) = 2, with the Hecke twisted Hilbert modular surfaces from two subfields F1 , F2 , say, give non-trivial independent algebraic classes of codimension 2.
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As a consequence, the decomposable part of CH3 (Xπ , 1) has rank at least 2 when π is a base change from Q and K biquadratic. (Here Xπ refers to the submotive of [X] in degree 4 cut out by π.) But since Z∗ = {±1}, such classes will not come from a regular, proper model X of X (when such a model exists). Now let N > 0 be defined by N Z = N ∩ Z. Then when the modular curve X0 (N ) has positive genus, there exists at least one (elliptic) newform h of weight 2 and level N , and the base change to K of the associated cuspidal automorphic representation on GL(2)/Q, assures the existence of a π as in Theorem 7.11.1. Hence Theorem 7.4.2 follows from Theorem 7.11.1.
7.12 What to expect A straight-forward calculation shows that −ords=2 L∞ (s, V (π)) = 3. In the biquadratic case, if a twist of π is a base change from Q, we have ran,Q (π ∨ ) = 2. Hence by the functional equation, ords=2 L(s, V (π)) = 1. In this case, Beilinson predicts (in [Be]) the existence of a non-trivial class β in CH3 (Xπ , 1)Q which comes from a proper model X over Z, to account for this simple zero of the L-function. In the cyclic case, ran,Q (π ∨ ) = 1, and so the we should have two independent classes in the higher Chow group. The general philosophy is that it is much harder to produce classes in a motivic cohomology group (or a Selmer group) which is conjecturally of rank bigger than one, and this is why we are not at present concentrating on this (cyclic) case.
7.13 Elements in CH3 (X, 1)Q Let K/Q be biquadratic with quadratic subfields F1 , F2 , F3 . For g1 , g2 , g3 ∈ GL+ 2 (K), consider the surfaces Zi = ZFi ,gi , 1 ≤ i ≤ 3, which are Hecke translates of the three Hilbert modular surfaces in X associated to {Fi |1 ≤ i ≤ 3}. Put Ci,j = Zi ∩ Zj ,
for 1 ≤ i 6= j ≤ 3.
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Theorem 7.13.1. One can find g1 , g2 , g3 in GL+ 2 (K), and an integer m > 0, such that, up to modifying the construction by decomposable classes supported on the boundary and on Xres , we have for each i ≤ 3 and a permutation (i, j, k) of (1, 2, 3), m(Ci,j − Ci,k ) = div(fi ) for some functions fi on Zi . Here is the basic idea: Each Zi is a Hecke translate of a Hilbert modular surface Si , which is simply connected. So P ic(Zi ) = N S(Zi ). We show that the homology class [gi−1 (Ci,j − Ci,k )] is trivial in H 2 (Si , Q(1)), up to modifying by the trivial classes coming from the boundary and the residual part. Thanks to the explicit description of the algebraic cycles (modulo homological equivalence) on Hilbert modular surfaces (see [H-L-R, Mu-Ra1]), one knows that in the situation we are in, the divisor classes are spanned by Hecke translates of modular curves. From this it is not too difficult to show (for suitable {gi }) the homological triviality (modulo trivial cycles) of [gi−1 (Ci,j − Ci,k )] when the Hilbert modular surface Si has geometric genus 1; in fact it is enough to know that there is a unique base changed newform πi of weight 2 over Fi for this. In general one has to deal with several newforms, and one uses a delicate refinement of an argument of Zagier [Z, page 243)]. The subtlety comes from the fact that one needs to deal with three quadratic fields at the same time. A simple example is when g1−1 g2 , g1−1 g3 are diagonal matrices in GL+ 2 (K), (Q), fixed by Gal(K/F ), Gal(K/F ) respectively. For each i ≤ 3, not in GL+ 1 3 2 Cij , Cik are Hecke translates of modular curves on Zi . Thanks to 7.13.1, the formal sum 3 X
(Zi , fi )
i=1
P
satisfies i div(fi ) = 0 as a codimension 2 cycle on X, and hence defines a class in CH3 (X, 1) ⊗ Q. Problem. Compute, for ω ∈ H (2,2) (X(C)), XZ log |fi | ω i
Zi
We can understand this period integral a bit better in the analogous situation where X is the four-fold product of a modular curve X0 (N ) for prime level N , the simplification arising from the fact that one can reduce to conP sidering f of the form rj=1 f1,j ⊗f2,j ⊗f3,j ⊗f4,j , with each fi,j a translate of
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a modular unit on X0 (N ). Here the Hilbert modular surfaces are replaced by the twisted images of X0 (N )2 → X0 (N )4 , which are not simply connected, but anyhow, their Pic0 is generated by elementary divisors of degree 0, and this suffices. Work is in progress to prove for N = 11, by a combination of theoretical and numerical arguments, that the integral is non-zero for a suitable choice; this will show that the class is of infinite order. We hope to investigate if such a class can give any information on the Bloch-Kato Selmer group at s = 0 of the Sym4 motive, twisted by Q(2), of X0 (N ). 7.14 The modular complex Let X be a smooth, toroidal compactification of a Shimura variety of dimension n over its natural field of definition k. Consider the class of closed irreducible subvarieties of X, called modular, generated by components of the twisted Hecke translates of Shimura subvarieties, the boundary components, the components of their intersections, and so on. For every p ≥ 0, p denote the set of such subvarieties of codimension p. The modlet Xmod ular points in this setting will be the CM points and the points arising from successive intersections of components at the boundary. To give a concrete example, consider the case of a Hilbert modular surface X which is obtained by blowing up each cusp into a cycle of rational curves. When Γ0 (N) is torsion-free, the modular points on X will be the CM points and the points where the rational curves over cusps intersect. We can now consider the modular analogue of the Gersten complex, namely a a a ··· → K2 (k(W )) → k(Z)∗ → Z.Y p+2 W ∈Xmod
p+1 Z∈Xmod
p Y ∈Xmod
We may look at the homology of this complex, and denote the resulting groups - the first two from the right – by B p (X) and B p+1 (X, 1). There are natural maps B p (X) → CHp (X),
and B p+1 (X, 1) → CHp+1 (X, 1)
Denote the respective images by CHpmod (X) and CHp+1 mod (X, 1). The nice thing about these groups is that since the modular subvarieties are all defined over number fields, the building blocks do not change from Q to C. It will be very interesting (exciting?) to try to verify, in some concrete cases of dimension ≥ 2, whether CHpmod (X) and CHp+1 mod (X, 1) are finitely generated over k. Note that the classes we consider in this article are modular in this sense.
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When n = 2m and when L(s, M ) has, for a simple submotive M of H n (X), a simple zero at s = m, resp. a simple pole at s = m + 1, it is tempting to ask, in view of the known examples, if there is an element of CHm+1 mod (X, 1), resp. CHm (X), which explains it. When n = 2m − 1 and L(s, M ) has a simple mod zero at s = m, one could again ask if it is explained by a modular element of CHm (X)0 , the homologically trivial part. For modular curves X0 (N )/Q, one has striking evidence for this in the work of Gross and Zagier. For Hilbert modular surfaces it is again true ([H-Z, H-L-R]). The situation is not the same when the order of pole or zero is 2 or more, especially over non-abelian extensions of k ([Mu-Ra1]).
References [Be] Beilinson, A.A.: Higher regulators and values of L-functions, Journal of Soviet Math. 30, No. 2, 2036–2070 (1985). [B] Bloch, S.: Algebraic cycles and higher K-theory. Advances in Math. 61, no. 3, 267–304 (1986). [Cu-R] Curtis, C.W. and I. Reiner: Methods of representation theory I, Wiley, NY (1981). [D-M-O-S] Deligne, P., J.S. Milne, A. Ogus and K-Y. Shih: Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics 900. SpringerVerlag, Berlin-New York (1982). [H-L-R] Harder, G.,R.P. Langlands and M. Rapoport: Algebraische Zykeln auf Hilbert-Blumenthal-Fl¨ achen, Crelles Journal 366 (1986), 53–120. [H-Z] Hirzebruch, F. and D. Zagier: Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Inventiones Math. 36, 57–113 (1976). [K] Kudla, S.: Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. Journal 86, no. 1, 39–78 (1997). [Mu-Ra1] Murty, V.K. and D. Ramakrishnan: Period relations and the Tate conjecture for Hilbert modular surfaces, Inventiones Math. 89, no. 2 (1987), 319–345. [Mu-Ra2] Murty, V.K. and D. Ramakrishnan:Comparison of Q-Hodge structures of Hilbert modular varieties and Quaternionic Shimura varieties, in preparation. [N] Nair, A.: Intersection cohomology, Shimura varieties, and motives, preprint (2003). [Ra1] Ramakrishnan, D.: Arithmetic of Hilbert-Blumenthal surfaces, CMS Conference Proceedings 7, 285–370 (1987). [Ra2] Ramakrishnan, D.: Periods of integrals arising from K1 of HilbertBlumenthal surfaces, preprint (1988); and Valeurs de fonctions L des surfaces d’Hilbert-Blumenthal en s = 1, C. R. Acad. Sci. Paris S´er. I Math. 301, no. 18, 809–812 (1985) [Ra3] Ramakrishnan, D.: Modularity of solvable Artin representations of GO(4)type, International Mathematics Research Notices (IMRN) 2002, No. 1 (2002), 1–54. [Ra4] Ramakrishnan, D.: Algebraic cycles on Hilbert modular fourfolds and poles
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of L-functions, in Algebraic Groups and Arithmetic, 221–274, Tata Institute of Fundamental Research, Mumbai (2004). [Sch] Scholl, A.J.: Integral elements in K-theory and products of modular curves, in The arithmetic and geometry of algebraic cycles, 467–489, NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer Acad. Publ., Dordrecht 2000. [Z] Zagier, D.: Modular points, modular curves, modular surfaces and modular forms, Workshop Bonn 1984, 225–248, Lecture Notes in Math. 1111, Springer, Berlin-New York (1985).
Part II Research Articles
8 Beilinson’s Hodge Conjecture with Coefficients for Open Complete Intersections Masanori Asakura and Shuji Saito
Dedicated to Professor J.P. Murre on the occasion of his 75th birthday 8.1 Introduction Let U be a smooth algebraic variety over C and let U an be the analytic site on U (C), the associated analytic space. An important object to study in algebraic geometry is the regulator map from the higher Chow group ([7]) to the singular cohomology of U (cf. [18]) √ q q p an q p an regp,q U : CH (U, 2q − p) ⊗ Q → (2π −1) W2q H (U , Q) ∩ F H (U , C), where F ∗ and W∗ denote the Hodge and the weight filtrations of the mixed Hodge structure on the singular cohomology defined by Deligne [8]. For the special case p = q, we get √ regqU : CH q (U, q)⊗Q → H q (U an , Q(q))∩F q H q (U an , C). (Q(q) = (2π −1)q Q) Beilinson’s Hodge conjecture claims the surjectivity of regqU (cf. [11, Conjecture 8.5]). In [4] we studied the problem in case U is an open complete intersection, namely U is the complement in a smooth complete intersection X of a simple normal crossing divisor Z = ∪sj=1 Zj on X such that Zj ⊂ X is a smooth hypersurface section. One of the main results affirms that regqU is surjective if the degree of the defining equations of X and Zj are sufficiently large and if U is general in an appropriate sense. Indeed, under the assumption we have shown a stronger assertion that regqU is surjective even restricted on the subgroup CH q (U, q)dec of decomposable elements in CH q (U, q), which is not true in general. In order to explain this, let KqM (O(U )) be the Milnor K-group of the ring O(U ) = Γ(U, OZar ) (see §8.2.3 for its definition). We have the natural map σU : KqM (O(U )) → CH q (U, q) 311
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induced by cup product and the natural isomorphism ∼ =
K1M (O(U )) = Γ(U, O∗Zar ) −→ CH 1 (U, 1) and CH q (U, q)dec is defined to be its image. Note that we have the following formula for the value of regqU on decomposable elements; regqU ({g1 , . . . , gq }) = [g1 ] ∪ · · · ∪ [gq ] ∈ H q (U an , Q(q)) = dlog g1 ∧ · · · ∧ dlog gq ∈ H 0 (X, ΩqX/C (log Z)) = F q H q (U an , C) where gj ∈ O(U )∗ for 1 ≤ j ≤ q and [gj ] ∈ H 1 (U an , Q(1)) is the image of gj under the map O(U )∗ → H 1 (U an , Z(1)) induced by the exponential sequence exp
0 → Z(1) → OU an −−→ O∗U an → 0. In what follows we are mainly concerned with the map regqU : KqM (O(U )) ⊗ Q → H q (U an , Q(q)) ∩ F q H q (U an , C)
(8.1)
which is the composition of the regulator map and σU . Now we consider the following variant of the above problem. Assume that we are given a smooth algebraic variety S over C and a smooth surjective morphism π : U → S over C. Let π∗an : U an → S an be the associated morphism of sites. Assume that the fibers of π are affine of dimension m. Then Rb π∗an Q = 0 for b > m and we have the natural map α : H m+q (U an , Q(m + q)) → H q (S an , Rm π∗an Q(m + q)) which is an edge homomorphism of the Leray spectral sequence E2a,b = H a (S an , Rb π∗an Q(m + q)) ⇒ H a+b (U an , Q(m + q)). Note that H a (S an , Rb π∗an Q(m+q)) carries in a canonical way a mixed Hodge structure and α is a morphism of mixed Hodge structures ([17] and [1]). Let M q an m an m+q regm+q U/S : Km+q (O(U )) ⊗ Q → H (S , R π∗ Q(m + q)) ∩ F
(8.2)
and α where F t ⊂ H q (S an , Rm π∗an C) denotes be the composition of regm+q U the Hodge filtration. In this paper we study regm+q U/S in case U/S is a family of open complete intersections, namely in case that the fibers of π are open complete intersections. Roughly speaking, our main results affirm that regm+q U/S is surjective for q = 0, 1 if π : U → S is the pullback of the universal family of open complete intersection of sufficiently high degree via a dominant smooth morphism from S to the moduli space. Let di , ej ≥ 0 (1 ≤ i ≤ r, 1 ≤ j ≤ s) be fixed integers. Let M = M(d1 , · · · , dr ; e1 , · · · , es )
Beilinson’s Hodge Conjecture with Coefficients
313
be the moduli space of the sets (X1,o , . . . , Xr,o ; Y1,o . . . , Ys,o ) of smooth hypersurfaces in Pn of degree d1 , · · · , dr ; e1 , · · · , es respectively which intersect transversally with each other. Let f : S → M be a morphism of finite type with S = Spec R nonsingular affine and let Xi → S and Yj → S be the pullback of the universal families of hypersurfaces over M. Put X = X1 ∩ · · · ∩ Xr
and
U = X\
∪
1≤j≤s
X ∩ Yj
with the natural morphisms π : U → S. Put d=
r X
di ,
δmin = min {di , ej }, 1≤i≤r
i=1
dmax = max {di }. 1≤i≤r
1≤j≤s
Theorem 8.1.1. (see §8.3) Assume f is dominant smooth. (1) Assuming δmin (n − r − 1) + d ≥ n + 1, M 0 an m an regm U/S : Km (O(U )) ⊗ Q → H (S , R π∗ QU (m + 1))
is surjective. (ii) Assuming δmin (n − r − 1) + d ≥ n + 2, δmin (n − r) + d ≥ n + 1 + dmax , δmin ≥ 2, M 1 an m an m+1 regm+1 U/S : Km+1 (O(U )) ⊗ Q → H (S , R π∗ QU (m + 1)) ∩ F
is surjective. The method of the study is the infinitesimal method in Hodge theory and is a natural generalization of that in [3] and [4]. To explain this, we now work over an arbitrary algebraically field k of characteristic zero which will be fixed in the whole paper. Let f : S → M and π : U → S be defined over k as above. Following Katz and Oda ([12]), we have the algebraic Gauss Manin connection on the de Rham cohomology (see §8.2.2) • • (U/S) ⊗R Ω1R/k . ∇ : HdR (U/S) −→ HdR
(8.3)
q+1 • (U/S) ⊗ Ωq • The map ∇ is extended to HdR R R/k −→ HdR (U/S) ⊗R ΩR/k by imposing the Leibniz rule
∇(e ⊗ ω) = ∇(e) ∧ ω + e ⊗ dω
(8.4)
and it induces the complex p q p−1 m m m Grp+1 HdR (U/S)⊗Ωq−1 HdR (U/S)⊗Ωq+1 F R/k −→ GrF HdR (U/S)⊗ΩR/k −→ GrF R/k ,
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where m = dim(U/S) and F • denotes the Hodge filtration: q q q F p HdR (U/S) = HZar (X, Ω≥p X/S (log Z)) ⊂ HdR (U/S).
The cohomology at the middle term of the complex has been studied in [3] when 1 ≤ p ≤ m − 1. In the study of the variant of Beilinson’s Hodge conjecture, a crucial role will be played by the kernel of the following map: m ∇q : F m HdR (U/S) ⊗ ΩqR/k −→ Grm−1 HdR (U/S) ⊗ Ωq+1 F R/k
(q ≥ 0).
which arises as the special case p = m in the above complex. The key result g i is, roughly speaking, that when f : S → M factors as S −→ T −→ M where g is smooth and i is a regular immersion of small codimension, then the kernel of ∇q is generated by the image of M m (U/S) ⊗ ΩqR/k dlog : Km+q (O(U )) −→ F m HdR
(see §8.2.3 for its definition). In case k = C it implies the surjectivity of regm+q U/S (8.2) for q = 0 and 1 by using the known surjectivity of the map (8.1) for U = S. The main tool for the proof of the above key result is the theory of generalized Jacobian rings developed by the authors in [3]. It describe the Hodge cohomology groups of U and the Gauss-Manin connection ∇q in terms of multiplication of the rings, so that the various problems can be translated into algebraic computations in Jacobian rings. We show several computational results on Jacobian rings in §8.4 and §8.5. The basic techniques for this were developed by M.Green, C.Voisin and Nori. We note that a key to the computational results is Proposition 8.5.5, which is proved in [3] as a generalization of Nori’s connectivity theorem ([14]) to open complete intersections. Notation and Conventions For an abelian group M , we write MQ = M ⊗Z Q.
8.2 The Main Theorem Throughout the paper, we work over an algebraically closed field k of characteristic zero.
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8.2.1 Setup We fix integers n ≥ 2, r, s ≥ 1, n ≥ r and d1 , · · · , dr , e1 , · · · , es ≥ 1. We put d =
r X
di ,
e=
i=1
dmax =
s X
ej ,
δmin = min {di , ej }, 1≤i≤r
j=1
max {di },
1≤j≤s
emax = max {ej }.
1≤i≤r
1≤j≤s
Let P = k[X0 , · · · , Xn ] be the polynomial ring over k and P d denote the subspace of the homogeneous polynomials of degree d. Then the space P d −{0} parametrizes hypersurfaces in Pn of degree d with a chosen defining equation. Let f = M(d f 1 , · · · , dr ; e1 , · · · , es ) ⊂ M
r s Y Y (P di − {0}) × (P ej − {0}) i=1
j=1
be the Zariski open subset such that the associated divisor X1,o +· · ·+Xr,o + f is a simple normal crossing divisor on Y1,o + · · · + Ys,o to any point o ∈ M n P , namely all Xi,o and Yj,o are nonsingular and they intersect transversally with each other. Put Xo = X1,o ∩ · · · ∩ Xr,o and Zj,o = Xo ∩ Yj,o . Then Xo P is a nonsingular complete intersection of dimension n − r, and sj=1 Zj,o is a simple normal crossing divisor on Xo . f be a morphism of finite type with S = Spec R nonsingular Let f : S → M affine. We write PR = P ⊗k R and PR` = P ` ⊗k R. Let Fi ∈ PRdi (1 ≤ i ≤ r)
and
e
Gj ∈ PRj (1 ≤ j ≤ s)
(8.5)
be the pullback of the universal polynomials over the moduli space. We denote by X, Xi , Yj and Zj the associated families of the complete intersections Xo , Xi,o , Yj,o and divisors Zj,o respectively. Thus we get the smooth morphisms: πX : X −→ S,
πXi : Xi −→ S,
πYj : Yj −→ S,
πZj : Zj −→ S. (8.6)
We write X∗ =
r X i=1
Xi ,
Y∗ =
s X j=1
Yj ,
Z∗ =
s X
Zj .
j=1
Put U = X − Z∗ and we get π : U → S, a family of open complete intersections.
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8.2.2 Gauss-Manin connection For an integer q ≥ 0 we have the Gauss-Manin connection • • ∇ : HdR (U/S) −→ HdR (U/S) ⊗ Ω1R/k .
(8.7)
• (U/S) is the de Rham cohomology defined as Here HdR k k HdR (U/S) = HZar (X, Ω•X/S (log Z∗ )) = Γ (S, Rk πX∗ Ω•X/S (log Z∗ )),
where the second equality follows from the assumption that S is affine. It is an integrable connection and satisfies the Griffiths transversality: • • ∇(F p HdR (U/S)) ⊂ F p−1 HdR (U/S) ⊗ Ω1R/k
(8.8)
with respect to the Hodge filtration • • F p HdR (U/S) := HZar (X, Ω≥p X/S (log Z∗ )).
(8.9)
n−r (U/S) (Since X is a complete intersection, the We are interested in HdR cohomology in other degrees is not interesting). We denote by n−r n−r ∇q : F n−r HdR (U/S) ⊗ ΩqR/k −→ F n−r−1 HdR (U/S) ⊗ Ωq+1 R/k
(8.10)
the map given by (8.4). Noting n−r n−r (U/S) ' H n−r−p (X, ΩpX/S (log Z∗ )), (U/S)/F p+1 HdR F p HdR
(8.8) implies that ∇q induces q q+1 n−r−1 1 ∇q : H 0 (X, Ωn−r X/S (log Z∗ )) ⊗ ΩR/k −→ H (X, ΩX/S (log Z∗ )) ⊗ ΩR/k . (8.11) Our main theorem gives an explicit description of Ker(∇q ) under suitable conditions. For its statement we need more notations.
8.2.3 Milnor K-theory We denote by the Milnor K-group of a commutative ring A ([13, 19]). By definition, it is the quotient of A∗⊗` by the subgroup generated by K`M (A)
a1 ⊗ · · · ⊗ a` ,
(ai + aj = 0 or 1 for some i 6= j).
The element represented by a1 ⊗ · · · ⊗ a` is called the Steinberg symbol, and written by {a1 , · · · , a` }. We have {a1 , · · · , ai , · · · , aj , · · · , a` } = −{a1 , · · · , aj , · · · , ai , · · · , a` }
for i 6= j
following from the expansion {ab, −ab} = {a, b} + {b, a} + {a, −a} + {b, −b}.
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Let O(U ) = Γ(UZar , OU ) be the ring of regular functions on U . We have the dlog map dlog : K`M (O(U )) −→ H 0 (Ω`X/k (log Z∗ )),
{h1 , · · · , h` } 7−→
dh1 dh` ∧···∧ h1 h` (8.12)
Assuming ` ≥ n − r = dim(X/S), there is the unique map n−r (log Z∗ ) ⊗ Ω`−n+r . υX : Ω`X/k (log Z∗ ) −→ ΩX/S S/k
(8.13)
n−r such that its composition with ΩX/k (log Z∗ ) ⊗ Ω`−n+r → Ω`X/k (log Z∗ ) is the S/k `−n+r `−n+r identity map Ωn−r → Ωn−r . Let X/k (log Z∗ ) ⊗ ΩS/k X/S (log Z∗ ) ⊗ ΩS/k `−n+r n−r `−n+r ψ`M : K`M (O(U )) −→ H 0 (Ωn−r = F n−r HdR (U/S)⊗ΩR/k . X/S (log Z∗ ))⊗ΩR/k
be the composition of υX and dlog. Its image is contained in Ker(∇`−n+r ) since it lies in the image of H 0 (Ω`X/k (log Z∗ )). Thus we get the map ψ`M : K`M (O(U )) −→ Ker(∇`−n+r )
(8.14)
We will also consider the induced maps M (O(U )) ⊗Z Ωq−`,d=0 −→ Ker(∇q ); K`+n−r R/k
M (ξ) ∧ ω, ξ ⊗ ω 7→ ψ`+n−r
M K`+n−r (O(U )) ⊗Z Ωq−` R/k −→ Ker(∇q );
M ξ ⊗ ω 7→ ψ`+n−r (ξ) ∧ ω,
• • where Ω•,d=0 R/k = Ker(d : ΩR/k → ΩR/k ) is the module of closed forms. Now we construct some special elements in K`M (O(U )). Let ` ≥ 1 be ` V an integer. We define (Gj ) as the Q-vector space spanned by symbols vJ indexed by multi-indices J = (j0 , · · · , j` ) (1 ≤ jk ≤ s) with relations
vj0 ···jp ···jq ···j` = −vj0 ···jq ···jp ···j`
for 0 ≤ p 6= q ≤ `
(8.15)
and `+1 X
(−1)k ejk vj0 ···bjk ···j`+1 = 0.
(8.16)
k=0
We formally put easily see dimQ and
` V
` ^
0 V
(Gj ) = Q. By convention,
(Gj ) =
s−1 `
` V
(Gj ) = 0 if s = 0 or 1. We
with basis {v1j1 ···j` ; 2 ≤ j1 < · · · < j` ≤ s}, e
(Gj ) = 0 if ` ≥ s. Let Gej i /Gi j |X be the restriction on X of a rational
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M. Asakura and S. Saito e
function Gej i /Gi j on PnR = Proj(R[X0 , . . . , Xn ]). Then we have a natural homomorphism sym` :
` ^
(Gj ) −→ K`M (O(U ))Q .
o n e ej` ej1 ej0 j0 vJ 7→ gJ := e−`+1 | G /G | , · · · , G /G j0 j0 X j1 j0 X j`
(8.17) (J = (j0 , . . . , j` ))
e
Putting gj = Gj /X0 j |X , a calculation shows dlog(gJ ) =
` X
(−1)ν ejν
ν=0
d dgj` dg dgj0 jν ∧···∧ ∧···∧ gj0 gjν gj`
on {X0 6= 0}. (8.18)
The maps ψ•M and sym` induce a homomorphism ΨqU/S :
q `+n−r M ^
(Gj ) ⊗Q Ωq−` R/k −→ Ker(∇q );
gI ⊗ η 7→ ψ`M (gI )
^
η.
`=0
(8.19) The main theorem affirms that this map is an isomorphism under suitable conditions. In order to give the precise statement we need to introduce some notations.
8.2.4 Statement of the Main Theorem Let TR/k be the derivation module of R over k which is the dual of Ω1R/k . A derivation θ ∈ TR/k acts on PR = P ⊗k R = R[X0 , . . . , Xn ] by idP ⊗ θ. Introducing indeterminants µ1 , . . . , µr , λ1 , . . . , λs , we define an R-linear homomorphism Θ = Θ(Fi ,Gj ) : TR/k −→ A1 (0),
θ 7→
r X
θ(Fi )µi +
i=1
s X
θ(Gj )λj .
(8.20)
j=1
where A1 (0) =
r M
PRdi µi
s M M
i=1
e
PRj λj
(PR` = P ` ⊗k R)
(8.21)
j=1
f We note that Θ is surjective (resp. an isomorphism) if f : S = Spec(R) → M is ´etale (resp. smooth). Put W = Im(Θ) ⊂ A1 (0). It is a finitely generated R-module.
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For an ideal I ⊂ PR we denote by A1 (0)/I the quotient of A1 (0) by the submodule r s M M M e (I ∩ PRdi )µi (I ∩ PRj )λj . i=1
j=1
For a variety V over k we denote by |V | the set of the closed points of V . Let α ∈ |PnR | and x ∈ S = Spec(R) be its image with κ(x), its residue field. Let mα,x ⊂ Px := P ⊗R κ(x) be the homogeneous ideal defining α in Proj(Px ) and let mα ⊂ PR be the inverse image of mα,x . The evaluation at α induces an isomorphism (note κ(x) = k) vα : A1 (0)/mα '
r M
k · µi
i=1
s M M
k · λj .
(8.22)
j=1
ΨqU/S
We now introduce the conditions for to be an isomorphism. We fix an integer q ≥ 0. Consider the following four conditions. (I) Both W and A1 (0)/W are locally free R-modules. We put c = rankR (A1 (0)/W ). (II) W has no base points: W → A1 (0)/mα is surjective for ∀α ∈ |PnR |. (III)q One of the following conditions holds: (i) q = 0 and δmin (n − r − 1) + d − n − 1 ≥ c, (ii) q = 1, δmin (n − r − 1) + d − n − 1 ≥ c + 1 and δmin (n − r) + d − n − 1 − dmax ≥ c, (iii) δmin (n − 1) − n − 1 ≥ c + q. (IV)q For any x ∈ |S| and any 1 ≤ j1 < · · · < jn−r ≤ s, there exist q + 1 points α0 , · · · , αq ∈ |X ∩ Yj1 ∩ · · · ∩ Yjn−r | lying over x such that the map M M W → A1 (0)/(J 0 + mα0 ) ··· A1 (0)/(J 0 + mαq ) is surjective. Here J 0 ⊂ A1 (0) denotes the R-submodule generated by the elements s r X X ∂Gj ∂Fi µi + λj ) L·( ∂Xk ∂Xν i=1
with 0 ≤ ν ≤ n and L ∈ PR1 .
j=1
g i f Remark 8.2.1. (I) holds if f factors as S − → T− →M where g is smooth and i is a regular immersion. In this case c = codimM f(T ).
Remark 8.2.2. In view of (8.22), (II) holds if Fi µi , Gj λj ∈ W for ∀i, j and J0 ⊂ W .
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Remark 8.2.3. (IV)q always holds if s ≤ n − r + 1. Indeed we will see (cf. 8.5.3) that for any 1 ≤ j1 < · · · < jn−r ≤ s and any α ∈ |X ∩ Yj1 ∩ · · · ∩ Yjn−r |, A1 (0)/(J 0 + mα ) is a k-vector space of dimension s − 1 − (n − r) and A1 (0)/(J 0 + mα ) = 0 if s − 1 ≤ n − r. Remark 8.2.4. (IV)q holds if W = A1 (0) and δmin ≥ q (cf. §8.2.1). In this case the natural map A1 (0) −→
q M
A1 (0)/mαi
(8.23)
i=0
is surjective for arbitrary (q + 1)-points αi ∈ |PnR | (0 ≤ i ≤ q) lying over a point x ∈ |S|. To see this it suffices to show that Pxq
−→
q M
Pxq /(mαi ,x ∩ Pxq )
(8.24)
i=0
is surjective. Let Hi ∈ Px1 (0 ≤ i ≤ q) be a linear form such that Hi (αj ) 6= 0 ci · · · Hq ∈ Pxq for j 6= i and Hi (αi ) = 0. Then the images of Hi0 := H0 · · · H for 0 ≤ i ≤ q generate the right hand side of (8.24). Main Theorem. Fix an integer q ≥ 0. i) Assuming (IV)q , ΨqU/S is injective. ii) Assuming (I), (II)q , (III) and (IV)q , ΨqU/S is an isomorphism. In order to clarify the technical conditions of the Main Theorem, we explain in the next section its implications on the image of the regulator map (8.2). The proof of the Main Theorem will be given in the sections following the next.
8.3 Implications of the Main Theorem Let d
•+1 • Ω•,d=0 R/k = Ker(ΩR/k −→ ΩR/k )
be the module of closed differential forms. Theorem 8.3.1. Fix an integer q ≥ 0 and assume (I), (II), (III)q and (IV)q+1 in the Main Theorem. Then the map ψ`M (cf. (8.14)) induces an isomorphism q `+n−r M ^ `=0
∼ =
(Gj ) ⊗Q Ωq−`,d=0 −→ Ker(∇q ), R/k
(8.25)
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321
n−r n−r where ∇q : F n−r HdR (U/S) ⊗ ΩqR/k → F n−r−1 HdR (U/S) ⊗ Ωq+1 R/k .
Proof We first note that (IV)q+1 =⇒ (IV)q by definition. Consider the following commutative diagram Lq
`=0
`+n−r V
(Gj ) ⊗Q Ωq−` R/k id⊗dy
Lq+1 `+n−r V `=0
Ψ0
n−r F n−r HdR (U/S) ⊗ ΩqR/k ∇ y q
−−−−→
Ψ00
n−r (Gj ) ⊗Q Ωq+1−` −−−−→ F n−r−1 HdR (U/S) ⊗ Ωq+1 R/k R/k .
Ψ0 is injective and its image is ker ∇q by the Main Theorem (ii). Ψ00 is injective by the Main Theorem (i). Thus the assertion follows by diagram chase. The second implication of the Main theorem concerns the Hodge filtration on cohomology with coefficients. The Gauss-Manin connection (cf. (8.7)) gives rise to the following complex of Zariski sheaves on S ∇
∇
∇
n−r n−r n−r S (U/S) ⊗ Ωdim (U/S) ⊗ Ω1S/k −→ · · · −→ HdR (U/S)−→ HdR HdR S/k .
(8.26)
n−r (U/S)⊗Ω•S/k . We define the de Rham cohomology which is denoted by HdR with coefficients as the hypercohomology q q n−r n−r (S, HdR (U/S) ⊗ Ω•S/k ). HdR (S, HdR (U/S)) = HZar
It is a finite dimensional k-vector space. It follows from the theory of mixed • (S, H n−r (U/S)) carries in Hodge modules by Morihiko Saito ([17]) that HdR dR a canonical way the Hodge filtration and the weight filtration W• denoted by n−r • (U/S)) F p HdR (S, HdR
and
n−r • (S, HdR (U/S)) Wp HdR
respectively. (Arapura [1] has recently given a simpler proof of this fact.) In case k = C there is the comparison isomorphism between the de Rham cohomology and the Betti cohomology ([9, Thm.6.2]) q n−r H q (S an , Rn−r π∗an CU ) ' HdR (S, HdR (U/S))
(π : U → S)
(8.27)
which preserves the Hodge and weight filtrations on both sides defined by M. Saito. It endows H • (S an , Rn−r π∗an QU ) with a mixed Hodge structure.. n−r Define the subcomplex Gi of HdR (U/S) ⊗ Ω•S/k as n−r n−r F i HdR (U/S) → F i−1 HdR (U/S) ⊗ Ω1S/k → · · · n−r S · · · → F i−dim S HdR (U/S) ⊗ Ωdim S/k
(8.28)
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n−r where F • HdR (U/S) is the Hodge filtration as in (8.9). If S were proper over k, we would have n−r • • F i HdR (S, HdR (U/S)) = HZar (S, Gi ).
When S is not proper, there is in general only a natural injection (cf. [2, Lemma 4.2]) n−r • • F i HdR (S, HdR (U/S)) ,→ HZar (S, Gi )
(∀i ≥ 0).
(8.29)
The precise description of the Hodge filtration on the de Rham cohomology with coefficients is more complicated in general. Theorem 8.3.2. Fix an integer q ≥ 0. Let S ⊂ S be a smooth compactification with ∂S := S − S, a normal crossing divisor on S. Assuming (I), (II), (III)q and (IV)q+1 in the main theorem, we have an isomorphism q `+n−r M ^
∼ =
q−` q n−r (Gj ) ⊗Q Γ (S, ΩS/k (U/S)). (S, HdR (log ∂S)) −→ F n−r+q HdR
(8.30)
`=0
Proof We have the following commutative diagram Lq
`=0
`+n−r V
Lq
`=0
Lq
`=0
φ
q n−r (Gj ) ⊗Q Γ(S, Ωq−` (U/S)) (S, HdR (log ∂S)) −−−−→ F n−r+q HdR S/k ey∩ ay∩
`+n−r V
`+n−r V
∼ =
(Gj ) ⊗Q Ωq−`,d=0 R/k by
−−−−→
H q (S, Gn−r+q ) y
q−` (S/k) (Gj ) ⊗Q HdR
−−−−→
c
q n−r HdR (S, HdR (U/S))
(8.31)
where by definition t (S/k) = H t (S, Ω•S/k (log ∂S)) HdR
and the map b comes from the isomorphism t−1 H t (S, Ω•S/k (log ∂S)) ' H t (S, Ω•S/k ) ' Ωt,d=0 R/k /dΩR/k
(8.32)
due to [8, II (3.1.11)] and it is surjective. The map a comes from (8.29). The map e comes from [8, II (3.2.14)]. The bijection in the middle row is the composition of the isomorphism in Theorem 8.3.1 and the isomorphism Ker(∇` ) ' H ` (S, G`+n−r )
for ∀` ≥ 0.
(8.33)
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323
The map c is induced by the composition q `+n−r M ^
q−` (Gj ) ⊗Q Ωq−`,d=0 −→ Ker(∇` ) ⊗Q HdR (S/k) R/k
`=0 n−r −→ H ` (S, HdR (U/S) ⊗ Ω•S/k ) ⊗ H q−` (S, Ω•S/k ) n−r −→ H q (S, HdR (U/S) ⊗ Ω•S/k ) M where the first map is induced by ψ`+n−r (8.14) and (8.32), the second by (8.33), and the last by cup product. We claim that there is a map φ which makes the upper square of the diagram (8.31) commute. Indeed let V be the source of c. Endowing V with the Hodge filtration defined by
Fp
t ^
t ^ u u (Gj ) ⊗Q HdR (S/k) = (Gj ) ⊗Q F p−t HdR (S/k)
u (S/k) = H u (S, Ω≥p−t (log ∂S)), c respects the Hodge filtrawith F p−t HdR S/k tions. Noting
F n−r+q V =
q `+n−r M ^
(Gj ) ⊗Q H 0 (S, Ωq−` (log ∂S)), S/k
`=0
we see that c induces φ as desired. The injectivity of φ follows from that of e. q n−r To show its surjectivity, note that Im(c) contains F n−r+q HdR (S, HdR (U/S)) n−r+q by the diagram. This shows F Coker(c) = 0. By strictness of the Hodge filtration, we get the surjectivity of φ. This completes the proof of the theorem. In what follows we assume k = C. Take S ⊂ S, a smooth compactification with ∂S := S − S, a normal crossing divisor on S. Write for t ≥ 0 HQt,0 (S) := H t (S an , Q(t))∩F t H t (S an , C) = H t (S an , Q(t))∩H 0 (S, ΩtS/C (log ∂S)) Write m = n − r = dim(U/S). Let M ` an m an m+` regm+` U/S : Km+` (O(U )) ⊗ Q → H (S , R π∗ Q(m + `)) ∩ F
be as (8.2). It induces for q ≥ 0 L M (O(U )) ⊗ H q−`,0 (S) → λq : q`=0 Km+` Q Q q → H (S an , Rm π∗an QU (m + q)) ∩ F m+q .
(8.34)
Theorem 8.1.1 follows from the following corollaries in view of Remarks 8.2.1, 8.2.2, 8.2.3.
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Corollary 8.3.3. Fix an integer q ≥ 1 and assume (I), (II), (III)q and (IV)q+1 in the Main Theorem. Then the map (8.2) M q an m an m+q regm+q U/S : Km+q (O(U )) ⊗ Q → H (S , R π∗ QU (m + q)) ∩ F
is surjective for q = 1. More generally regm+q U/S is surjective if the regulator map for S: regtS : KtM (O(S)) ⊗ Q → HQ (S)t,0 is surjective for 1 ≤ ∀t ≤ q. Proof The first assertion of 8.3.3 follows from the second in view of the fact that reg1S is surjective, namely HQ1,0 (S) is generated by dlog O(S)∗ . The second assertion is a direct consequence of the following isomorphism induced by λq : q `+m M ^ ∼ = (Gj ) ⊗Q HQ (S)q−`,0 −→ H q (S an , Rm π∗an QU (m + q)) ∩ F m+q .
(8.35)
`=0
which follows from Theorem 8.3.2. Corollary 8.3.4. Assuming (I), (II), (III)q and (IV)q+1 for q = 0, λ0 induces an isomorphism m ^
∼ =
(Gj ) −→ H 0 (S an , Rm π∗an QU (m)).
Proof Applying (8.35), we have an isomorphism m ^
∼ =
(Gj ) −→ H 0 (S an , Rm π∗an QU (m)) ∩ F m .
(8.36)
We need show that the right hand side is equal to H 0 (S an , Rm π∗an QU ). It suffices to show that H 0 (S an , Rm π∗an Q) is pure of type (m, m). We need a result from [3, Theorem (III)], which implies that the map m m ∇ : GrpF HdR (U/S) −→ GrFp−1 HdR (U/S) ⊗ Ω1R/C
is injective for all 1 ≤ p ≤ m − 1 under the assumption of Corollary 8.3.4. It implies m m Ker(∇) ∩ F 1 HdR (U/S) = Ker(∇) ∩ F m HdR (U/S), m (U/S) → H m (U/S) ⊗ Ω1 where ∇ : HdR dR R/C is the algebraic Gauss-Manin ∼
connection. Noting H 0 (S an , Rm π∗an CU ) → Ker(∇) under the comparison
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325
isomorphism (8.27), it implies F 1 H 0 (S an , Rm π∗an CU ) = F m H 0 (S an , Rm π∗an CU ).
(8.37)
Consider the mixed Hodge structure H := H 0 (S an , Rm π∗an QU ). By the Hodge symmetry (8.37) implies W H p,q := GrFp GrFq Grp+q H=0
unless (p, q) = (m, 0), (m, m), (0, m).
an C ) Hence it suffices to show H m,0 = 0. Putting V = F m H 0 (S an , Rm πX∗ X m,0 where πX : X → S is as in (8.6), we have the surjection V → H while V = 0 by Theorem 8.3.2 applied to the case s = 0. This completes the proof.
8.4 Theory of Generalized Jacobian Rings We introduce the generalized Jacobian ring. It describes the Hodge cohomology groups H • (Ω•X/S (log Z∗ )) of open complete intersections, and enables us to identify the Gauss-Manin connection (cf. (8.11)) with the multiplication of rings. The computational results in this section will play a key role in the proof of the Main Theorem (see §8.5.2).
8.4.1 Fundamental results on generalized Jacobian ring Recall the notations in §8.2.1. Let A = PR [µ1 , · · · , µr , λ1 , · · · , λs ] = R[X0 , . . . , Xn , µ1 , · · · , µr , λ1 , · · · , λs ] be the polynomial ring over PR with indeterminants µ1 , · · · , µr , λ1 , · · · , λs . For q ∈ Z and ` ∈ Z, we put M m(a,b,`) Aq (`) = PR · µa11 · · · µar r λb11 · · · λbss a1 +···+ar +b1 +···+bs =q
P P with m(a, b, `) = ri=1 ai di + sj=1 bj ej + `. Here ai and bj run over nonnegative integers satisfying a1 + · · · + ar + b1 + · · · + bs = q. By convention, Aq (`) = 0 for q < 0. Note that the notation in 8.21 is compatible with the above definition. The Jacobian ideal J = J(F1 , · · · , Fr , G1 , · · · , Gs ) is defined to be the ideal of A generated by r s X X ∂Gj ∂Fi µi + λj , ∂Xk ∂Xk i=1
j=1
F` ,
G `0 λ ` 0
(0 ≤ k ≤ n, 1 ≤ ` ≤ r, 1 ≤ `0 ≤ s).
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The quotient ring B = A/J is called the generalized Jacobian ring ([3]). We put Jq (`) = Aq (`) ∩ J
and
Bq (`) = Aq (`)/Jq (`).
We now recall some fundamental results from [3]. Theorem 8.4.1 ([3], Theorem (I)). Suppose n ≥ r + 1. For each integer 0 ≤ p ≤ n − r there is a natural isomorphism ∼ =
φ : Bp (d + e − n − 1) −→ H p (X, Ωn−r−p X/S (log Z∗ )) and the following diagram is commutative up to a scalar in R× : Bp (d + e − n − 1) φy
−−−−→
Bp+1 (d + e − n − 1) ⊗ A1 (0)∗ y1⊗Θ∗
(8.38)
∇
H p (X, Ωn−r−p −−−→ H p+1 (X, Ωn−r−p−1 (log Z∗ )) ⊗ Ω1R/k X/S (log Z∗ )) − X/S where ∇ is induced by the Gauss-Manin connection: n−r n−r ∇ : F n−r−p HdR (U/S) −→ F n−r−p−1 HdR (U/S) ⊗ Ω1R/k
and is induced from the multiplication Bp (d + e − n − 1) ⊗ A1 (0) → Bp+1 (d + e − n − 1) and Θ∗ is the dual of the map (8.20) Θ : TR/k → A1 (0);
θ 7→
r X i=1
θ(Fi )µi +
s X
θ(Gj )λj .
j=1
The second fundamental result is the duality theorem for generalized Jacobian rings. For an R-module we denote M ∗ = HomR (M, R). Theorem 8.4.2 ( [3], Theorem (II)). There is a natural map (called the trace map) τ : Bn−r (2(d − n − 1) + e) → R. Let hp : Bp (d − n − 1) → Bn−r−p (d + e − n − 1)∗ be the map induced by the following pairing induced by the multiplication τ
Bp (d − n − 1) ⊗ Bn−r−p (d + e − n − 1) → Bn−r (2(d − n − 1) + e)−→ R. Then hp is bijective if 1 ≤ p ≤ n − r, andsurjective if p = 0. The kernel of s−1 h0 is a locally free R-module of rank n−r .
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8.4.2 Generalized Jacobian rings ` a la M. Green We review from [3, §2] a “sheaf theoretic” definition of generalized Jacobian ring. This sophisticated definition originates from M.Green ([10]). It is useful for various computations (cf. §8.4.3 and §8.5.2). Put E = E0
M
with E0 =
E1
r M
O(di ) and E1 =
i=1
s M
O(ej )
j=1
which is a locally free sheaf on Pn = PnR . We consider the projective space bundle π : P := P(E) −→ Pn . Let L = OP(E) (1) be the tautological line bundle. We have the Euler exact sequence 0 −→ OP −→ π ∗ E∗ ⊗ L −→ TP/Pn −→ 0. We consider the effective divisors M M Qi := P( O(dα ) E1 ) ,→ P(E)
(8.39)
for 1 ≤ i ≤ r,
1≤α6=i≤r
Pj := P(E0
M
M
O(eβ )) ,→ P(E)
for 1 ≤ j ≤ s,
1≤β6=j≤r
and let µi ∈ H 0 (P, L ⊗ π ∗ O(−di )),
λj ∈ H 0 (P, L ⊗ π ∗ O(−ej ))
be the global sections associated to these. We put σ=
r X
Fi µi +
i=1
s X
Gj λj ∈ Γ(P, L).
j=1
Let ΣL be the sheaf of differential operators on L of order ≤ 1, defined as: ΣL = Diff
≤1
(L) = {P ∈ Endk (L) ; P f − f P is OP -linear for ∀f ∈ OP } ' L ⊗ Diff
≤1
(OP ) ⊗ L∗ .
(It might be helpful to mention that ΣL is a prolongation bundle.) By definition it fits into an exact sequence 0 −→ OP −→ ΣL −→ TP −→ 0
(8.40)
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with extension class −c1 (L) ∈ Ext1 (TP , OP ) ' Ext1 (OP , Ω1P ⊗ OP ) ' H 1 (P, Ω1P ). Letting U ⊂ Pn be an affine subspace and x1 , · · · , xn be its coordinates, Γ (π −1 (U ), ΣL ) is generated by the following sections ∂ ∂ ∂ ∂ ∂ , λi , λi , µi , µi , OP -linear maps. ∂xi ∂λj ∂µj ∂λj ∂µj
(8.41)
The section σ defines a map j(σ) : ΣL −→ L,
P 7−→ P (σ),
which is surjective by the assumption that X∗ +Y∗ is a simple normal crossing divisor. It gives rise to the exact sequence j(σ)
0 −→ TP (− log Z) −→ ΣL −→ L −→ 0,
(8.42)
where Z ⊂ P is the zero divisor of σ. Put Q∗ = Q1 + · · · + Qr
and
P∗ = P1 + · · · + Ps ,
and define ΣL (− log P∗ ) to be the inverse image of TP (− log P∗ ) via the map in (8.40). We then have the exact sequence 0 −→ OP −→ ΣL (− log P∗ ) −→ TP (− log P∗ ) −→ 0.
(8.43)
Moreover (8.42) gives rise to an exact sequence j(σ)
0 −→ TP (− log(Z + P∗ )) −→ ΣL (− log P∗ ) −→ L −→ 0.
(8.44)
Lemma 8.4.3. For integers k and `, put Ak (`)Σ = H 0 (Lk ⊗ π ∗ O(`)) and j(σ)⊗1 Jk (`)Σ = Im H 0 (ΣL (− log P∗ ) ⊗ Lk−1 ⊗ π ∗ O(`)) −→ H 0 (Lk ⊗ π ∗ O(`)) . Then we have Ak (`) = Ak (`)Σ ,
Jk (`) = Jk (`)Σ .
Proof See [3, Lem.(2-2)]. Thus we have obtained another definition of the generalized Jacobian ring.
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8.4.3 Some computational results We keep the notations in §8.4.2. In what follows we simply write Σ = ΣL (− log P∗ ). Lemma 8.4.4. We have H w (P,
p ^
Σ∗ ⊗ Lν ⊗ π ∗ O(`)) = 0
if one of the following conditions holds: (i) p − ν ≤ r + s − 1 and ν ≤ −1, (ii) p − ν ≥ n + 1 and ν ≥ −s + 1. (iii) w > 0, ν ≥ −s + 1, ` ≥ 0 and (ν, `) 6= (0, 0). Proof See [3, Thm.(4-1)]. Proposition 8.4.5. Let k be an integer. k V (i) H ν ( Σ∗ ) = 0 for any k ≥ 0 and ν 6= 0, n. k V (ii) H 0 ( Σ∗ ) is a locally free R-module of rank k V (iii) H n ( Σ∗ ) is a locally free R-module of rank for ` < 0 by convention.)
s−1 k
.
s−1 k−n−1
. (Note
x `
=0
The rest of this section is devoted to the proof of Proposition 8.4.5. Recall that there is an exact sequence 0 −→ Ω1P (log P∗ ) −→ Σ∗ −→ OP −→ 0
(8.45)
with the extension class c1 (L)|P−P∗ ∈ Ext1 (OP , Ω1P (log P∗ )) = H 1 (Ω1P (log P∗ )). It gives rise to the short exact sequence 0 −→ Ω•P (log P∗ ) −→
• ^
Σ∗ −→ Ω•−1 P (log P∗ ) −→ 0,
(8.46)
and we have the long exact sequence k V · · · H ν (ΩkP (log P∗ )) −→ H ν ( Σ∗ ) −→ δ
ν+1 (Ωk (log P )) · · · , H ν (Ωk−1 ∗ P P (log P∗ )) −→ H
(8.47)
where δ is induced by the cup-product with c1 (L)|P−P∗ ∈ H 1 (Ω1P (log P∗ )).
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The first step is to write down H • (Ω•P (log P∗ )) and δ explicitly. We prepare some notations. For an integer k ≥ 1 we put ∆k = {I = (i1 , · · · , ik )| 1 ≤ i1 < · · · < ik ≤ s}. For I = (i1 , · · · , ik ) ∈ ∆k we write PI = Pi1 ∩ · · · ∩ Pi1 . For k = 0 we put ∆0 = {∅} and P∅ = P by convention. To compute H ν (ΩkP (log P∗ )) we first note the isomorphisms ν+k H ν (ΩkP (log P∗ )) ' GrkF H ν+k (P − P∗ ) ' GrW (P − P∗ ), 2k H
where H ∗ (P − P∗ ) = H ∗ (P, Ω•P/S (log P∗ )), and ∗ • F p H ∗ (P − P∗ ) = H ∗ (P, Ω≥p P/S (log P∗ )) ⊂ H (P, ΩP/S (log P∗ ))
is the Hodge filtration and Wp H ∗ (P − P∗ ) ⊂ H ∗ (P − P∗ ) denotes the weight filtration induced by the spectral sequence M H p (PI ) =⇒ H p+q (P − P∗ ) (8.48) E2pq = I∈∆q
where H ∗ (PI ) = H ∗ (PI , Ω•PI /S ) (cf. [8]). We note E2p,q = 0 unless 0 ≤ q ≤ s. Since the spectral sequence (8.48) degenerates at E3 , H ν (ΩkP (log P∗ )) is isomorphic to the cohomology at the middle term of the following complex M M M H 2ν−2 (PI1 ) −→ H 2ν (PI2 ) −→ H 2ν+2 (PI3 ). (8.49) I1 ∈∆k−ν+1
I2 ∈∆k−ν
I3 ∈∆k−ν−1
The arrows in (8.49) are described as follows. Let I1 = (i1 , · · · , ik ) ∈ ∆k and I2 ∈ ∆k−1 . If I2 6⊂ I1 , then H 2ν−2 (PI1 ) → H 2ν (PI2 ) is the zero map. If I2 = (i1 , · · · , ibp , · · · , ik ), then it is (−1)p−1 φI1 I2 where φI1 I2 is the Gysin map. In order to describe it in more convenient way we introduce some notations. Let SR = R[x, y] ν be the set of homogeneous polynomials of be the polynomial ring and SR degree ν. We put r Y Y (x − di y) · (x − ej y) if I 6= ∅, i=1 j6∈I QI (x, y) = s r Y Y (x − di y) · (x − ej y) if I = ∅, i=1
j=1
Beilinson’s Hodge Conjecture with Coefficients
Lemma 8.4.6.
331
i) There is an isomorphism of graded rings: ∼ =
SR /(QI (x, y), y n+1 ) −→ H ∗ (PI ); x 7→ c1 (L)|PI , y 7→ π ∗ c1 (O(1))|PI , where we recall π : P = P(E) → Pn . ii) For I ⊂ I 0 we have the commutative diagram ∼ =
H ∗ (PI ) −−−−→ R[x, y]/(QI (x, y), y n+1 ) ψII 0 y y ∼ =
H ∗ (PI 0 ) −−−−→ R[x, y]/(QI 0 (x, y), y n+1 ) where the left vertical map is the restriction map and the right vertical map is the natural surjection. iii) If I 0 = I ∪ {j} with j 6∈ I we have the commutative diagram ∼ =
H ∗ (PI 0 ) −−−−→ R[x, y]/(QI 0 (x, y), y n+1 ) x−ej y y φII 0 y ∼ =
H ∗ (PI ) −−−−→ R[x, y]/(QI (x, y), y n+1 ) where the left vertical map is the Gysin map and the right vertical map is the multiplication by x − ej y. Proof The first assertion is well-known, and the second assertion follows immediately from the first. To show the last assertion, we note that φII 0 is the Poincar´e dual of ψII 0 and the composite φII 0 ψII 0 : H ∗ (PI ) → H ∗+2 (PI ) is the multiplication by the class c1 (PI 0 )|PI of the divisor PI 0 in PI . Hence the assertion follows by noting c1 (PI 0 )|PI = c1 (Pj )|PI = x − ej y. For I = (i1 , . . . , i` ) ∈ ∆` write λI = λi1 ∧ · · · ∧ λi` and λI = 1 if I = ∅. Lemma 8.4.6 provides us with an isomorphism M M p ∼ = SR /(QI (x, y), y n+1 ) ⊗ λI −→ H 2p (PI ) for each p ≥ 0. I∈∆`
I∈∆`
Under this isomorphism, the arrows in (8.49) are identified with dλ : ξ ⊗ λI 7−→
` X
(−1)k−1 (x − eik y)ξ ⊗ λi1 ∧ · · · ∧ λc ik ∧ · · · ∧ λi` .
k=1
Thus we have obtained the following result.
(8.50)
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Lemma 8.4.7. For any integers k and ν, H ν (ΩkP (log P∗ )) is isomorphic to the cohomology at the middle term of the complex M
d
M
λ ν−1 SR /(QI1 (x, y), y n+1 )⊗λI1 →
I1 ∈∆k−ν+1
ν SR /(QI2 (x, y), y n+1 )⊗λI2
I2 ∈∆k−ν d
M
λ →
ν+1 SR /(QI3 (x, y), y n+1 ) ⊗ λI3
(8.51)
I3 ∈∆k−ν−1
with dλ defined as in (8.50) (Note that by convention 0 ≤ ` ≤ s).
L
I∈∆` (· · · )
= 0 unless
In order to calculate the cohomology at the middle term of the complex (8.51), we introduce new symbols ε1 , · · · , εs and write εI = εi1 ∧ · · · ∧ εi` for I = (i1 , . . . , i` ) ∈ ∆` . Consider the following diagram 0 y L
0 y
SR ⊗ εI aI y L L SR ⊗ εI I∈∆` SR ⊗ λI bI y L I∈∆` SR ⊗ λI y L n+1 ) ⊗ λ I I∈∆` SR /(QI , y y I∈∆`
d
−−−ε−→
L
SR ⊗ ε I 0 a 0 y I L d +dε L −−λ−−→ SR ⊗ εI 0 I 0 ∈∆`−1 SR ⊗ λI 0 b y I0 L d −−−λ−→ I 0 ∈∆`−1 SR ⊗ λI 0 y L d n+1 ) ⊗ λ 0 −−−λ−→ I I 0 ∈∆`−1 SR /(QI 0 , y y I 0 ∈∆`−1
0
(8.52)
0
where dε is given by: dε : ξ ⊗ εI 7−→
` X
(−1)k−1 ξ ⊗ εi1 ∧ · · · ∧ εc ik ∧ · · · ∧ εi` ,
(8.53)
k=1
aI and bI are given by: aI : ξ⊗εI 7→ ξQI ⊗λI +ξy n+1 ⊗εI
and
bI : ξ1 ⊗λI +ξ2 ⊗εI 7→ (ξ1 y n+1 −ξ2 QI )⊗λI .
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One can easily check that the diagram is commutative and the vertical sequences are exact. Put M M dε 0 0 E ` = Ker SR ⊗ εI −→ SR ⊗ εI . (8.54) I∈∆`
I∈∆`−1
Note E ` = 0 unless 0 ≤ ` ≤ s by convention. Claim 8.4.8. i) The following sequence is exact: M M M dε dε 0 −→ SR ⊗ εI −→ SR ⊗ εI −→ · · · −→ SR ⊗ 1 −→ 0. I∈∆s
I∈∆s−1
I=∅
ii) Assume s ≥ 2 and ei 6= ej for some i 6= j. Then, for an integer ` ≥ 0, the following sequence is exact: M M M dλ dλ 0 1 ` SR ⊗ λI −→ SR ⊗ λI −→ · · · −→ SR ⊗ 1 −→ 0. I∈∆`
I∈∆`−1
I=∅
iii) Assume e = e1 = · · · = es . Then the cohomology at the middle term of the complex M M dλ M dλ p−1 p p+1 SR ⊗ λI1 −→ SR ⊗ λI2 −→ SR ⊗ λI3 (8.55) I1 ∈∆`+1
I2 ∈∆`
I3 ∈∆`−1
p /(x − ey) ⊗ E ` . is isomorphic to SR
Proof (i) Easy (and well-known). (ii) Let V0 be a free R-module with basis λ1 , · · · , λs . Let c : OP1 ⊗ V0 → OP1 (1) be the map of locally free sheaves on P1 = Proj(SR ), defined by λj 7→ x − ej y. This is surjective by the assumption. It gives rise to the Koszul complex 0 → OP1 (` − s) ⊗
s V
s−1 V V0 → OP1 (` − s + 1) ⊗ V0 → · · · · · · → OP1 (` − 1) ⊗ V0 → OP1 (`) → 0.
(8.56)
We decompose (8.56) into the following sequences: 0 → OP1 (` − s) ⊗ 0 → V1 −→ OP1 ⊗
s ^
` ^
V0 → · · · → OP1 (−1) ⊗
`+1 ^
V0 → V1 → 0.
V0 → · · · → OP1 (` − 1) ⊗ V0 → OP1 (`) → 0.
(8.57)
(8.58)
` and that (8.58) gives an acyclic resolution of V , it Noting H 0 (OP1 (`)) = SR 1
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suffices to show H i (P1 , V1 ) = 0 for i ≥ 1 which is obvious for i ≥ 2. To show H 1 (V1 ) = 0 it suffices to prove H 0 (O(−2) ⊗ V1∗ ) = 0 by the Serre duality. `+1 V By (8.57) there is an injection O(−2) ⊗ V1∗ ,→ O(−1) ⊗ V0∗ . The assertion follows from this. (iii) Since e = e1 = · · · = es , we have the following commutative diagram L
L
d
p−1 SR ⊗ εI1 −−−−→ ι0 y
I1 ∈∆`+1
L
d
I1 ∈∆`+1
p−1 SR ⊗ λI1 −−−λ−→
d
p−1 SR ⊗ εI2 −−−−→ ι1 y
I2 ∈∆`
L
d
I2 ∈∆`
p SR ⊗ λI2 −−−λ−→
L
L
p−1 SR ⊗ εI3 ι2 y
I3 ∈∆`−1
I3 ∈∆`−1
p+1 SR ⊗ λI3
where ιi : ξ ⊗ εI 7→ (x − ey)i ξ ⊗ λI . Note that ι0 is bijective and ιi are injective for i > 0. Due to (i), the cohomology group of the complex (8.55) is isomorphic to M p M dλ p+1 Ker SR /(x − ey) ⊗ λI2 −→ SR /(x − ey)2 ⊗ λI3 . (8.59) I2 ∈∆`
I3 ∈∆`−1
p+1 p /(x−ey)2 , given by multiplication with (x−ey), /(x−ey) → SR The map SR is injective. Hence (8.59) is isomorphic to M M p dε p p SR /(x − ey) ⊗ εI3 = SR /(x−ey)⊗E ` . Ker SR /(x − ey) ⊗ εI2 −→ I3 ∈∆`−1
I2 ∈∆`
This completes the proof. Combining Lemmas 8.4.7, 8.4.8 and (8.52), we get the following explicit description of H ν (ΩkP (log P∗ )). Lemma 8.4.9. Let E • be as in (8.54), and put M M dλ 0 1 Λ` = Ker SR ⊗ λI −→ SR ⊗ λI . I∈∆`
I∈∆`−1
i) Assume s ≥ 2 and ei 6= ej for some i 6= j. Then we have k if ν = 0 Λ ν k H (ΩP (log P∗ )) ' Λk−n−1 if ν = n otherwise 0 (Note Λ` = 0 unless 0 ≤ ` ≤ s by convention.)
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335
ii) Assume e = e1 = · · · = es . Then we have for ∀k, ν ≥ 0 ν H ν (ΩkP (log P∗ )) ' SR /(y n+1 , x − ey) ⊗ E k−ν ' R[y]ν /(y n+1 ) ⊗ E k−ν .
In order to complete the proof of Proposition 8.4.5 we need the following Lemma, which gives an explicit description of the map δ in (8.47). Lemma 8.4.10. i) Assume s ≥ 2 and ei 6= ej for some i 6= j. Then c1 (L)|P−P∗ = 0 and δ = 0. ii) Assume e = e1 = · · · = es . Then π ∗ : H 1 (Ω1Pn ) −→ H 1 (Ω1P (log P∗ )) is injective and c1 (L)|P−P∗ = π ∗ c1 (O(e)). The map δ in (8.47) is identified with the multiplication by ey ⊗ 1 under the isomorphisms in Lemma 8.4.7. Proof (i) Noting O(Pj ) ' L ⊗ π ∗ O(−ej ) and c1 (Pj )|P−P∗ = 0, we have c1 (L)|P−P∗ = π ∗ c1 (O(ej )) = ej π ∗ c1 (O(1))
for 1 ≤ ∀j ≤ s.
By the assumption this implies c1 (L)|P−P∗ = 0 and π ∗ c1 (O(1)) = 0. (ii) The first assertion follows from the existence of an isomorphism P ' Pn × Pr+s−1 such that Pj corresponds to Pn × Hj with Hj a hyperplane. The second assertion has been already shown in (i). To show the last we first note that the cup product for H ν (ΩkP (log P∗ )) is induced by the cup product H 2i (PI ) ⊗ H 2j (PI ) → H 2(i+j) (PI+J ) when one identifies H ν (ΩkP (log P∗ )) with the cohomology at the middle term of the complex (8.49). Here PI+J = PI ∩PJ if I ∩J = ∅ and H 2(i+j) (PI+J ) = 0 otherwise by convention. Under the isomorphisms of Lemma 8.4.6, it is identified with j i+j i SR /(QI (x, y), y n+1 ) ⊗ SR /(QJ (x, y), y n+1 ) → SR /(QI+J (x, y), y n+1 ),
(f ⊗ λI ) ⊗ (g ⊗ λJ ) 7→ f g ⊗ (λI
^
λJ ).
Since δ is induced by the cup product with c1 (L)|P−P∗ ∈ H 1 (Ω1P (log P∗ )), the desired assertion follows by noting c1 (L)|P−P∗ corresponds to ey under the 1 /(Q (x, y), y n+1 ) due to Lemma 8.4.7. isomorphism H 1 (Ω1P (log P∗ )) ' SR ∅
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Finally we can complete the proof of Proposition 8.4.5. First assume s ≥ 2 and ei 6= ej for some i 6= j. The assertion follows from (8.47), Lemma 8.4.10 (i) and Lemma 8.4.9 (i) by noting that the R-module Λ` is s−2 locally free of rank ` due to Claim 8.4.8 (ii). (Compare the coefficients P P s p s−2 s −2 p of (1 − x) = (1 − x) · (1 − x) = ( p (−1) p x ) · ( q qxq ) to get s−2 s s = − 2 ` ` `−1 + · · · .) Next assume e = e1 = · · · = es . By (8.47), Lemma 8.4.10 (ii) and Lemma 8.4.9 (ii), we have an exact sequence ν−1
R[y]
/(y
n+1 ey⊗1
ν
) −→ R[y] /(y
n+1
)⊗E
k−ν
k ^ −→ H ( Σ∗ ) ν
ey⊗1
−→ R[y]ν /(y n+1 ) ⊗ E k−ν−1 −→ R[y]ν+1 /(y n+1 ) ⊗ E k−ν−1 . The desiredassertion follows by noting that the R-module E ` is locally free of rank s−1 due to Claim 8.4.8 (i). This completes the proof of Proposition ` 8.4.5.
8.5 Proof of the Main Theorem In this section we prove the Main Theorem stated in §8.2.4.
8.5.1 Proof of (i) Let ΨqU/S :
q `+n−r M ^
q n−r 0 (Gj )X ⊗Q Ωq−` R/k −→ H (ΩX/S (log Z∗ )) ⊗ ΩR/k
(8.60)
`=0
be the map (8.19). We show the stronger assertion that ΨqU/S ⊗R κ(x) is injective for any x ∈ |S| assuming (IV)q . We fix x ∈ |S|. Without loss of generality we may assume j1 = 1, · · · , jn−r = n − r in (IV)q . We may work in an ´etale neighbourhood of x to assume that R is a strict henselian local ring with the closed point x ∈ Spec(R) and that the 0-dimensional scheme defined by F1 = · · · = Fr = G1 = · · · = Gn−r = 0 in PnR is a disjoint union of copies of Spec(R). ` ` V V For an integer ` ≥ 1 let (Gj )j6=1 be the subspace of (Gj ) generated by such gj0 ···j` that jν 6= 1 for 0 ≤ ∀ν ≤ ` (cf. (8.17)). We have the exact sequence 0→
` ^
(Gj )j6=1 →
` ^
τ
(Gj ) −→
`−1 ^
(Gj )j6=1 → 0,
where τ is characterized by the condition that τ (g1j1 ···j` ) = −gj1 ···j` and
Beilinson’s Hodge Conjecture with Coefficients
337
` V (1) that it annihilates (Gj )j6=1 . Put Z∗ = Z2 + · · · + Zs where we recall that Zj ⊂ X is a smooth hypersurface section defined by Gj . Consider the residue map along Z1 : (1)
n−r−1 Res : Ωn−r X/S (log Z∗ ) → ΩZ1 /S (log Z∗ ∩ Z1 ); dg1 /g1 ∧ ω 7→ ω|Z1 ,
where g1 is a local equation of Z1 . By (8.18) one sees that Res ◦ ΨqU/S factors through τ and we get the following commutative diagram: 0 y q `+n−r L V `=0
(Gj )j6=1 ⊗Q Ωq−` R/k y
q `+n−r L V `=0
q `+n−r−1 L V `=0
0 y
(Gj ) ⊗Q Ωq−` R/k yτ ⊗id
−−−−→
ΨqU/S
−−−−→
(1)
n−r H 0 (ΩX/S (log Z∗ )) ⊗R ΩqR/k y q H 0 (Ωn−r X/S (log Z∗ )) ⊗R ΩR/k yRes⊗id (1)
q (Gj )j6=1 ⊗Q Ωq−` −−−→ H 0 (Ωn−r−1 R/k − Z1 /S (log Z∗ ∩ Z1 )) ⊗R ΩR/k y y
0
0
By the diagram and induction we are reduced to show the injectivity of ΨqU/S ⊗R κ(x) in case s = 1 or n − r = 0. If s = 1, the assertion is clear ` V because (Gj ) = 0 by convention. We consider the case n − r = 0. Then X = {F1 = · · · = Fn = 0} ⊂ PnR By the assumption we have X =
and `
Yj = {Gj = 0} ⊂ PnR
(1 ≤ j ≤ s).
Spec(R) where X(R) is the set of
β∈X(R)
sections of X → Spec(R). The map (8.60) becomes Ψ:
q ^ ` M
q−` (Gj ) ⊗Q ΩR/k −→ H 0 (OX ) ⊗R ΩqR/k
`=0
By Nakayama’s lemma, condition (IV)q in the Main Theorem implies: (∗) There are q + 1 points β0 , · · · , βq ∈ X(R) such that the map Θ
W −→ A1 (0)/(J 0 + mβ0 ) ⊕ · · · ⊕ A1 (0)/(J 0 + mβq ),
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is surjective, where mβ ⊂ PR denotes the homogeneous ideal defining β in PnR = Proj(PR ). We note M R · [β] H 0 (OX ) = β∈X(R)
and put H 0 (OX )0 =
M
R · [βν ].
0≤ν≤q
It suffices to show the injectivity of Ψ0 ⊗R κ(x) where 0
Ψ :
q ^ ` M
q−` (Gj ) ⊗Q ΩR/k −→ H 0 (OX )0 ⊗R ΩqR/k
(8.61)
`=0
is the composite of Ψ with the projection H 0 (OX ) → H 0 (OX )0 . We have 0
Ψ (v1j1 ···j` ⊗ η) =
q X
[βν ] ⊗ j1 ν ∧ · · · ∧ j` ν ∧ η
(η ∈ Ωq−` R/k ),
ν=0
with
e
jν := dlog((Gej 1 /G1j )(βν )) ∈ Ω1R/k .
Hence the desired assertion follows from the following two lemmas. Lemma 8.5.1. The log forms jν (2 ≤ j ≤ s, 0 ≤ ν ≤ q) are linearly independent in Ω1R/k ⊗ κ(x). Lemma 8.5.2. Let Ω be a finite dimensional vector space over a field k of characteristic zero. Suppose that jν ∈ Ω (1 ≤ j ≤ s, 0 ≤ ν ≤ q) are linearly t V independent. For given ηt ∈ Ω with 0 ≤ t ≤ q, put for 0 ≤ ν ≤ q (the product is wedge product) P P ) ων = ηq + sj=1 ηq−1 jν + 1≤j1 <j2 ≤s ηq−2 j1 ν j2 ν + · · · q (8.62) P V · · · + 1≤j1 <···<jq ≤s η0 j1 ν · · · jq ν ∈ Ω. If ων = 0 for 0 ≤ ∀ν ≤ q, then ηt = 0 for 0 ≤ ∀t ≤ q. Now we prove the above lemmas. Lemma 8.5.1 follows from the condition (∗) by noting the following: Lemma 8.5.3. For β ∈ X(R), A1 (0)/(J 0 + mβ ) is a free R-module of rank s − 1, and the dual of TR/k → A1 (0)/(J 0 + mβ ) induced by Θ is given by the matrix Ge1 Ge1 (dlog e22 (β), · · · , dlog ses (β)) G1 G1
Beilinson’s Hodge Conjecture with Coefficients
339
for a suitable choice of a basis of A1 (0)/(J 0 + mβ ). Proof Giving β = (β0 : · · · : βn ) in the homogeneous coordinate of PnR , mβ = (βi Xj − βj Xi )0≤i<j≤n ⊂ PR = R[X0 , · · · , Xn ]. We may assume without loss of generality that β0 = 1. We write ∂` := ∂/∂X` for 0 ≤ ` ≤ n and L(β) := L(1, β1 , · · · , βn ) ∈ R for any homogeneous polynomial L. Then we have an isomorphism n M
A1 (0)/(J + mβ ) ∼ = 0
R·
i=1
with
Jβ0
=
n X
X0di µi
s MM
e R · X0 j λj /Jβ0
j=1
n s X X e di R·( ∂` Fi (β)X0 µi + ∂` Gj (β)X0 j λj ).
`=0
i=1
j=1
Using the fact
∂1 F1 (β) · · · .. det . ∂n F1 (β) · · ·
∂1 Fn (β) .. ∗ ∈R , . ∂n Fn (β)
we get
A1 (0)/(J 0 + mβ ) ∼ =
s M
s X e e R · X0 j λj /R · ( ej Gj (β)X0 j λj ),
j=1
j=1
(8.63)
which is a free R-module of rank (s − 1). To prove the second assertion we first show for θ ∈ TR/k Θ(θ) ≡ θ(G1 (β)) · X0e1 λ1 + · · · + θ(Gs (β)) · X0es λs mod J 0 + (X` − β` X0 )1≤`≤n . (8.64)
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In fact s s s X n X X X ∂Gj e e e (β) · θ(β` ) · X0 j λj θ(Gj (β)) · X0 j λj = (θGj )(β) · X0 j λj + ∂X` j=1
j=1
≡
s X
=
j=1 s X
(∗∗)
≡
e
(θGj )(β) · X0 j λj − e
j=1 `=0 n n XX ∂Fi i=1 `=0 n X
n X
j=1
i=1
(β) · θ(β` ) · X0di µi
mod J 0
(θFi )(β) · X0di µi
(θGj )(β) · X0 j λj +
j=1 s X
∂X`
i=1
(θGj )λj +
(θFi )µi
mod (X` − βl X0 )1≤`≤n
= Θ(θ). P Here (∗∗) follows from 0 = θ(Fi (β)) = (θFi )(β) + n`=0 (∂` Fi )(β) · θ(β` ) e (note Fi (β) = 0 since β ∈ X(R)). Let {(X0 j λj )∗ }1≤j≤s be the dual basis of Ls ej j=1 R · X0 λj . Then 1 1 e (X0 j λj )∗ − (X e1 λ1 )∗ ej Gj (β) e1 G1 (β) 0
(2 ≤ j ≤ s)
is a basis of the dual module of the right hand side of (8.63). By (8.64), we see that the dual of TR/k → A1 (0)/(J 0 + mβ ) is given by Gej 1 1 1 1 e (X0 j λj )∗ − (X0e1 λ1 )∗ 7−→ dlog ej (β). ej Gj (β) e1 G1 (β) e1 ej G1 This completes the proof of Lemma 8.5.1. Finally we prove Lemma 8.5.2. We prove the assertion by induction on q ≥ 0. If q = 0, it is clear. Let ∗il (1 ≤ i ≤ s, 0 ≤ ` ≤ q) be a linear form on Ω such that ∗il (i0 `0 ) = 0 if (i, `) 6= (i0 , `0 ) and ∗il (il ) = 1. For 1 ≤ ν 6= ` ≤ q we have s X X (ηq−1 , ∗il )jν + · · · + (η1 , ∗il )j1 ν · · · jq−1 ν (ων , ∗il ) = (ηq , ∗il ) + 1≤j1 <···<jq−1 ≤s
j=1
=0∈
q−1 ^
Ω.
By induction this implies (ηt , ∗il ) = 0 for 0 ≤ ∀t, ∀` ≤ q and 1 ≤ ∀i ≤ s. Then 0 = (ων , ∗j1 ν · · · ∗jq−t ν ) = ηt . This completes the proof of Lemma 8.5.2.
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341
8.5.2 Proof of (ii) : Case TR/k ' W Since we have proved the injectivity of ΨqU/S ⊗R κ(x) for ∀x ∈ |S|, it suffices to show that the kernel of ∇q is a locally free R-module of the same rank as the source of ΨqU/S . More precisely we want to show the following. Lemma 8.5.4. Assuming (I), (II) and (III)q in the Main Theorem, Ker(∇q ) is a locally free R-module of rank q−k ^ X s − 1 · rank TR/k . n−r+k
(8.65)
k≥0
In this subsection we show Lemma 8.5.4 assuming Θ : TR/k → W is an isomorphism. We note that W is a locally free R-module by (I). By Theorem 8.4.1, ∇q is identified with the map q ^
B0 (d + e − n − 1) ⊗
∗
W −→ B1 (d + e − n − 1) ⊗
q+1 ^
W∗
induced by the multiplication Bq (`)⊗W → Bq+1 (`). By the duality theorem (Theorem 8.4.2), the dual of the map fits into the commutative diagram B1 (d + e − n − 1)∗ ⊗ x ∼ = Bn−r−1 (d − n − 1) ⊗
q+1 V
q+1 V
W −−−−→ B0 (d + e − n − 1)∗ ⊗ x ι Φ
W −−−−→
Bn−r (d − n − 1) ⊗
q V
q V
W (8.66)
W.
The diagram induces an exact sequence ∗ 0 → Coker(Φ) → Ker(∇q ) → Coker(ι) → 0. Due to Theorem 8.4.2 ι is injective and its cokernel is a locally free R-module q V s−1 · rank W . Therefore it suffices to show that the cokernel of of rank n−r the map Φ is a locally free R-module of rank q−k ^ X s − 1 · rank W. n−r+k
(8.67)
k≥1
In order to show this we recall the notations in §8.4.2 and §8.4.3. For integers k, h and ` we put Mk,h (`) =
n+r+s−h ^
Σ∗ ⊗ Lr+k−h ⊗ π ∗ O(` − d + n + 1)
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and Ck,h (`) = H 0 (P, Mk,h (`)). Due to Lemma 8.4.3, there is an exact sequence Ck,1 (`) −→ Ck,0 (`) −→ Bk (`) −→ 0.
(8.68)
We now need the following two results Lemmas 8.5.5 and 8.5.6. The first result is a direct consequence of [3, Lem.(7-4)]. It is a generalization of Nori’s connectivity [14] to the case of open complete intersections and is the key to the proof of the Main Theorem. Proposition 8.5.5. Putting Ck,h = Ck,h (d − n − 1), the Koszul complex
Cn−r+h−1,h ⊗
q−h+1 ^
W
→ Cn−r+h,h ⊗
q−h ^
→ Cn−r+h+1,h ⊗
W
q−h−1 ^
W → Cn−r+h+2,h ⊗
q−h−2 ^
W
is exact for ∀h ≥ 0 assuming (II) and (III)q in the Main Theorem. Putting Mk,h = Mk,h (d − n − 1), (8.44) induces the following exact sequence (cf. [3, Lem.(5-1)]) 0 → Mn−r+k,n+r+s → Mn−r+k,n+r+s−1 → · · · → Mn−r+k,0 → 0.
(8.69)
Lemma 8.5.6. Let k ≥ 1 be an integer. Then the complex 0 → Cn−r+k,n+r+s → Cn−r+k,n+r+s−1 → · · · → Cn−r+k,0 → 0 induced by (8.69) is exact except at the term Cn−r+k,k−1 , and the cohomology r+s−k V ∗ group at this term is isomorphic to H n ( Σ ). Before proving Lemma 8.5.6, we complete the proof of Lemma 8.5.4 assuming TR/k ' W . We write B• = B• (d − n − 1). Consider the following
Beilinson’s Hodge Conjecture with Coefficients
343
commutative diagram: q+1 V Cn−r−1,1 ⊗ W ↓ q V Cn−r,1 ⊗ W
→ →
q+1 V Cn−r−1,0 ⊗ W ↓ q V Cn−r,0 ⊗ W
↓ Cn−r+1,2 ⊗
q−1 V
W
→
Cn−r+2,2 ⊗
q−1 V
W
→
Cn−r+1,0 ⊗
↓ q−2 V
W
→
q−1 V
W
↓
Cn−r+2,1 ⊗
↓ Cn−r+3,2 ⊗
↓
Cn−r+1,1 ⊗
↓
q+1 V j1 → Bn−r−1 ⊗ W ↓Φ q V j2 → Bn−r ⊗ W
q−2 V
W
.. .
→
↓ q−3 V
W
→
.. .
↓ .. . Starting from the third row, each horizontal sequence is exact except at q−k V the term Cn−r+k,k−1 ⊗ W (k ≥ 1) by Lemma 8.5.6. The horizontal sequences in the first and second row are exact, and the maps j1 and j2 are surjective (cf. (8.68)). The vertical sequences are exact at the boxed terms by Proposition 8.5.5. A diagram chase now shows that there is a finite decreasing filtration U • on Coker Φ such that U 0 Coker Φ = Coker Φ and that r+s−k ^
Coker Φ ' H n ( Grk−1 U
Σ∗ ) ⊗
q−k ^
W
(k ≥ 1).
This shows that the cokernel of Φ is a locally free R-module. Moreover, by Proposition 8.4.5 (iii), we have rank(Coker Φ) =
X k≥1
q−k ^ s−1 · rank W, r+s−k−n−1
(8.70)
which is equal to (8.67). (Note x` = 0 for ` < 0 by convention.) This completes the proof of Lemma 8.5.4 assuming TR/k ' W . r+s−k V ∗ Now we prove Lemma 8.5.6. Noting Mn−r+k,n+k = Σ , we decompose (8.69) into the following exact sequences: 0 → Mn−r+k,n+r+s → · · · → Mn−r+k,n+k+1 → N1 → 0,
(8.71)
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0 −→ N1 −→
r+s−k ^
Σ∗ −→ N2 −→ 0,
(8.72)
0 → N2 → Mn−r+k,n+k−1 → · · · → Mn−r+k,0 → 0.
(8.73)
By Lemma 8.4.4 i), H w (Mn−r+k,h ) = 0 for ∀w ≥ 0 and ∀h ≥ n + k + 1. r+s−k V ∗ Hence H w (N1 ) = 0 and H w ( Σ ) = H w (N2 ) for ∀w ≥ 0. On the other • V hand, since H w ( Σ∗ ⊗ Lν ) = 0 if ν > 0 and w > 0 by Lemma 8.4.4 (iii), we have H w (Mn−r+k,h ) = 0 for ∀w > 0 and 0 ≤ ∀h ≤ n + k − 1. This means that (8.73) is a flabby resolution of N2 . Therefore, for 0 ≤ h ≤ n + k − 2, the r+s−k V ∗ cohomology group at Cn−r+k,h is isomorphic to H n+k−1−h ( Σ ). Now the assertion follows from Proposition 8.4.5 (i). 8.5.3 Proof of (ii) : General Case It remains to show 8.5.4 in case Θ : TR/k → W is not necessarily an isomorphism. Setting I = Ker(TR/k → W ), we get the exact sequence: 0 −→ I −→ TR/k −→ W −→ 0.
(8.74)
Since W is a locally free R-module, so is I. By the argument in §8.5.2 it suffices to show that the cokernel of the map q+1 ^
Bn−r−1 (d − n − 1) ⊗
TR/k −→ Bn−r (d − n − 1) ⊗
q ^
TR/k
is a locally free R-module of rank q−k X s − 1 ^ · rank TR/k . n−r+k
(8.75)
(8.76)
k≥1
q q−i V V (8.74) gives rise to a filtration U • on TR/k such that U i /U i+1 = ( W ) ⊗ i V ( I). Since I annihilates B• (d − n − 1), the map (8.75) admits a filtration whose graded quotients for 0 ≤ i ≤ q are given by
Bn−r−1 (d−n−1)⊗
q+1−i ^
W⊗
i ^
I −→ Bn−r (d−n−1)⊗
q−i ^
W⊗
i ^
I. (8.77)
By what we have shown in §8.5.2 the cokernel of the map (8.75) is a locally free R-module of rank q q−i−k i X X s − 1 ^ ^ · rank( W ) · rank( I). (8.78) n−r+k i=0
k≥1
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It is easy to see that the numbers (8.76) and (8.78) are equal (left to the reader). This completes the proof of the Main Theorem. References [1] Arapura, D: The Leray spectral sequence is motivic, Invent. Math. 160, pp.567–589 (2005). [2] Asakura, M: On the K1 -groups of algebraic curves, Invent. Math. 149, pp.661– 685 (2002). [3] Asakura, M. and S. Saito: Generalized Jacobian rings for open complete intersections, Math. Nachr. 2004. [4] Asakura, M. and S. Saito: Noether-Lefschetz locus for Beilinson-Hodge cycles I, Math. Zeit. 2004. [5] Beilinson, A: Notes on absolute Hodge cohomology, in: Applications of Algebraic K-theory to Algebraic Geometry and Number theory, Contemp. Math. 55, 1986. pp.35-68, 27–41, Springer, 1987. [6] Beilinson, A: Height pairings between algebraic cyles, Lecture Notes in Math. 1289, 1–26, Springer, 1987. [7] Bloch, B.: Algebraic cycles and higher K-theory. Advances in Math. 61, 1986, 267–304 [8] Deligne, D.: Th´eorie de Hodge. II, III, Publ. Math. IHES, 40 (1971), 5–58, Publ. Math. IHES, 44 (1975), 5–77. [9] Deligne, D.: Equations diff´erentielles `a points singuliers r´eguliers, Lecture note in Math. 163, Springer, 1970. [10] Green, M.: The period map for hypersurface sections of high degree on an arbitrary variety, Compositio Math. 55 (1984), 135–156. [11] Jannsen, U.: Mixed Motives and Algebraic K-theory, Lecture note in Math. 1400, Springer, 1990. [12] Katz, N and T. Oda: On the differentiation of De Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199–213. [13] Milnor, J.: Introduction to Algebraic K-theory, Ann. of Math Studies 72, Princeton 1970. [14] Nori, M. S: Algebraic cycles and Hodge theoretic connectivity, Invent. Math., 111 (1993), 349–373 [15] Quillen, D.: Higher algebraic K-theory. I, 85–147, Lecture Notes in Math. Vol. 341, Springer, Berlin 1973. [16] Saito, N.: Modules de Hodge polarisables, Publ. RIMS. Kyoto Univ. 24 (1988), 849-995. [17] Saito, N.: Mixed Hodge modules, Publ. RIMS. Kyoto Univ. 26 (1990), 221333. [18] Schneider, P.: Introduction to the Beilinson conjectures, in: Beilinson’s Conjectures on Special Values of L-functions (editors: M. Rapoport, N. Schappacher and P. Schneider), Perspectives in Math. 4, Academic Press [19] Srinivas, V.: Algebraic K-theory, Progress in Math. 90 (1991), Birkh¨auser. [20] Steenbrink, J. and S. Zucker: Variation of mixed Hodge structure, Invent. math. 80 (1985) no.3, 489-542.
9 On the Splitting of the Bloch-Beilinson Filtration Arnaud BEAUVILLE Institut Universitaire de France
Laboratoire J.-A. Dieudonn´e ´ DE NICE UMR 6621 du CNRS, UNIVERSITE Parc Valrose F-06108 NICE Cedex 02
Introduction This paper deals with the Chow ring CH(X) (with rational coefficients) of a smooth projective variety X – that is, the Q-algebra of algebraic cycles on X, modulo rational equivalence. This is a basic invariant of the variety X, which may be thought of as an algebraic counterpart of the cohomology ring of a compact manifold; in fact there is a Q-algebra homomorphism cX : CH(X) → H(X, Q), the cycle class map. But unlike the cohomology ring, the Chow ring, and in particular the kernel of cX , is poorly understood. Still some insight into the structure of this ring is provided by the deep conjectures of Bloch and Beilinson. They predict the existence of a functorial ring filtration (F j )j≥0 of CH(X), with CHp (X) = F 0 CHp (X) ⊃ . . . ⊃ F p+1 (X) = 0 and F 1 CH(X) = Ker cX . We refer to [J] for a discussion of the various candidates for such a filtration and the consequences of its existence. The existence of that filtration is not even known for an abelian variety A. In that case, however, there is a canonical ring graduation given by CHp (A) =⊕ CHps (A), where CHps (A) is the subspace of elements α ∈ CHp (A) s
∗ α = k 2p−s α for all k ∈ Z (k denotes the endomorphism a 7→ ka of with kA A A) [B2]. Unfortunately this does not define the required filtration because the vanishing of the terms CHps (A) for s < 0 is not known in general – in fact, this vanishing is essentially equivalent to the existence of the BlochBeilinson filtration (the precise relationship is thoroughly analyzed in [Mu]). So if the Bloch-Beilinson filtration indeed exists, it splits in the sense that it is the filtration associated to a graduation of CH(A). In [B-V] we observed that this also happens for a K3 surface S. Here the filtration is essentially trivial; the fact that it splits means that the image of the intersection product CH1 (S) ⊗ CH1 (S) → CH2 (S) is always one-dimensional – an easy but somewhat surprising property.
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On the Splitting of the Bloch-Beilinson Filtration
347
The motivation for this paper was to understand whether the splitting of the Bloch-Beilinson filtration for abelian varieties and K3 surfaces is accidental or part of a more general framework. Now asking for a conjectural splitting of a conjectural filtration may look like a rather idle occupation. The point we want to make is that the mere existence of such a splitting has quite concrete consequences, which at least in some cases can be tested. We will restrict for simplicity to the case of regular varieties, that is, varieties X for which F 1 CH1 (X) = 0. Then if the filtration comes from a graduation, any product of divisors must have degree 0; therefore, if we denote by DCH(X) the sub-algebra of CH(X) spanned by divisor classes, the cycle class map cX : DCH(X) −→ H(X) is injective. In other words, any polynomial relation P (D1 , . . . , Ds ) = 0 between divisor classes which hold in cohomology must hold in CH(X). We will call this property the weak splitting property. Despite its name it is rather restrictive: it implies for instance the existence of a class ξX ∈ CHn (X), with n = dim X, such that D1 · . . . · Dn = deg(D1 · . . . · Dn ) · ξX
in CHn (X)
for any divisor classes D1 , . . . , Dn in CH1 (X). What kind of varieties can we expect to have the weak splitting property? A natural class containing abelian varieties and K3 surfaces is that of CalabiYau varieties, but that turns out to be too optimistic – it is quite easy to give counter-examples (Example 9.1.5. b)). A more restricted class is that of holomorphic symplectic manifolds – projective manifolds admitting an everywhere non-degenerate holomorphic 2-form. We want to propose the following conjecture: Conjecture. — A symplectic (projective) manifold satisfies the weak splitting property. We have to admit that the evidence we are able to provide is not overwhelming. We will prove that the weak splitting property is invariant under some simple birational transformations called Mukai flops (Proposition 9.2.4). We will also prove that the conjecture holds for the simplest examples of symplectic manifolds, the Hilbert schemes S [2] and S [3] associated to a K3 surface S (Proposition 9.3.1). Already for S [3] the proof is intricate, and makes use of some nontrivial relations in the Chow rings of S 2 and S 3 established in [B-V]. We hope that this might indicate a deep connection between the symplectic structure and the Bloch-Beilinson filtration, but we have not even a conjectural formulation of what this connection could be.
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9.1 Intersection of divisors Let X be a projective (complex) manifold. We denote by CH(X) and H(X) the Chow and cohomology rings with rational coefficients, and by CH(X, C) and H(X, C) the corresponding rings with complex coefficients. We denote by DCH(X) the sub-algebra of CH(X) spanned by divisor classes. We will say that X has the weak splitting property if the cycle class map cX : DCH(X) → H(X) is injective. Remark 9.1.1. The property as stated implies that CH1 (X) is finite-dimensional, that is, X is regular in the sense that H 1 (X, OX ) = 0. For irregular varieties the definition should be adapted, either by considering cycles modulo algebraic equivalence, or by picking up an appropriate subspace of CH1 (X). We will restrict ourselves to regular varieties in what follows. Examples 9.1.2. a) A regular surface S satisfies the weak splitting property if and only if the image of the intersection map CH1 (S) ⊗ CH1 (S) → CH2 (S) has rank 1; in other words, there exists a class ξS ∈ CH2 (S), of degree 1, such that C · D = deg(C.D) ξS for all curves C, D on S. This is the case when S is a K3 surface, or also an elliptic surface over P1 with a section [B-V]. b) Let S be a K3 surface, p a point of S with[p] 6= ξS in CH2 (S). Let ε : b is spanned by Sb → S be the blowing-up of S at p. The space DCH2 (S) ∗ b ε ξS and [q], where q is any point of S above p. Since the pushforward b → CH2 (S) is an isomorphism, theses classes are map ε∗ : CH2 (S) b so the map c2 : DCH2 (S) b → CH2 (S) b linearly independent in CH2 (S), b S is not injective. Observe that we get a family of surfaces parameterized by p ∈ S, for which the weak splitting property fails generically, but holds when p lies in the union of countably many subvarieties of the parameter space. c) We will give later (9.1.5) examples of Fano and Calabi-Yau threefolds which do not satisfy the weak splitting property. Proposition 9.1.3. Let X, Y be two smooth projective regular varieties. a) We have DCHp (X × Y ) = ⊕
r+s=p
pr∗1 DCHr (X) ⊗ pr∗2 DCHs (Y ). In
particular, X × Y satisfies the weak splitting property if and only if X and Y do. b) Let f : X → Y be a surjective map. If cpX : DCHp (X) → H2p (X) is injective, then so is cpY : DCHp (Y ) → H2p (Y ).
On the Splitting of the Bloch-Beilinson Filtration
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Proof a) We have CH1 (X × Y ) = pr∗1 CH1 (X) ⊕ pr∗2 CH1 (Y ) since X and Y are regular; the assertion a) follows at once. b) Follows from the commutative diagram cp
DCHxp (X) −−−X−→ H2px(X) f∗ f∗ cpY p DCH (Y ) −−−−→ H2p (Y ) and the injectivity of f ∗ : CHp (Y ) → CHp (X) (if h is an ample class in CH1 (X) and d = dim X − dim Y , we have f∗ (hd ) = r · 1Y , with r ∈ Q∗ , and f∗ (hd · f ∗ ξ) = r ξ for ξ in CH(Y )). We now consider the behaviour of the weak splitting property when the variety X is blown up along a smooth subvariety B. We will use the notation summarized in the following diagram: E η y B
i
,−−−−→ j
,−−−−→
b X ε y X .
(9.1)
We denote by c the codimension of B in X and by N its normal bundle. Lemma 9.1.4. Let p be an integer. Assume i) The cycle class map cqB : DCHq (B) → H2q (B) is injective for p − c < q < p; ii) The Chern classes ci (N ) belong to DCH(B) ; iii) The map cpX : CHp (X) → H2p (X) restricted to DCHp (X)+j∗ DCHp−c (B) is injective. b → H2p (X) b is injective. Then the cycle class map cpb : DCHp (X) X
Proof The projection p : E → B identifies E to PB (N ∨ ). Let h ∈ CH1 (E) be the class of the tautological bundle OE (1); we have i∗ [E] = −h, and therefore, for ξ ∈ CH(X), [E]p · ε∗ ξ = i∗ (i∗ [E]p−1 · i∗ ε∗ ξ) = (−1)p−1 i∗ (hp−1 · η ∗ j ∗ ξ). b = ε∗ CH1 (X) ⊕ Q[E], we get Since CH1 (X) b = ε∗ DCHp (X) + [E] · ε∗ DCHp−1 (X) + . . . + Q[E]p DCHp (X) ⊂ ε∗ DCHp (X) + i∗ η ∗ DCHp−1 (B) + i∗ (h · η ∗ DCHp−2 (B)) + . . . + Qi∗ hp−1 . For q ≥ c we have a relation hq = hc−1 · η ∗ cq,c−1 + . . . + η ∗ cq,0 , where the
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ci,j are polynomial in the Chern classes of N ; by our hypothesis (ii) these classes lie in DCH(B). Moreover the “key formula” [F, 6.7] i∗ (γ · η ∗ ξ) = ε∗ j∗ ξ
for ξ ∈ CH(B) ,
with γ = hc−1 + hc−2 · η ∗ c1 (N ) + . . . + η ∗ cc−1 (N ), implies i∗ (hc−1 · η ∗ DCHp−c (B)) ⊂ ε∗ j∗ DCHp−c (B) +
c−2 X
i∗ (hk · η ∗ DCHp−k−1 (B)) ,
k=0
so that we finally get c−2 X ∗ p p−c b DCH (X) ⊂ ε DCH (X) + j∗ DCH (B) + i∗ (hk · η ∗ DCHp−k−1 (B)) p
k=0
Since the map
2p
H (X) ⊕
c−2 X
H2(p−k−1) (B)
−→
b H2p (X)
7−→
ε∗ α +
k=0
(α ; β0 , . . . , βc−2 )
X
i∗ (hk · η ∗ βk )
k
is an isomorphism (see for instance [Jo]), our hypotheses (i) and (iii) ensure that cpb is injective. X
Examples 9.1.5. a) Take X = P3 , and let B be a smooth curve, of degree d and genus g. Let ` be the class of a hyperplane in P3 , `B its b is generated by pull back to B. The space DCH2 (X) ε∗ `2
,
ε∗ ` · [E] = i∗ p∗ `B
,
[E]2 = −i∗ h = i∗ η ∗ c1 (N ) − ε∗ [B]
b contains the elements i∗ η ∗ `B We have c1 (N ) = 4`B +KB , so DCH2 (X) and i∗ η ∗ KB . The map i∗ η ∗ : CH1 (B) → CH2 (X) induces an isomorphism of the subspace of degree 0 divisor classes on B onto the subspace of homologically trivial classes in CH2 (X). If we choose `B non proportional to KB in CH1 (B), the class i∗ η ∗ (d KB − (2g − 2)`B ) in b is homologically trivial, but non-trivial. Thus the map DCH2 (X) 2 b → H4 (X) b is not injective. c b : DCH2 (X) X b is a Fano vaIf B is a scheme-theoretical intersection of cubics, X riety [M-M] – we can take for instance B of genus 2 and `B a general divisor class of degree 5 (or B of genus 3 and `B general of degree 6, or B of genus 5 and `B ≡ KB − p for p a general point of B). Note
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that by making the linear system vary we get again families where the general member does not satisfy the weak splitting property, while countably many special members of the family do satisfy it. b) Going on with the Fano case, let D be a smooth divisor in | − 2KX |, and let V → X be the double covering of X ramified along D. Then by the above example and Proposition 9.1.3. b), V is a Calabi-Yau threefold which does not satisfy the weak splitting property.
9.2 The weak splitting property for symplectic manifolds By a symplectic manifold we mean here a simply-connected projective manifold which admits a holomorphic, everywhere non-degenerate 2-form. The manifold is said to be irreducible if the 2-form is unique up to a scalar; any symplectic manifold admits a canonical decomposition as a product of irreducible ones. In view of Proposition 9.1.3. a), we may restrict ourselves to irreducible symplectic manifolds. Let X be an irreducible symplectic manifold, of dimension 2r. Recall that the space H 2 (X) admits a canonical quadratic form q ([B1], [H]) with the following properties: • every class x ∈ H 2 (X, C) with q(x) = 0 satisfies xr+1 = 0; R • Rthere exists λ ∈ Q such that X x2r = λ q(x)r for all x ∈ H 2 (X, C), where ∼ 2r X is the canonical isomorphism H (X, C) −→ C. In fact the following more precise statement has been proved by Bogomolov: Proposition 9.2.1. Let V be a subspace of H2 (X, C) such that the restriction of q to V is non-degenerate (for instance V = H2 (X, C) or V = CH1 (X, C)). The kernel of the map Sym V → H(X, C) is the ideal of Sym V spanned by the elements xr+1 for x ∈ V, q(x) = 0. Proof The case V = H2 (X, C) is the main result of [Bo], but the proof given there implies the slightly more general statement 9.2.1. Namely, define A(V ) as the quotient of Sym V by the ideal spanned by the elements xr+1 for x ∈ V, q(x) = 0. Then Lemma 2.5 in [Bo] says that A(V ) is a finitedimensional graded Gorenstein C- algebra, with socle in degree 2r – in other words, A2r (V ) is one-dimensional, and the multiplication pairing Ad (V ) × A2r−d (V ) → A2r (V ) ∼ = C is a perfect duality. Since any element x of H2 (X, C) with q(x) = 0 satisfies xr+1 = 0, we get a C- algebra homomorphism u : A(V ) → H(X, C). The kernel of u is an ideal of A(V ); if it is non-zero, it contains the minimal ideal A2r (V ) of A(V ). But
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this is impossible because V contains an element h with q(h) 6= 0, hence with h2r 6= 0. Corollary 9.2.2. The following conditions are equivalent: i) The cycle class map cX : DCH(X) → H(X) is injective (that is, X satisfies the weak splitting property); r+1 ii) The map cr+1 (X) → H 2r+2 (X) is injective; X : DCH
iii) Every element x of CH1 (X, C) with q(x) = 0 satisfies xr+1 = 0 (in CHr+1 (X, C)). Proof Consider the diagram Sym CH1 (X, C)
PPP PPPv PPP u PPP ( / H(X, C) . DCH(X, C) c X
The injectivity of c is equivalent to Ker v ⊂ Ker u. In view of the Proposition, this is exactly condition (iii), and it is equivalent to Ker v r+1 ⊂ Ker ur+1 .
Remark 9.2.3. Assume that there is an element α ∈ CH1 (X) with q(α) = 0 – this is the case for instance if dimQ CH1 (X) ≥ 5. Then the set of such elements is Zariski dense in the quadric q = 0 of CH1 (X, C). Thus the conditions of the Corollary are also equivalent to: (iii0 ) Every element x of CH1 (X) with q(x) = 0 satisfies xr+1 = 0. A possible proof of (iii0 ) could be as follows. It seems plausible that the subset of nef classes x ∈ CH1 (X) with q(x) = 0 is Zariski dense in the quadric q = 0 (this holds at least when X is a K3 surface). If this is the case, it would be enough to prove (iii0 ) for nef classes. Now it is a standard conjecture (see [S]) that a nef class x ∈ CH1 (X) with q(x) = 0 should be the pull back of the class of a hyperplane in Pr under a Lagrangian fibration f : X → Pr , so that xr+1 = f ∗ (hr+1 ) = 0. We will now consider the behaviour of the weak splitting property under a Mukai flop. Let X be an irreducible symplectic manifold, of dimension 2r; assume that X contains a subvariety P isomorphic to Pr . Then P is a Lagrangian subvariety, and its normal bundle in X is isomorphic to Ω1P . We
On the Splitting of the Bloch-Beilinson Filtration
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blow up P in X, getting our standard diagram i
,−−−−→
E η y P
b X ε y X .
j
,−−−−→
The exceptional divisor E is the cotangent bundle P(TP ), which can be identified with the incidence divisor in P × P ∨ , where P ∨ is the projective space dual to P . The projection η ∨ : E → P ∨ identifies E to P(TP ∨ ), and b → X 0 , where X 0 is a smooth E can be blown down to P ∨ by a map ϕ : X algebraic space. To remain in our previous framework we will assume that X 0 is projective, so that X 0 is again an irreducible symplectic manifold. The diagram
X
ε
b@ X @
@@ ϕ @@ @@
X0
is called a Mukai flop. There are many concrete examples of such flops, see [M]. Proposition 9.2.4. If X satisfies the weak splitting property, so does X 0 . Proof Consider the Q- linear map ϕ∗ ε∗ : CH1 (X) → CH1 (X 0 ). It is bijective and preserves the canonical quadratic forms (see e.g. [H, Lemma 2.6]. In view of Corollary 9.2.2, the Proposition will follow from Lemma 9.2.5. Let α ∈ CH1 (X), and α0 := ϕ∗ ε∗ α. Then α0 r+1 = ϕ∗ ε∗ (αr+1 ). b be Proof We have ϕ∗ α0 = ε∗ α + m[E] for some m ∈ Q. Let ` ∈ CH2r−1 (X) ∨ the class of a line contained in a fibre of η ; we have deg([E] · `) = −1, and ε∗ ` is the class of a line in P . Intersecting the above equality with ` gives m = deg(α|P ), or equivalently α|P = mk in CH1 (P ), where k is the class of a hyperplane in P . Then ∗ 0 r+1
ϕ α
∗
r+1
= (ε α + m[E])
=
r+1 X r+1 p=0
p
mr+1−p ε∗ αp · [E]r+1−p .
As in (9.1.4), let h ∈ CH1 (E) be the class of OE (1). For p ≤ r we have p ε∗ αp · [E]r+1−p = (−1)r−p i∗ (hr−p · i∗ ε∗ αp ) = (−1)r−p i∗ (hr−p · η ∗ α|P ),
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A. Beauville
∗ 0 r+1
ϕ α
∗ r+1
=ε α
+m
r+1
i∗
r X r+1 p=0
p
(−1)r−p hr−p η ∗ k p .
Now since the total Chern class of TP is (1 + k)r+1 we have in CHr (E) r r X X r+1 p r−p ∗ p (−1) h η k = (−1)p hr−p η ∗ cp (TP ) = 0 , p p=0
p=0
hence ϕ∗ α0 r+1 = ε∗ αr+1 . Applying ϕ∗ gives the lemma, hence the Proposition. Corollary 9.2.6. Let X, X 0 be birationally equivalent projective symplectic fourfolds. Then X satisfies the weak splitting property if and only if X 0 does. Indeed any birational map between projective symplectic fourfolds is a composition of Mukai flops [W]. 9.3 The weak splitting property for S[2] and S[3] . The simplest symplectic manifolds are K3 surfaces, for which we have already seen that the weak splitting property holds (Example 9.1.2). More precisely [B-V], let S be a K3 surface and o a point of S lying on a (singular) rational curve R. The class of o in CH2 (S) is independent of the choice of R, and we have, for every α, β ∈ CH1 (S), α · β = deg(α · β) [o]
in CH2 (S) .
Let ∆ : S ,−→ S × S be the diagonal embedding. For α ∈ CH1 (S), we have in CH3 (S × S) ([B-V, Prop. 1.6],) ∆∗ α = pr∗1 α · pr∗2 [o] + pr∗1 [o] · pr∗2 α .
(9.2)
K3 surfaces are the first instance of a famous series of symplectic manifolds, the Hilbert schemes S [r] parameterizing finite subschemes of length r on the K3 surface S. Proposition 9.3.1. Let S be a K3 surface. The symplectic varieties S [2] and S [3] satisfy the weak splitting property. Proof — Let us warm up with the easy case of S [2] . Let S {2} be the variety obtained by blowing up the diagonal of S × S. The Hilbert scheme S [2] is the quotient of S {2} by the involution which exchanges the factors. In
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view of Corollary 9.2.1 and Proposition 9.1.3. b) it suffices to prove that the cycle class map c3S {2} : DCH3 (S {2} ) → H6 (S {2} ) is injective. We will check that the hypotheses of Lemma 9.1.4 are satisfied. Condition (i) is the weak splitting property for S. The normal bundle to the diagonal in S × S is TS , so (ii) means that the class c2 (TS ) belongs to DCH2 (S); this is proved in ([B-V, thm. 1 c]). Formula (9.2) implies ∆∗ CH1 (S) ⊂ DCH3 (S × S), so condition (iii) reduces to the injectivity of c3S×S , which follows from Proposition 9.1.3. a) and the corresponding result for S. — Let us pass to the more difficult case of S [3] . The Hilbert scheme S [3] is dominated by the nested Hilbert scheme S [2,3] which parameterizes pairs (Z, Z 0 ) ∈ S [2] × S [3] with Z ⊂ Z 0 ; it is isomorphic to the blow-up of S × S [2] along the incidence subvariety I = {(x, Z) | x ∈ Z}. Let π : S {2} → S [2] be the quotient map, and p : S {2} → S the first projection. Then the map j = (p, π) : S {2} ,−→ S × S [2] induces an isomorphism of S {2} onto I (see for instance [L, 1.2]). To prove the theorem, it suffices, by Corollary 9.2.2 and Proposition 9.1.3. b), to prove that the cycle class map DCH4 (S [2,3] ) → H8 (S [2,3] ) is injective. We will again check that the hypotheses of Lemma 9.1.4 are satisfied. Condition (i) is the injectivity of the cycle class map c3S {2} : DCH3 (S {2} ) → H6 (S {2} ), which has just been proved. Let N be the normal bundle to the embedding j : S {2} ,−→ S × S [2] , and E ⊂ S {2} the exceptional divisor, which is the ramification locus of π. From the exact sequences 0 → N ∨ −→ p∗ Ω1S ⊕ π ∗ Ω1S [2] −→ Ω1S {2} → 0 0 → π ∗ Ω1S [2] −→ Ω1S {2} −→ OE (−E) → 0
0 → OS {2} (−2E) −→ OS {2} (−E) −→ OE (−E) → 0 we obtain the equality in K-theory [N ∨ ] = [p∗ Ω1S ]+[OS {2} (−2E)]−[OS {2} (−E)]. We conclude that c2 (N ) = c2 (N ∨ ) belongs to DCH2 (S {2} ), so that condition (ii) holds. — The rest of the proof will be devoted to check condition (iii), namely the injectivity of DCH4 (S × S [2] ) + j∗ DCH2 (S {2} ) −→ H8 (S × S [2] ) . Let us fix some notation. We will use our standard diagram (9.1) E η y S
i
{2} ,−−−−→ S ε y ∆ ,−−−−→ S 2 .
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We denote by p and q the two projections of S {2} onto S. We define an injective Q-linear map ι : CH(S) → CH(S [2] ) by ι(ξ) := π∗ p∗ ξ; we will use the same notation for cohomology classes. We have π ∗ ι(ξ) = p∗ ξ + q ∗ ξ for ξ in CH(S) or H(S). Finally if α ∈ CH(S) and ξ ∈ CH(S [2] ) we put α ξ := pr∗1 α ⊗ pr∗2 ξ. We have CH1 (S {2} ) = p∗ CH1 (S)⊕q ∗ CH1 (S)⊕Q[E]. In CH2 (S {2} ) we have [E] · ε∗ α = i∗ η ∗ ∆∗ α for α ∈ CH1 (S 2 ), and [E]2 = −ε∗ [∆(S)]. Therefore: DCH2 (S {2} ) = Q p∗ [o]+Q q ∗ [o] + p∗ CH1 (S)⊗q ∗ CH1 (S) + i∗ η ∗ CH1 (S)+Q ε∗ [∆(S)] . We want to describe the space j∗ DCH2 (S) + DCH4 (S × S {2} ). Lemma 9.3.2. Let α, β ∈ CH1 (S). The classes j∗ p∗ [o], j∗ ( p∗ α · q ∗ β), and j∗ i∗ η ∗ α belong to DCH4 (S × S {2} ) + Q ([o] ι([o])). Proof Let j 0 : S {2} ,−→ S ×S {2} be the embedding given by j 0 (z) = (p(z), z), so that j = (1, π) ◦ j 0 . From the cartesian diagram j0
{2} S p y S
(9.3.2)
{2} ,−−−−→ S × S (1,p) y ∆ ,−−−−→ S × S
we obtain j∗0 p∗ [o] = (1, p)∗ ∆∗ [o] = [o] p∗ [o], hence j∗ p∗ [o] = [o] the same way we have j∗0 p∗ α = (1, p)∗ ∆∗ α, hence, using (9.2) , j∗0 p∗ α = α
p∗ [o] + [o]
ι([o]). In
p∗ α .
Multiplying by pr∗2 q ∗ β and using pr2 ◦ j 0 = Id we obtain
hence
j∗0 (p∗ α · q ∗ β) = α
(p∗ [o] · q ∗ β) + [o]
j∗ (p∗ α · q ∗ β) = α
π∗ (p∗ [o] · q ∗ β) + [o]
(p∗ α · q ∗ β) ,
π∗ (p∗ α · q ∗ β) .
For α, β ∈ CH1 (S), put hα, βi := deg(α · β). Then π ∗ π∗ (p∗ α · q ∗ β) = p∗ α · q ∗ β + p∗ β · q ∗ α = (p∗ α + q ∗ α)(p∗ β + q ∗ β) − hα, βi(p∗ [o] + q ∗ [o]) = π ∗ ι(α)ι(β) − hα, βiι([o]) ; we find similarly π ∗ π∗ (p∗ [o] · q ∗ β) = π ∗ ι([o])ι(β), and finally j∗ (p∗ α · q ∗ β) = α
ι([o])ι([β]) + [o]
ι(α)ι(β) − hα, βi ι([o]) .
Let γ ∈ CH1 (S). We have ι(β)2 · ι(γ) = hβ 2 i ι([o])ι(γ) + 2hβ · γi ι([o])ι(β) (this is easily checked by applying π ∗ as above). If hβ 2 i = 6 0 we conclude
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by taking γ = β that ι([o])ι(β) is proportional to ι(β)3 . If hβ 2 i = 0 we can choose γ so that (β · γ) 6= 0; then ι([o])ι(β) is proportional to ι(β)2 ι(γ). In each case we see that ι([o])ι(β) belongs to DCH3 (S [2] ), hence the assertion of the lemma about j∗ ( p∗ α · q ∗ β). Consider finally the cartesian diagram E η y S
k
,−−−−→ S × E (1,η) y ∆ ,−−−−→ S × S
with k(e) = (η(e), e). Using again (9.2) we get k∗ η ∗ α = (1, η)∗ ∆∗ α = α
η ∗ [o] + [o]
η∗α .
Pushing forward in S × S [2] we obtain j∗ i∗ η ∗ α = α i0∗ η ∗ [o] + [o] i0∗ η ∗ α, where i0 = π ◦ i is the embedding of E in S [2] . ¯ the image of E in S [2] , so that To avoid confusion let us denote by E ¯ = 2[E]. We have i0∗ η ∗ α = π∗ ([E] · p∗ α) = 1 [E] ¯ · ι(α) ∈ DCH2 (S [2] ). π ∗ [E] 2 1 ¯ 3 Likewise [E]3 = i∗ h2 = −24i∗ η ∗ [o], hence i0∗ η ∗ [o] = − 96 [E] ∈ DCH2 (S [2] ). This finishes the proof of the lemma. The lemma and the formula for DCH2 (S {2} ) show that j∗ DCH2 (S {2} ) is spanned modulo DCH4 (S × S [2] ) by the classes [o]
ι([o]) ,
j∗ q ∗ [o] ,
j∗ ε∗ [∆(S)] .
In fact there is one more relation, much more subtle, between these classes modulo DCH4 (S × S [2] ). Lemma 9.3.3. We have 2[o]
ι([o]) − 2j∗ q ∗ [o] + j∗ ε∗ [∆(S)] ∈ DCH4 (S × S [2] ) .
Proof We start from a relation in CH4 (S 3 ), proved in [B-V, Prop. 3.2]. For 1 ≤ i < j ≤ 3, let us denote by pij : S 3 → S 2 the projection onto the i- th and j- th factors, and by pi : S 3 → S the projection onto the i- th factor. We will write simply ∆ for the diagonal ∆(S) ⊂ S 2 , and δ ⊂ S 3 for the small diagonal, that is, the subvariety of triples (x, x, x) for x ∈ S. Then: X X [δ] − p∗ij [∆] · p∗k [o] + p∗i [o] · p∗j [o] = 0 . i<j,k6=i,j
i<j
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Pull back this relation by the map εS = (1S , ε) : S × S {2} → S × S 2 . Since p1 ◦ εS = pr1 , p2 ◦ εS = p ◦ pr2 , p3 ◦ εS = q ◦ pr2 , p23 ◦ εS = ε , we obtain ε∗S [δ] = j∗0 ε∗ [∆] , ε∗S (p∗1 [o] · p∗23 [∆]) = [o] ε∗ [∆] , ∗ ∗ ∗ ∗ ∗ εS (p2 [o] · p3 [o]) = 1 p [o] · q [o] , ε∗S (p∗1 [o] · p∗2 [o]) = [o] p∗ [o] , ε∗S (p∗1 [o] · p∗3 [o]) = [o] q ∗ [o] . We have p12 ◦ εS = (1, p), hence ε∗S p∗12 [∆] = j∗0 1 (see diagram 9.3.2) and ε∗S (p∗3 [o] · p∗12 [∆]) = j∗0 q ∗ [o]. Let j 00 = (q, 1) : S {2} → S × S {2} ; the same argument gives ε∗S p∗13 [∆] = j∗00 1 and ε∗S (p∗2 [o] · p∗13 [∆]) = j∗00 p∗ [o] . Finally we have kj∗0 q ∗ [o] + j∗00 p∗ [o] = πS∗ j∗ q ∗ [o]. Pushing forward by πS we obtain in CH4 (S × S [2] ): j∗ ε∗ [∆] − 2j∗ q ∗ [o] − [o]
π∗ ε∗ [∆] + 2[o]
ι([o]) + 1
ι([o])2 = 0
It remains to observe that [o] π∗ ε∗ [∆] and 1 ι([o])2 belong to DCH4 (S × ¯ 2 ∈ DCH2 (S [2] ). S [2] ). Indeed from [E]2 = −ε∗ [∆] we deduce π∗ ε∗ [∆] = − 21 [E] 1 2 And if h is any element of CH (S) with h = d 6= 0, we have π ∗ ι(h)4 = 6d2 p∗ [o] · q ∗ [o] = 3d2 π ∗ ι([o])2 , hence ι([o])2 ∈ DCH4 (S [2] ) . For a smooth projective variety X, let us denote by DH(X) the (graded) subspace of H(X) spanned by intersection of divisor classes – that is, the image of DCH(X) by the cycle class map. It remains to prove that the cycle class map c4S×S [2] is injective on DCH4 (S × S [2] ) + Q ([o] ι([o])) + Q j∗ q ∗ [o]. Since we know by (9.3) and (9.1.3. a)) that it is injective on DCH4 (S × S [2] ), this amounts to : Lemma 9.3.4. There is no non-trivial relation a [o]
ι([o]) + b j∗ q ∗ [o] ∈ DH8 (S × S [2] ) ,
with a, b ∈ Q. To prove this, suppose that such a relation holds. Let ω be a non-zero class in H 2,0 (S); for any class ξ in H8 (S ×S [2] , C) put h(ξ) := (pr2 )∗ (pr∗1 ω·ξ). Since the product of ω with any algebraic class in H 2 (S) is zero, h is zero on DH8 (S ×S [2] ). Clearly h([o] ι([o])) = 0, while h(j∗ q ∗ [o]) = π∗ (p∗ ω·q ∗ [o]) = ι(ω)ι([o]). This class is nonzero, for instance because hι(ω)ι([o]), ι(¯ ω )i = hω, ω ¯ i > 0. Thus b = 0, and our relation reduces to [o] ι([o]) ∈ DH8 (S × S [2] ). Since DH8 (S × S [2] ) = ⊕ DHi (S) DHj (S [2] ) (see Proposition 9.1.3. a)), this is i+j=8
equivalent to ι([o]) ∈ DH4 (S [2] ). Thus the proof reduces to the following assertion:
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Lemma 9.3.5. The class ι([o]) does not belong to DH4 (S [2] ). Proof We have H4 (S {2} ) = ε∗ H4 (S 2 ) ⊕ i∗ η ∗ H2 (S) = Q p∗ [o] ⊕ Q q ∗ [o] ⊕ (p∗ H2 (S) ⊗ q ∗ H2 (S)) ⊕ i∗ η ∗ H2 (S) . Taking the invariants under the involution of S {2} which exchanges the factors, we find H4 (S [2] ) = Q ι([o]) ⊕ Sym2 H2 (S) ⊕ i0∗ η ∗ H2 (S) , where Sym2 H2 (S) is identified with a subspace of H4 (S [2] ) by α·β 7→ π∗ (p∗ α· q ∗ β), and i0 := π ◦ i is the natural embedding of E in S [2] . Since CH1 (S [2] ) = ι(CH1 (S)) ⊕ Q · [E], the subspace DH4 (S [2] ) is spanned by the classes ι(α)ι(β) ,
ι(α) · [E] = 2i0∗ η ∗ α ,
[E]2 = −2π∗ ε∗ [∆]
Suppose that we have a relation X ι([o]) = mij ι(αi )ι(αj ) + i0∗ η ∗ γ + m π∗ ε∗ [∆]
for α, β ∈ CH1 (S) .
in H4 (S [2] ) ,
i<j
where (αi ) is a basis of CH1 (S). This gives in H4 (S {2} ): X p∗ [o] + q ∗ [o] = mij (p∗ αi + q ∗ αi )(p∗ αj + q ∗ αj ) + 2i∗ η ∗ γ + 2m ε∗ [∆] . i<j
Projecting onto the direct summand i∗ η ∗ H2 (S) of H4 (S {2} ) we find i∗ η ∗ γ = 0. Multiplying by p∗ ω and pushing forward by q we find as in the proof of (9.3.4) that all terms but ε∗ [∆] give 0, so m = 0. Finally the equality X p∗ [o] + q ∗ [o] = mij (p∗ αi + q ∗ αi )(p∗ αj + q ∗ αj ) i<j
projected onto Sym2 H2 (S) gives mij = 0 for all i, j. This achieves the proof of the lemma and therefore of the Proposition. Comments. A variation of this method can be used to prove that the generalized Kummer variety K2 associated to an abelian surface A [B1] has the weak splitting property; one must replace CH1 (A) by the subspace of symmetric divisor classes. We leave the details to the reader. We should point out, however, that even among symplectic fourfolds these examples are quite particular. Indeed for each integer g ≥ 2, the projective K3 surfaces of genus g (that is, embedded in Pg with degree 2g − 2) form
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an irreducible 19-dimensional family; the corresponding family of Hilbert schemes S [2] is contained in a 20-dimensional irreducible family of projective symplectic manifolds (see [B1]). Since the weak splitting property is not invariant under deformation, we do not know whether it holds for the general member of such a family. It would be interesting, in particular, to check whether the property holds for the variety of lines contained in a smooth cubic hypersurface in P5 . References [B1] Beauville, A.: Vari´et´es k¨ahl´eriennes dont la premi`ere classe de Chern est nulle, J. of Diff. Geometry 18, 755–782 (1983). [B2] Beauville, A.: Sur l’anneau de Chow d’une vari´et´e ab´elienne, Math. Annalen 273, 647–651 (1986). [B-V] Beauville, A, C. Voisin: On the Chow ring of a K3 surface, J. Algebraic Geom., to appear. [Bo] Bogomolov, F.: On the cohomology ring of a simple hyper-K¨ahler manifold (on the results of Verbitsky), Geom. Funct. Anal. 6 (1996), 612–618. [F] Fulton, W.: Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer-Verlag, Berlin (1984). [J] Jannsen, U.: Motivic sheaves and filtrations on Chow groups, in Motives (Seattle, 1991), 245–302, Proc. Sympos. Pure Math. 55 (Part 1), Amer. Math. Soc., Providence, RI, 1994. [Jo] Jouanoulou, J.-P.: Cohomologie de quelques sch´emas classiques et th´eorie cohomologique des classes de Chern, SGA 5 (Cohomologie l- adique et fonctions L), Lecture Notes in Math. 589, 282–350, Springer-Verlag, Berlin (1977). [H] Huybrechts, D.: Compact hyper-K¨ahler manifolds: basic results, Invent. Math. 135 (1999), 63–113. [L] Lehn, M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), 157–207. [M] Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), no. 1, 101–116. [M-M] Mori, S. and S. Mukai: Classification of Fano 3-folds with B2 ≥ 2. Manuscripta Math. 36 (1981/82), 147–162. [Mu] Murre, J. P.: On a conjectural filtration on the Chow groups of an algebraic variety, I: The general conjectures and some examples, Indag. Math. (N.S.) 4 (1993), 177–188. [S] Sawon, J.: Abelian fibred holomorphic symplectic manifolds, Turk. J. Math. 27 (2003), 197-230. [W] Wierzba, J.: Birational Geometry of Symplectic 4-folds, Preprint (2002).
10 K¨unneth Projectors for Open Varieties Spencer Bloch Dept. of Mathematics, University of Chicago, Chicago, IL 60637, USA
[email protected]
H´el`ene Esnault Mathematik, Universit¨ at Duisburg-Essen FB6, Mathematik, 45117 Essen, Germany
[email protected]
To Jacob Murre
Abstract We consider correspondences on smooth quasiprojective varieties U . An algebraic cycle inducing the K¨ unneth projector onto H 1 (U ) is constructed. Assuming normal crossings at infinity, the existence of relative motivic cohomology is shown to imply the independence of ` for traces of open correspondences.
10.1 Introduction Let X be a smooth, projective algebraic variety over an algebraically closed field k, and let H ∗ (X) denote a Weil cohomology theory. The existence of algebraic cycles on X × X inducing as correspondences the various K¨ unneth i ∗ i projectors π : H (X) → H (X) is one of the standard conjectures of Grothendieck, [10, 11]. It is known in general only for the cases i = 0, 1, 2d− 1, 2d where d = dim X. The purpose of this note is to consider correspondences on smooth quasi-projective varieties U . In the first section we prove the existence of an “algebraic” K¨ unneth projector π 1 : H ∗ (U ) → H 1 (U ) assuming that U admits a smooth, projective completion X. The word algebraic is placed in quotes here because in fact the algebraic cycle on X × U inducing π 1 is not, as one might imagine, trivialized on (X − U ) × U . It is only partially trivialized. This partial trivialization is sufficient to define a class in Hc2d−1 (U ) ⊗ H 1 (U ) giving the desired projection. Of course, our cycle on X × U will be trivialized on (X − U ) × V for V ⊂ U suitably small nonempty open, but our method does not in any obvious way yield a full trivialization on (X − U ) × U . We finish this first section with some 361
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comments on π i for i > 1 and some speculation, mostly coming from discussions with A. Beilinson, on how these ideas might be applied to study the Milnor conjecture that the Galois cohomology ring of the function field H ∗ (k(X), Z/nZ) is generated by H 1 . In the last section, we use the existence of relative motivic cohomology [12] to prove an integrality and independence of ` result for the trace of an algebraic correspondence Γ on U × U . We are indebted to G. Laumon for pointing out that one may endeavor to prove this using results already in the literature([5, 14, 15], and [7]) by reduction mod p and composition with a high power of Frobenius. Our objective in what follows is to show how techniques in motivic cohomology can apply to such questions, at least when the divisor at infinity D = X − U is a normal crossings divisor. When the Zariski closure of the correspondence stabilizes the various strata DI at infinity (e.g. when the correspondence is the graph of Frobenius) then the trace on H ∗ (U ) is realized as an alternating sum of traces on H ∗ (DI ). When in addition all the intersections with the diagonals are transverse, the contribution to the alternating sum coming from points lying off U cancels, and the trace on H ∗ (U ) is just the sum of the fixed points on U. We would like to acknowledge helpful correspondence with A. Beilinson, M. Levine, J. Murre, T. Saito, and V. Srinivas. We thank G. Laumon and L. Lafforgue for explaining to us [7], and the referee for very useful comments and advice.
10.2 The first K¨ unneth component Let k be an algebraically closed field. We work in the category of algebraic varieties over k. H ∗ (X) will denote ´etale cohomology with Q` -coefficients for some ` prime to the characteristic of k. If k = C, we take Betti cohomology with Q-coefficients. Let C be a smooth, complete curve over k, and let δ ⊂ C be a nonempty finite set of reduced points. Let J(C) be the Jacobian of C, and let J(C, δ) be the semiabelian variety which represents the functor X 7→ {(L, φ) | L line bundle on C × X, deg L|C×k(X) = 0 φ : L|δ×X ∼ = Oδ×X }/ ∼ =,
(10.1)
where the equivalence relation ∼ = consists of the line bundle isomorphisms commuting with φ. There is an exact sequence 0 → T → J(C, δ) → J(C) → 0
(10.2)
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where T is the torus Γ(δ, O× )/Γ(C, O× ). By abuse of notation we shall write J(C, δ) rather than J(C, δ)(k). We can identify the character group Hom(T, Gm ) with Div0δ (C), the group of 0-cycles of degree 0 supported on δ. A split subgroup ∆ ⊂ Div0δ (C) corresponds to a quotient T T∆ = T / ker ∆, where ker ∆ ⊂ T is the subtorus killed by all characters in ∆. We may push out (10.2) and define J(C, ∆) := J(C, δ)/ ker ∆: 0 → T∆ → J(C, ∆) → J(C) → 0.
(10.3)
The functor represented by J(C, ∆) is the following quotient of (10.1) X 7→ {(L, φ) | L line bundle on X × C, deg L|k(X)×C = 0 X φ : ⊗i L⊗ni |X×{ci } ∼ ni ci ∈ ∆}/ ∼ = OX for all =.
(10.4)
These trivializations should be compatible in an evident way with the group law on ∆. Lemma 10.2.1. We write H 1 (C, δ) = Hc1 (C − δ). Define H 1 (C, ∆) := (H 1 (C, δ)/∆⊥ ) ⊗ Q` , where ∆⊥ ⊗ Q` ⊂ Q` [δ]/Q` ⊂ H 1 (C, δ; Q` ) is perpendicular to ∆ ⊂ Div0δ (C) under the evident coordinatewise duality. There is a well defined first Chern class c1 (L∆ ) of the universal Poincar´e bundle L∆ on J(C, ∆) × C which lies in H 1 (J(C, ∆))(1) ⊗ H 1 (C, ∆). Proof Let Iδ ⊂ OJ(C,∆)×C be the ideal of J(C, ∆) × δ. Let π : C → C 0 be the singular curve obtained from C by gluing all the points of δ to a single × point δ 0 ∈ C 0 . Define M∆ ⊂ (1 × π)∗ (O× J(C,∆)×C )/k to be the pullback as indicated: 0 − → (1 × π)∗ (1 + Iδ ) − →
0 − → (1 × π)∗ (1 + Iδ ) − →
M∆ y (1×π)∗ O× J(C,∆)×C k×
→ (ker ∆)J(C,∆)×δ0 − − →0 y → −
(1×π)∗ O× J(C,∆)×δ k×
(10.5)
→ 0. −
Using that R1 (1 × π)∗ Gm = (0), it is straightforward to check that pairs (L, φ) in (10.4) with X = J(C, ∆) correspond to M∆ torsors on J(C, ∆)×C 0 . In particular, we have a class [L∆ ] ∈ H 1 (J(C, ∆) × C 0 , M∆ ) corresponding to the Poincar´e bundle.
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One gets a diagram of Kummer sequences of sheaves on J(C, ∆) × C 0 (Here j : C − δ ,→ C) 0 y
0 y
0 y `n
0 − → (1 × π)∗ (1 × j)! µ`n − → (1 × π)∗ (1 + Iδ ) −→ (1 × π)∗ (1 + Iδ ) − →0 y y y 0 − →
M∆,`n y
− →
M∆ y
`n
−→
M∆ y
→0 −
`n
0 − → ((ker ∆)J(C,∆)×δ0 )`n − → (ker ∆)J(C,∆)×δ0 −→ (ker ∆)J(C,∆)×δ0 − →0 y y y 0
0
0
(10.6) We have [L∆ ] ∈ ∆ ) and so by the Kummer coboundary, c1 (L∆ ) ∈ limn H 2 (J(C, ∆) × C 0 , M∆,`n ). But M∆,`n ∼ = Z/`nJ(C,∆) ψ`n , ←− where ψ`n fits into an exact sequence of sheaves on C 0 H 1 (J(C, ∆) × C 0 , M
0 → π∗ j! µ`n → ψ`n → (ker ∆)δ0 ,`n → 0.
(10.7)
We can identify ∆⊥ ⊗ µ`n with (ker ∆)`n . the exact cohomology sequence from (10.7) yields (ker ∆)µ`n → H 1 (C, δ; µ`n ) → H 1 (C 0 , ψ`n ) → 0.
(10.8)
Passing to the limit over n, it now follows that we may define c1 (L∆ ) ∈ H 1 (J(C, ∆)) ⊗ H 1 (C, ∆)(1) as in the statement of the lemma. (Note that L∆ is trivial on (0)×C. Further we are free to replace L∆ by L∆ ⊗(MOC ) for M a line bundle on J(C, ∆). We may therefore assume the K¨ unneth components of c1 (L∆ ) in degrees (2, 0) and (0, 2) vanish.) Lemma 10.2.2. Suppose given a morphism ρ : X → J(C). Let Ξ be a Cartier divisor on C × X representing ρ. We assume Ξ is flat over C so we may define a correspondence Ξ∗ : Div(C) → Div(X). Let U ⊂ X be nonempty open in X. Then there exists a lifting ρU,∆ : U → J(C, ∆) of ρ if and only if (Ξ|C×U )∗ (∆) ⊂ Div(U ) consists of principal divisors. The set of such liftings is a torsor under Hom(∆, Γ(U, O× U )). Proof Choose a basis zi =
P
j
nij cj for the free abelian group ∆. Write
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OC×X (Ξ)zi ×X := ⊗j OC×X (Ξ)⊗nij |{cj }×X . The assumption that Ξ∗ (∆) consists of principal divisors is precisely the assumption that all the line bundles OC×X (Ξ)zi ×X |U are trivial. The choice of the trivializations for a basis of ∆ yields the choice of the desired lifting ρU,∆ . Lemma 10.2.3. Assume X is a smooth variety, and let ρ : X → J(C) be as above. Suppose U ⊂ X is a dense open set such that ρ|U admits a lifting ρU,∆ : U → J(C, ∆). Let Div0X−U (X) be the free abelian group on Cartier divisors supported on X − U which are homologous to 0 on X. Then we get a commutative diagram on cohomology 0 − → H 1 (J(C)) − → H 1 (J(C, ∆)) − → ρ∗ ρ∗ y y U,∆ 0 − → H 1 (X)
→ −
H 1 (U )
∆ ⊗ Q` (−1) a y
→0 − (10.9)
→ Div0X−U (X) ⊗ Q` (−1) − − → 0.
Proof The left hand square is commutative by functoriality. That the cokernels on the top and bottom row are as indicated follows on the top row from the Leray spectral sequence for the projection π : J(C, ∆) → J(C) and on the bottom from the localization sequence which may be written 2 0 → H 1 (X) → H 1 (U ) → HX−U (X) → H 2 (X).
(10.10)
2 The identification HX−U (X) ∼ = DivX−U (X) ⊗ Q` (−1) is saying that by purity, the Gysin homomorphism is an isomorphism.
Remark 10.2.4. (i) Fixing ρU,∆ amounts to fixing trivializations of the restriction OC×X (Ξ)zi ×X |U as above. Such trivializations exhibit OC×X (Ξ)zi ×X ∼ = OX (Di ) for some divisor Di with support on X − U . The map labeled a in (10.9) sends zi 7→ Di . (ii) The diagram ∆ −−−−→ J(C) a ρ∗ (10.11) y y Div0X−U (X) −−−−→ Pic0 (X) is commutative, where the horizontal arrows are cycle classes. Indeed, both a and ρ∗ are defined by the divisor on C × X. Note that a depends on the choice of ρU,∆ but only up to rational equivalence. Now suppose X is smooth, projective, of dimension d. Let U ⊂ X be a dense open subset. Write X − U = D ∪ Z where D ⊂ X is a divisor and
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codim(Z ⊂ X) ≥ 2. We have H 1 (X − D) ∼ = H 1 (U ). Since we are interested 1 in H (U ), we may assume U = X − D is the complement of a divisor. Let i : C ,→ X be a general linear space section of dimension 1, and let δ = C ∩ D. We may choose ρ : X → J(C) such that the composition i∗
ρ∗
Pic0 (X) − → J(C) −→ Pic0 (X)
(10.12)
is multiplication by an integer N 6= 0. Indeed, let H be a very ample line bundle so that C is the (d − 1)-fold product of general sections of H. Intersection with H yields an isogeny Pic0 (X) → Alb(X), which defines an inverse isogeny Alb(X) → Pic0 (X) of degree N . We pull back the Poincar´e bundle from J(C) × J(C) to C × X via the composite map C × X → J(C) × Alb(X) → J(C) × Pic0 (X) → J(C) × J(C), where the first map is the cycle map, the second one is 1×isogeny, the third one is 1× restriction. We define OC×X (Ξ) to be the inverse image of the Poincar´e bundle. The morphism ρ : X → J(C) is the correspondence x 7→ OC×X (Ξ)|C×{x} and does not depend on the choice of the section Ξ. Consider the diagram 0 − →
Q` [δ]/Q` yb
→ −
Hc1 (C − δ) i y∗
→ −
H 1 (C) i y∗
→0 −
2d−2
H (D) → Hc2d−1 (U )(d − 1) − → H 2d−1 (X)(d − 1) 0 − →H 2d−2 (X) (d − 1) −
(10.13) Here, the rows are long exact sequences associated to restriction to closed subsets, and the vertical arrows are Gysin maps. The map b can be described as follows. The Q` -vector space H 2d−2 (D)(d − 1) has basis the irreducible components of D, and b(x) is the basis element [Dx ] associated to the unique component Dx of D containing x. We have dual exact sequences (defining Div0D (X)) 0 → Div0D (X) → H2d−2 (D)(1 − d) → H 2 (X)(1) H 2d−2 (X)(d − 1) → H 2d−2 (D)(d − 1) →
H 2d−2 (D) H 2d−2 (X)
(10.14)
(d − 1) → 0.
If we view Q` [δ] and H 2d−2 (D)(d − 1) as endowed with symmetric pairings with orthonormal bases the points x ∈ δ and the cohomology classes of irreducible components Di ⊂ D, then b is adjoint to the map Di 7→ Di · δ. We conclude Lemma 10.2.5. Define Div0D (X) to be the Q` -vector space spanned by divisors on X supported on D and homologous to 0 on X. Define ∆ ⊂ Div0δ (C)
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to be the image of Div0D (X) under pullback i∗ . Then there is a commutative diagram 0 − →
Q` [δ]/∆⊥ yb
→ −
Hc1 (C, ∆) i y∗
→ −
H 1 (C) i y∗
→0 −
2d−2
H (D) → Hc2d−1 (U )(d − 1) − → H 2d−1 (X)(d − 1). 0 − →H 2d−2 (X) (d − 1) −
(10.15) i∗
Proof The map b is dual to the restriction map Div0D (X) − → Div0δ (C). By definition ∆⊥ is orthogonal to the image of i∗ , i.e. ∆⊥ = ker b. Lemma 10.2.6. Let ∆ = i∗ (Div0D (X)) ⊂ Div0δ (C) be as in Lemma 10.2.5. Then ρ defined in (10.12) lifts to some ρU,∆ : U → J(C, ∆). Proof The correspondence defined by OC×X (Ξ) in (10.12) carries OC (z) for z ∈ ∆ = i∗ (Div0D (X)) to line bundles in Pic0 (X), the classes of which fall in the image of ρ∗ i∗ (Div0D (X)) ≡ N · Div0D (X) in Pic0 (X). To be more precise, let Dp be a basis for Div0D (X), and set zp = i∗ Dp . This is a basis of ∆. Then OC×X (Ξ)|zp ×X = OX (Dp ). Thus choose ρU,∆ in Lemma 10.2.2 using this trivialization on U . Using the Lemmas 10.2.1, 10.2.5, 10.2.6 together with (10.9), we pull back i∗ ⊗ρ∗U,∆
c1 (L∆ ) ∈ H 1 (C, ∆) ⊗ H 1 (J(C, ∆))(1) −−−−−→ Hc2d−1 (U )(d) ⊗ H 1 (U ) ∼ = H 1 (U )∨ ⊗ H 1 (U ) (10.16) and define a correspondence Φ : H 1 (U ) → H 1 (U ). Lemma 10.2.7. The map Φ is the multiplication by N . Proof We consider Φ. It acts on H 1 (U ), compatibly with the exact sequence 0 → H 1 (X) → H 1 (U ) → Div0D (X)(−1) → 0
(10.17)
By definition of ρU,∆ , it is equal to N · Id on H 1 (X) and on Div0D (X)(−1). Thus Φ − N · Id is a correspondence from Div0D (X)(−1) to H 1 (X). We use purity in the sense of Deligne. There is no nontrivial correspondence Div0D (X)(−1) → H 1 (X). If k = C and we consider Betti cohomology, Div0D (X)(−1) is pure of weight 2 while H 1 (X) is pure of weight 1. If k is the algebraic closure of a finite field, we have the same conclusion. Otherwise, all the objects used are defined over a finitely generated field k over a finite ˇ field k0 . By Cebotarev’s theorem, the Galois group of k/k0 is generated
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by the Frobenius maps, so we may make sense of the notion of weight for H 1 (U ). We conclude as in the complex case. We now express in terms of cycles the trivialization of OC×X (Ξ)p ×X = OX (Dp ) used in the proof of Lemma 10.2.6. Theorem 10.2.8. With notation as above, there exists a cycle Γ on X × U of dimension d = dim X together with rational functions fµ on X for each divisor µ homologically equivalent to 0 on X and supported on D = XU such that pr2∗ (Γ · (µ × U )) = (fµ ). The data (Γ, {fµ }) define a class in Hc2d−1 (U ) ⊗ H 1 (U ) which gives the identity map on H 1 (U ). We close this section with a comment about K¨ unneth projectors π i : H ∗ (U ) → H i (U ) for i > 1. We consider the somewhat weaker question of the existence of an algebraic projector when we localize at the generic point of the target, i.e. we consider H ∗ (U ) → H i (U ) → limV ⊂U H i (V ). We −→ assume U = X − D with X smooth, projective, and D a Cartier divisor. Proposition 10.2.9. Let n < dim X be an integer. Let Y ⊂ X be a multiple hyperplane section of dimension n which is general with respect to D. Write δ = Y ∩ D. Then the restriction map n+1 (X) → Hδn+1 (Y ) HD
(10.18)
is injective. Proof Let d = dim(X). By duality, we have to show surjectivity of the Gysin map H n−1 (δ) → H 2d−(n+1) (D)(d − n). More generally, one has Theorem 10.2.10 (P. Deligne). Let F be a `-adic sheaf on PN . Then there exists a non-empty open set U ⊂ (PN )∨ such that for ι : A ,→ PN a hyperplane section corresponding to a point of U , the Gysin homomorphism i H i−2 (A, ι∗ F)(−1) → HA (PN , F),
is an isomorphism for all i. In particular, if V ⊂ PN is a projective variety, then the Gysin homomorphism H i (A ∩ V ) → H i+2 (V )(1) is an isomorphism for i > dim(A ∩ V ) and surjective for i = dim(A ∩ V ) for a non-empty open set of A. The proof of the general theorem is written in [6, Theorem 2.1], Applied to F = a∗ Q` , where a : V → PN is the projective embedding, it shows that i+2 the Gysin isomorphism H i (A ∩ V ) → HA∩V (V )(1) is an isomorphism. Then the application follows from Artin’s vanishing theorem H i (V − (A ∩ V )) = 0 for i > dim(V ).
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Let L be the Lefschetz operator on H ∗ (X). One of the standard conjectures (B(X, L) in [11]) is the existence of an algebraic correspondence Λ which is a “weak inverse” to L. Assume now that this standard conjecture B is true for X and for all smooth linear space sections Y ⊂ X. The ∼ = strong Lefschetz theorem implies that Ld−n : H n (X) − → H 2d−n (X)(d − n). d−n d−n −1 2d−n Assuming B(X, L), Λ = (L ) :H (X)(d − n) ∼ = H n (X). Write P = Λd−n |Y ×X . It is easy to check that the composition i∗
P
H n (X) − → H n (Y ) − → H n (X)
(10.19)
is the identity, so H n (Y ) = Image(i∗ ) ⊕ ker(P ). Consider the diagram H n (X) − → ∗ yi d
H n (U ) y
a
n+1 − → HD (X) yb
(10.20)
c
H n (Y ) − → H n (Y − δ) − → Hδn+1 (Y ) Define H n (Y − δ)0 = {x ∈ H n (Y − δ) | c(x) ∈ Im(b ◦ a)}.
(10.21)
As a consequence of proposition 10.2.9 and (10.20) we see that H n (U ) H n (Y − δ)0 /d(ker(P )), and the kernel of this map is the image in H n (U ) of elements x ∈ H n (X) such that i∗ x ∈ ker(P ) ⊕ Image(Hδn (Y ) → H n (Y )). For such an x, it will necessarily be the case that x = P (i∗ x) is supported on a proper closed subset of X. In particular, for some V ⊂ U open dense, P will induce a map PU : H n (Y − δ)0 → H n (V ).
(10.22)
The map i∗ : H n (U ) → H n (Y − δ)0 dualises to i∗ : (H n (Y − δ)0 )∨ → Hc2d−n (U ), so we may define (i∗ ⊗ PU ) : H n (Y − δ)0 )∨ ⊗ H n (Y − δ)0 → Hc2d−n (U ) ⊗ H n (V ). (10.23) Let T ⊂ H n (Y − δ)0 be the subgroup of cohomology classes supported in codimension 1. Assuming inductively that we are able to define an algebraic correspondence on Y which carries a class γ ∈ (H n (Y − δ)0 )∨ ⊗ (H n (Y − δ)0 /T )
(10.24)
corresponding to the evident map H n (Y − δ)0 H n (Y − δ)0 /T , it would follow since PU (T ) ⊂ ker(H n (V ) → limV ⊂U H n (V )) that we could view −→ (i∗ ⊗ PU )(γ) ∈ Hc2d−n (U ) ⊗ lim H n (V ). −→ V ⊂U
(10.25)
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This correspondence would have the desired properties. One interest in pursuing this line of investigation concerns the Milnor conjecture that the Galois cohomology with Z/`Z-coefficients prime to the residue characteristic is generated as an algebra by H 1 . There is a geometric proof of this result in top degree [2], so, for example, elements in H n (Y − δ) lie in the subalgebra generated by H 1 after localization. If PU exists as an algebraic correspondence, then using the existence of a norm in Milnor K-theory, one could show that the Milnor conjecture was true for H ∗ (k(X), Z/`Z) for almost all `. (The condition on ` arises because the standard conjectures only make sense after tensoring with Q.) Here, the idea that cohomology classes in degree n might come by correspondence from an algebraic variety of dimension ≤ n was suggested to us by Alexander Beilinson
10.3 Open correspondences The aim of this section is to give a simple motivic proof of the independence of ` or of a complex embedding of a ground field k of the trace for open correspondences. If we assume that k is finite, then, as conjectured by Deligne, high Frobenius power twists move the correspondence to a general position correspondence and the local factors have been computed in [5], [14], [15], [7]. Surely in this case the simple observations which follow are weaker. We consider open correspondences. This means the following. Let X be a smooth projective variety of dim d over an algebraically closed field k, and let U ⊂ X be a nontrivial open subvariety, with complement D = X − U . One considers codim d cycles Γ ⊂ U ×U which have the property that they induce a correspondence Γ∗ : H i (U ) → H i (U ) or equivalently Hci (U ) → Hci (U ) for all i. Here cohomology is ´etale Q` cohomology or Betti cohomology if k = C and we denote by pi : X × X → X the two projections. We write X Γ= nj Γj (10.26) where Γj is irreducible, nj ∈ Z and define X ¯ := ¯ j , Γj ⊂ U × U Γ nj Γ
(10.27)
where ¯ is the Zariski closure in X × X. We will use the following facts ([4], Th´eor`eme 2.9). Proposition 10.3.1. Let q : Γ → U be a proper map of quasi-projective
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varieties. Then one has a pull-back map q ∗ : Hci (U ) → Hci (Γ).
(10.28)
Let ι
Γ → − Γ0 py p0 y
(10.29)
=
U − → U be a commutative diagram of quasi-projective varieties of dimension d, with ι an open embedding, p0 proper and U smooth. Then there are push-down maps p∗ , p0∗ making the following diagram commutative ι∗
Hci (Γ) −→ Hci (Γ0 ) p∗ y p0∗ y
(10.30)
=
Hci (U ) − → Hci (U ) Definition 10.3.2. If p2 |(Γ)j : Γj → U is proper for all j, or equivalently Γj ⊂ X × U is Zariski closed or equivalently if ¯ j ∩ (D × X) ⊂ X × D ∀ j, Γ
(10.31)
then one defines p∗
(p1 )∗
2 (Γj )∗ : Hci (U ) −→ Hci (Γj ) −−−→ Hci (U ),
(10.32)
and call it the open correspondence defined by Γj . The correspondence P defined by Γ is then by definition Γ∗ = nj (Γj )∗ . (The ordering (p1 , p2 ) here is chosen as in [9].) Remark 10.3.3. If we compare this condition to the one yielding to K¨ unneth correspondences in section 2, then it is much stronger. Indeed, Theorem 10.2.8 yields a cycle Γ which meets physically D × U , but cohomologically it washes out, while in this section we handle the case where there is physically no intersection. ¯ j − (X × D) and we set Γ0 = Γ ¯ j − (D × X). Remark 10.3.4. We have Γj = Γ j 0 Thus p1 |Γ0j : Γj → U is projective. One has the following commutative
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diagram p∗
2 Hci (U ) −→ =y
Hci (Γj ) y
ι∗
−→
i+2d Hci (U ) − → Hc,Γ (X × U, d) j =y =y p∗
Hci (Γ0j ) y
(p1 )∗
−−−→ Hci (U ) =y
i+2d Hc,Γ 0 (U × X, d) j y ι∗
→ −
Hci (U ) (10.33) =y
(p1 )∗
2 i+2d i+2d Hci (U ) −→ Hc,Γ (U × U, d) −→ Hc,Γ −−→ Hci (U ) 0 (U × X, d) − j j
with factorization of the upper horizontal composition of maps, which is Γj,∗ , through the lower horizontal composition of maps. So from far left to far right on the bottom line, this is the correspondence Γj,∗ . We assume now that we have the assumption as in Definition 10.3.2 and we wish to give conditions under which one can compute the trace of Γ∗ which is defined by
Tr(Γ∗ ) :=
2d X
(−1)i Tr(Γ∗ |Hci (U ) ).
(10.34)
i=0
As it stands, the trace of Γ∗ depends a priori on `, or, for varieties defined over a field k of characteristic 0, and Betti cohomology taken with respect to a complex embedding ι : k → C, it depends on ι. One has
Theorem 10.3.5. Let X be a smooth projective variety of dimension d defined over a field k, together with a strict normal crossings divisor D ⊂ X of open complement U = X −D. Let Γ ⊂ U ×U be a dimension d cycle defining an open correspondence Γ∗ on `-adic cohomology or Betti cohomology as in Definition 10.3.2. Then Tr(Γ∗ ) does not depend on ` in `-adic cohomology or on the complex embedding of k in Betti cohomology. 2d (X × U, D × U, Z(d)), as Proof We use the relative motivic cohomology HM m (X ×U, D×U, Z(n)) defined in [12], chapter 4, 2.2 and p. 209. The group HM is the homology H2n−m (Zn (X × U, D × U, ∗)), where Zn (X × U, D × U, ∗) is
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the single complex associated to the double higher Chow cycle complex ··· ∂y
··· ∂y
··· y
rest
rest
rest
rest
Zn (X × U, 1) −−→ Zn (D(1) × U, 1) −−→ Zn (D(2) × U, 1) ∂y ∂y y
(10.35)
Zn (X × U, 0) −−→ Zn (D(1) × U, 0) −−→ Zn (D(2) × U, 0). Here D(a) is the normalization of all the strata of codimension a, Zn (D(a) × U, b) is a group of cycles on D(a) × U × S b where S • is the cosimplicial P scheme S n = Spec(k[t0 , . . . , tn ]/( ni=0 ti − 1)) with face maps S n ,→ S n+1 defined by ti = 0. More precisely, Zn (D(a) × U, b) is generated by the codimension n subvarieties Z ⊂ D(a) × U × S b such that, for each face F of S b , and each irreducible component F 0 ⊂ D(a) of the strata of D we have codimF 0 ×U ×F (Z ∩ (F 0 × U × F )) ≥ n. The horizontal restriction maps are the intersection with the smaller strata, the vertical ∂’s are the boundary maps. This relative motivic cohomology acts as correspondences on Hc∗ (U ), where Hc∗ (U ) is `-adic or Betti cohomology ([3], section 4). Let us write Γ = P nj Γj . By Definition 10.3.2, one has Γj ⊂ X ×U closed with Γj ∩(D×U ) = ∅, thus in particular, Γ ∈ Zd (X × U, 0) with rest(Γ) = 0 in Zd (D(1) × U, 0), thus it defines a class 2d [Γ] ∈ HM (X × U, D × U, d).
(10.36)
Similarly, we consider the restriction ∆U ⊂ U ×X of the diagonal ∆ ⊂ X×X. This defines a class in Zd (U × X, 0). As rest(ΓU ) = 0 in Zd (U × D(1) , 0), it defines a class 2d [∆U ] ∈ HM (U × X, U × D, d).
(10.37)
We want to pair [Γ] with [∆U ]. We argue using M. Levine’s work. Let Y be a N -dimensional smooth projective variety defined over k, with two strict normal crossings divisors A, B so that A + B is a strict normal crossings divisor. By [12], Chapter IV, lemma 2.3.5 and lemma 2.3.6, the motive M (Y − A, B − B ∩ A) is dual to the motive M (Y − B, A − B ∩ A). It yields a cup product 2N −a a HM (Y − A, B − B ∩ A, b) × HM (Y − B, A − A ∩ B, N − b) → Z. (10.38)
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This cup product is compatible with the cup product in `-adic or Betti cohomology. We apply this to Y = X × X, A = D × X, B = X × D
(10.39)
2d (Y − A, B − B ∩ A, d) and [Γ] ∈ H 2d (Y − B, A − so we can cup [∆U ] ∈ HM M A ∩ B, d)
[∆U ] ∪ [Γ] ∈ Z.
(10.40)
The theorem is then the consequence of the following proposition. Proposition 10.3.6. Tr(Γ∗ ) = [∆U ] ∪ [Γ] ∈ Z. Proof By the compatibility of the cup product (10.38) with cohomology, we just have to prove the proposition with [∆U ] and [Γ] replaced by their classes cl(∆U ) ∈ H 2d (U × X, U × D, d) and cl(Γ) ∈ H 2d (X × U, D × U, d) in cohomology. We may assume that Γ ⊂ X × U is irreducible. On the other hand, by (10.33), the map p1∗
Hci (Γ) −−→ Hci (U )
(10.41)
factors through Gysin
i+2d i+2d (U × U, d) (X × U, d) = HΓ,c Hci (Γ) −−−→ HΓ,c
(10.42)
p1∗
→ Hci+2d (U × U, d) → Hci+2d (U × X, d) −−→ Hci (U ) so the correspondence p∗
p1∗
2 Hci (Γ) −−→ Hci (U ) Γ∗ : Hci (U ) −→
(10.43)
is just defined on α ∈ Hci (U ) as follows. As the cup product from Hci (U ) with H j (U ) is well defined with values in Hci+j (U ), so a fortiori in H i+j (X), i 2d−i (U )(d), one has and cl(Γ) ∈ H 2d (X × U, D × U, d) = ⊕2d i=0 H (X, D) ⊗ H a well defined cup-product p∗2 (α) ∪ cl(Γ) ∈ H i+2d (X × X, D × X, d) = Hci+2d (U × X, d).
(10.44)
Then one applies p1∗ to the value of this cup-product. Now we argue as in the classical case. Let eia be a basis of Hci (U ), and (eia )∨ be its dual basis in P P P i i H 2d−i (U )(d). Write cl(Γ) = i a fai ⊗ (eia )∨ , fai = fab eb ∈ Hci (U ). So P P P P i , and Tr(Γ ) = i . On the other hand, one Γ∗ (eia ) = b fab (−1)i a b fab P P i ∨∗ P P P i i has cl(∆U ) = i a (ea ) ⊗ (ea ). Thus cl(∆U ∪ Γ) = (−1)i a b fab .
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The proposition finishes the proof of the theorem. The rest of the section is devoted to giving a concrete expression for (10.34) under stronger geometric assumptions on Γ. We define [ X o = X − (Di ∩ Dj ), (10.45) 0 o
i<j 0
¯ ∩ (X × X ), (Γ ) = Γ ∩ (X o × X o ), Do = X o ∩ D Γ =Γ o
o
o
and similarly for the components j . One has the following compatibility. Lemma 10.3.7. One has a commutative diagram Hci (U ) p∗2 y
→ H i (X o ) − p∗2 y
Hci (Γj ) ι∗ y
→ H i (Γoj ) − ι∗ y
Hci ((Γj )0 ) − → H i ((Γ0 )oj ) p1∗ y p1∗ y Hci (U )
→ H i (X o ) −
where the composition of the left vertical arrows is the correspondence (Γj )∗ . We deem (Γoj )∗ the composition of the right vertical arrows. Proof Given (10.30) the lemma follows directly from Definition (10.27). We remark that the Grothendieck-Lefschetz trace formula allows to compute ¯ ∗ on X the trace of the correspondence Γ ¯ ∗ ) = deg(Γ ¯ · ∆X ). Tr(Γ
(10.46)
Thus, in the corollary, we would like to complete the commutative diagram in an exact sequence of commutative diagrams, so that we can apply the trace formula on all the terms but the one we seek. In the sequel, we give a strong geometric condition under which it is possible. Definition 10.3.8. We assume that D is a strict normal crossings divisor. The dim d cycle Γ ⊂ U × U is said to be in good position with respect to D × X if the following two conditions are fulfilled. ¯ j cuts each stratum DI ×X in codim ≥ d, where DI = Di ∩. . .∩Dir i) Each Γ 1 for I = {i1 , . . . , ir } with |I| = r.
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ii) ¯ j ∩ (DI × X) ⊂ DI × DI Γ set theoretically. In this case, for all I we define the cycles ¯ j · (DI × X) ⊂ DI × DI ZjI = Γ
(10.47)
a (b). We Let us be more precise. We denote motivic cohomology by HM drop the subscript j , thus Γ = Γj . This defines X ZI = mI,a ZI,a (10.48)
as in (10.39), where the ZI,a are the reduced irreducible components of ZI . One has the Gysin isomorphisms ∼ =
2(d−r)
⊕a Q` [ZI,a ] − → HZI
∼ =
(DI × DI , d − r) − → HZ2dI (DI × X, d)
(10.49)
This yields the commutative diagram ⊕a Q · [ZI,a ] ∼ = y
⊗Q Q`
−−−→
⊕a Q` · [ZI,a ] ∼ y=
⊗Q Q`
2(d−r)
2(d−r)
HM,ZI (DI × DI , d − r) −−−→ HZI ∼ =y 2d (D × X, d) HM,Z I I
⊗Q Q`
(DI × DI , d − r) ∼ y=
(10.50)
HZ2dI (DI × X, d)
−−−→
So we conclude that ZI is a well defined cycle 2(d−r)
2(d−r)
ZI ∈ HM,ZI (DI × DI , d − r) → HM
(DI × DI , d − r),
which defines a correspondence X [ [ (ZIo )∗ = mI,a (ZI,a )∗ : Hci (DI − DI,i ) → Hci (DI − DI,i ) a
i∈I /
ZIo
= ZI ∩ (DI −
[ i∈I /
DI,i × DI −
(10.51)
(10.52)
i∈I /
[
DI,i )
i∈I /
by the Definition 10.3.2. We use the notations from (10.45), setting Γ = Γj , together with [ Z= Γ ∩ (Dio × Dio ). (10.53) i
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Lemma 10.3.9. One has a commutative diagram Hci−1 (Do ) − → Hci (U ) =y p∗2 y Hci−1 (Do ) − → Hci (Γ) p∗2 y ι∗ y Hci−1 (Z) − → Hci (Γ0 ) (p1 )∗ y (p1 )∗ y Hci−1 (Do ) − → Hci (U ) Proof Indeed, by assumption, one has Γ0 ⊂ X o ×U , thus, with the notations of (10.45), one has Γ0 = Γo − Z. In order to have a unified notation, we set for |I| = 0 ¯ ∆I = ∆X . ZI = Γ,
(10.54)
One has Theorem 10.3.10. Let X be a smooth projective variety over an algebraically closed field k, D ⊂ X be a strict normal crossing divisor, with complement U = X − D, and Γ ⊂ U × U be a dim d correspondence, with p2 |Γ : Γ → U proper, and in good position with respect to D × X in the sense of Definition 10.3.2. Then one has Tr(Γ∗ ) =
d X r=0
(−1)r
X
deg(ZI · ∆I ).
|I|=r
Proof We consider the long exact sequence . . . → Hci−1 (Do ) → Hci (U ) → Hci (X o ) → . . .
(10.55)
and apply Lemma 10.3.7 and Lemma 10.3.9. We find that the trace on U is the sum of the traces on X o and on Do . If D was smooth, this would finish the proof applying Grothendieck’s formula (10.46). If there are higher codimensional strata, we argue as follows. We know the trace on Dio by induction on the dimension. We have to understand the trace on X o . We define X oo = X − ∪i<j
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(10.55) with Do replaced by X oo − X o , U replaced by X o and X o replaced by X oo . Again this shifts the trace computation to X oo . We continue like this till the highest codimensional strata and finish with the trace on X, for which we of course apply Grothendieck formula (10.46). We now give a scheme-theoretic condition under which the expression given in Theorem 10.3.10 depends only on local contributions in U . This condition is inspired by [9], Lemma 2.3.1. Definition 10.3.11. We assume that D is a strict normal crossings divisor. The dim d cycle Γ ⊂ U × U is said to be in scheme theoretic good position with respect to D × X if it is in good position in the sense of Definition 10.3.8 and ii) is replaced by ¯ j ∩ (DI × X) ⊂ DI × DI Γ scheme theoretically, that is all the intersections multiplicities are equal to 1. Proposition 10.3.12. Let X be a smooth projective variety over an algebraically closed field k, D ⊂ X be a strict normal crossing divisor, with complement U = X − D, and Γ ⊂ U × U be a dim d correspondence, with p2 |Γ : Γ → U proper, and in scheme theoretic good position with respect to ¯ j and D × X in the sense of Definition 10.3.11. We assume moreover that Γ ∆|U cut transversally. Then Tr(Γ∗ ) = deg(∆U · Γ). Proof Due to the good position assumption, all intersection multiplicities are 1 and the contributions lying on (D × X) ∪ (X × D) cancel in Theorem 10.3.10. Example 10.3.13. One case where the conditions of Proposition 10.3.12 hold is in characteristic p when Γ is the graph of Frobenius. In this case, of course, the result is known by other methods. Example 10.3.14. This example is inspired by [9, Remark 2.3.6]. We take X = P1 , D = {∞}, U = A1 , Γ = Γpq = {xp − y q = 0} ⊂ A1 × A1 . Then Γ defines an open correspondence with Tr(Γ∗ ) = p. On the other hand, one has ( max(p, q) if p 6= q p q deg(Γpq · ∆U ) = dimk[t]/(t − t ) = ∞ if p = q
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and one has ¯ pq · (∞ × P1 )) = q, deg(Γ ¯ pq · (P1 × ∞)) = p. deg(Γ
(10.56)
Thus Γpq is in scheme theoretic good position with respect to ∞ × P1 if and only if p > q, and is always in good position with respect to P1 × ¯ pq ) = deg(O(1, 1) · O(p, q)) = p + q, we see exactly ∞. Since deg(∆P1 · Γ how the formula of Theorem 10.3.10 works both in Theorem 10.3.10 and in Proposition 10.3.12. References [1] Beilinson, A.: Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036–2070. [2] Bloch, S.: Lecture on Algebraic cycles, Duke University Mathematics Series, IV, (1980). [3] Bloch, S., H. Esnault and Levine, M.: Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces, Am. J. of Mathematics, 127 no1 (2005), 193-207. ´ globale, SGA 4, XVIII, Lecture [4] Deligne, P.: La formule de dualite Notes in Mathematics 305, Springer Verlag. [5] Deligne, P.: Rapport sur la formule des traces, SGA 4,5, Lecture Notes in Mathematics 569, Springer Verlag. [6] Esnault, H.: Eigenvalues of Frobenius acting on the `-adic cohomology of complete intersections of low degree, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 317–320. [7] Fujiwara, K.: Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture. [8] Jannsen, U.: Motivic Sheaves and Filtrations on Chow Groups, in Motives, Proc. Symp. Pure Math., Vol. 55 (1994), 245–302. [9] Kato, K., Saito, T.: Ramification theory for varieties over a perfect field, preprint 59 pages, 2004. [10] Kleiman, S: Algebraic Cycles and the Weil Conjectures, in Dix Expos´es sur la Cohomologie des Sch´emas, North-Holland, Amsterdam, 1968, 359–386. [11] Kleiman, S: The Standard Conjectures, in Motives, Proc. Symp. Pure Math., Vol. 55 (1994), 13–20. [12] Levine, M.: Mixed Motives, Mathematical Surveys and Monographs 57 (1998), American Mathematical Society. [13] Murre, J. P.: On a conjectural filtration on the Chow groups of an algebraic variety, (I and II), Indag. Math. New Series Vol. 4 (1993), 177–201. [14] Pink, R.: On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, Ann. of Math. 135 (1992), 483–525. [15] Zink, T.: The Lefschetz trace formula for open surfaces, In: Automorphic Forms, Shimura Varieties and L-functions, vol. II, Prospectives in Math. (1989), 337–376.
11 The Brill-Noether Curve of a stable Vector Bundle on a Genus Two Curve Sonia Brivio Dipartimento di Matematica, Universit´ a di Pavia, via Ferrata 1, 27100 Pavia,
[email protected]
Alessandro Verra † Dipartimento di Matematica, Terza Universit´ a di Roma, Largo S.Murialdo, 00146 Roma,
[email protected]
†
11.1 Introduction. In this note we deal with the moduli space Ur of semistable vector bundle of rank r and degree r(g − 1) over a smooth, irreducible complex projective curve of genus g ≥ 2. Ur contains the Brill-Noether locus Θr := { [E] ∈ Ur | h0 (E) ≥ 1 } which is an integral Cartier divisor and it is known as the generalized theta divisor of Ur , see [6], [5]. Moreover the tensor product defines a morphism f : Ur × Pic0 (C) → Ur and we can consider the pull-back f ∗ Θr of Θr . Let [E] ∈ Ur be the moduli point of the vector bundle E and let det E ∼ = M ⊗r . It is well known that then O (f ∗ Θr ) ∼ 0 = O 0 (rΘM ), [E]×Pic (C)
Pic (C)
where ΘM := {N ∈ Pic0 (C) | h0 (M ⊗ N ) ≥ 1}. Note that M is a line bundle of degree g − 1 and that ΘM is a theta divisor on Pic0 (C). We define ΘE := f ∗ Θr · [E] × Pic0 (C) † Partially supported by the research programs ’Moduli Spaces and Lie Theory’ and ’Geometry on Algebraic Varieties’ of the Italian Ministry of Education. ‡ 2000 Mathematics Subject Classification: 14H60
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if the intersection is proper. In this case we will say that ΘE is the theta divisor of E. The construction of ΘE allows us to define a rational map as follows. Consider [ Tr := |rΘM |. M ∈Picg−1 (C)
It is a standard fact that Tr has a natural structure of projective bundle over Picr(g−1) (C). So we omit its construction; we only mention that the corresponding projection p : Tr → Picr(g−1) (C) is defined as follows: p(D) = M ⊗r if and only if D ∈ |rΘM |. Note that the only elements of multiplicity r in |rΘM | are exactly the divisors rΘM ⊗η , where η varies in the set of the elements of order r in Pic0 (C). Therefore the map p is well defined. In the following we will study the rational map θr : Ur → Tr which associates to a general [E] ∈ Ur the corresponding theta divisor ΘE ∈ Tr . Let det : Ur → Picr(g−1) (C) be the determinant map. It is well known that Tr is the projectivization of det∗ OUr (Θr )∗ and that θr is the induced tautological map. In particular it follows that p · θr = det. We will say that θr is the theta map. Too many questions are still unsettled about the theta map, except if r ≤ 2: see e.g. [4] for a general survey. This situation is probably related to the fact that the following basic question is still largely unsolved. Question. Is θr generically finite onto its image? Actually the main difficulty here is that in most of the cases θr is not a morphism [9]. Thus, in spite of the ampleness of Θr , it is not a priori granted that θr is generically finite onto its image. In this paper we give a natural geometric interpretation of the fibres of the map θr for a curve C of genus two. A very special feature in this case is that dim Ur = dim Tr = r2 + 1, so the generic finiteness of θr is even more expected. Applying our description of the fibres we prove the generic finiteness of θr . Such a result is not new: Beauville recently proved it using a different,
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relatively simple method, see [3]. We believe that our description has some interest in itself and we hope to use it for further applications, in particular to compute the degree of θr . Our approach relies on Brill-Noether theory for curves contained in a genus two Jacobian. Let D ∈ θr (Ur ) be a sufficiently general element, then D is a smooth curve of genus r2 + 1 in Pic0 (C) (§ 11.2). Consider the Brill-Noether locus 2
Wrr−1 (D) = {L ∈ Picr (D) | h0 (L) ≥ r} 2 and observe that its expected dimension is one, i.e. the Brill-Noether number ρ(r − 1, r2 , r2 + 1) is one. Our main result can be summarized as follows: Theorem. Each point [E] ∈ θr−1 (D) defines an irreducible component CE ⊂ Wrr−1 (D) 2 biregular to C. Let Z be the set of all irreducible components of Wrr−1 (D) 2 and let iD : θr−1 (D) → Z be the map sending [E] to CE . Then iD is injective. The statement clearly implies that θr is generically finite. We define CE 2 as the Brill-Noether curve of E. Fixing a Poincar´e bundle P on D×Picr (D) in an appropriate way, it turns out that E is the restriction of ν∗ P to CE , 2 where ν is the projection onto Picr (D). In particular the family of the fibres of E is just the family of the spaces H 0 (L), L ∈ CE . Note that the choice of P, hence of det E, depends on the embedding D ⊂ Pic0 (C). This is essentially explained in the final part of this note. To have a typical example of what happens, the reader can consider the case r = 2. In this case D is a curve of genus 5 endowed with a fixed point free involution which is induced by the −1 multiplication of Pic0 (C). Since r = 2, the Brill-Noether locus W41 (D) is exactly the singular locus of the theta divisor of Pic4 (D). It follows from the theory of Prym varieties that W41 (D) is the union of two irreducible curves: one of them has genus 4, the other one is just a copy of C (see also [10]). This is the Brill-Noether curve of a stable rank two vector bundle E such that θ2 ([E]) = D. In higher rank the general theory of Prym-Tjurin varieties can certainly provide further information on Wrr−1 (D) and hence on the fibres of θr . However, in order to 2 reconstruct them, a very explicit description is needed for the Prym-Tjurin realizations of a genus two Jacobian.
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On Jacobians of higher genus several extensions of the above constructions are possible and perhaps deserve to be considered in the study of the theta maps. We hope to have underlined with this note the manifold links between moduli of vector bundles on a curve C, Prym-Tjurin realizations of its Jacobian JC and Brill-Noether theory for curves in JC. We wish to thank the referee for some helpful comments. The second author wishes to express his gratitude to Jacob Murre, on the occasion of the Proceedings of the Conference in his honour, for the friendship and for the scientific attention received during three decades.
11.2 Notation and preliminary results. From now on C is a smooth, irreducible, complex projective curve of genus 2. Let C (2) be the 2-symmetric product of C. A point of such a surface is a divisor x + y with x, y ∈ C. We consider the map a : C (2) → Pic0 (C) sending x + y ∈ C (2) to ωC (−x − y). Of course a is the composition of the Abel map defined by ωC with −1 multiplication on Pic0 (C). Therefore a = −σ, where σ : C (2) → Pic0 (C) is the blowing up of the zero point. For each fibre |rΘM | of the projective bundle Tr we have the linear isomorphism a∗M : |rΘM | → |a∗ rΘM | defined by the pull-back. With ΘE the theta divisor of [E], we will keep the following notation DE := a∗ ΘE . DE is an effective divisor in C (2) which is supported on the set {x + y ∈ C (2) | h0 (E ⊗ ωC (−x − y)) ≥ 1}. DE is biregular to ΘE if the zero point is not in ΘE , otherwise DE is the union of the projective line |ωC | and of a curve birational to ΘE . We find it more convenient to deal with DE than with ΘE . First we want to prove that a general DE is smooth. To do so, we need Laszlo’s singularity theorem. Theorem 11.2.1 ([8]). The multiplicity of Θr at its stable point [E] is h0 (E). Next, we show:
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Proposition 11.2.2. Let [E] be a general stable point of Ur then h0 (E ⊗ ωC (−x − y)) ≤ 1, ∀x + y ∈ C (2) . Proof By induction on r. Let r = 1 then DE = C and E is a general line bundle of degree 1, in particular |E ⊗ ωC | is a base-point-free pencil and this implies the statement. Let r ≥ 2. We can assume by induction that there exist general [B] ∈ Ur−1 and [A] ∈ U1 = Pic1 (C) satisfying the statement. We consider the exact sequence 0→B→E→A→0 defined by the vector e ∈ Ext1 (A, B). Tensoring such a sequence by ωC (−x− y) and passing to the long exact sequence we obtain the coboundary map ex+y : H 0 (ωC ⊗ A(−x − y)) → H 1 (ωC ⊗ B(−x − y)). Claim The statement holds for E if and only if ex+y has maximal rank for every x + y. Proof We have h0 (ωC ⊗ A(−x − y)) ≤ 1 and h0 (ωC ⊗ B(−x − y)) = h1 (ωC ⊗ B(−x − y)) ≤ 1. Then the statement follows from the above mentioned long exact sequence. Finally it is obvious that ex+y has maximal rank except possibly for points x + y with h0 (ωC ⊗ A(−x − y)) = h0 (ωC ⊗ B(−x − y)) = 1. The set of these points is DA ∩ DB . Since A is general we can assume that DA ∩ DB is finite. Let x + y ∈ DA ∩ DB then ex+y has not maximal rank if and only if it is the zero map. It is a standard property that, in the present case, the locus Hx+y = {e ∈ Ext1 (A, B) | ex+y is the zero map} S is a hyperplane. Let H := Hx+y , x + y ∈ DA ∩ DB , then a general 1 e ∈ Ext (A, B) − H defines a semistable E satisfying the condition of the statement. Since this condition is open on Ur the result follows. Corollary 11.2.3. Let [E] be a general stable point of Ur , then DE is smooth. Proof Let f : Ur × Pic0 (C) → Ur be the map defined via tensor product. Recall that ΘE = f ∗ Θr · [E] × Pic0 (C) ⊂ Ur × Pic0 (C). Therefore ΘE is the fibre of the projection q : f ∗ Θr → Ur . Then, by generic smoothness, a general ΘE is smooth if ΘE ∩ Sing f ∗ Θr = ∅. On the other
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hand f is smooth, with fibres biregular to Pic0 (C). The smoothness of f implies that Sing f ∗ Θr = f ∗ Sing Θr . Therefore, by Laszlo’s singularity theorem (Theorem 11.2.1) and the definition of f , we have Sing f ∗ Θr = {([E], ξ) ∈ Ur × Pic0 (C) | h0 (E(ξ)) ≥ 2}. But the previous proposition implies that h0 (E(ξ)) ≤ 1, for all ξ ∈ Pic0 (C). Then it follows that ΘE ∩ Sing f ∗ Θr = ∅ and hence a general ΘE is smooth. The same holds for DE .
11.3 The tautological model PE Now we want to see that the above curve DE appears as the singular locus of some natural tautological model of PE ∗ in P2r−1 . Proposition 11.3.1. Let E be any semistable bundle of degree r then 1) h1 (ωC ⊗ E) = 0 and h0 (ωC ⊗ E) = 2r. 2) ωC ⊗ E is globally generated unless E is not stable and Hom(E, OC (x)) is non zero for some point x ∈ C. Proof 1) By Serre duality h1 (ωC ⊗ E) = h0 (E ∗ ). Since E ∗ is semistable of slope −1 it follows h0 (E ∗ ) = 0. Then we have h0 (ωC ⊗ E) = 2r by Riemann-Roch. 2) By 1) E is globally generated if and only if h0 (ωC ⊗ E(−x)) = r, ∀ x ∈ C. By Serre duality this is equivalent to Hom(E, OC (x)) = 0, ∀ x ∈ C. Note also that Hom(E, OC (x)) 6= 0 implies that E is not stable. This completes the proof. In this section we assume that [E] ∈ Ur has the following properties (satisfied by a general [E]): • ωC ⊗ E is globally generated, • DE exists i.e. [E] is not in the indeterminacy locus of θr : Ur → Tr , • DE is smooth, To simplify notation we put F := ωC ⊗ E and PE := PF ∗ . Lemma 11.3.2. Let F be general and let F be defined by the standard exact sequence 0 → F ∗ → H 0 (F )∗ ⊗ OC → F → 0
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induced by the evaluation map. Then F is stable. In particular the map j : U r → Ur −1 −1 sending [ωC ⊗ F ] to [ωC ⊗ F ] is a birational involution.
Proof First of all we claim that F is semistable for F general enough. Let Fo = Lr , where L ∈ Pic3 (C) is globally generated, then h0 (Fo ) = 2r and F o = Fo . Up to a base change there exists an integral variety T and a vector bundle F over T × C such that the family of vector bundles {Ft := F ⊗ Ot×C , t ∈ T } dominates Ur and contains Fo . By semicontinuity we can assume, up to replacing T by a non empty open subset, that h0 (Ft ) = h0 (Fo ) = 2r and Ft is globally generated. So it is standard to construct from F a vector bundle F on T × C with the following property: F ⊗ Ot×C = F t , for each t ∈ T . Since F ⊗ Oo×C = Lr is semistable, the same holds for a general vector bundle F ⊗Ot×C . Hence the claim follows. Let F be a general stable bundle: F is semistable, by lemma 11.3.1 h0 (F ) = 2r, moreover since h0 (F ∗ ) = 0, we have H 0 (F )∗ ' H 0 (F ) and F is globally generated. So j is defined at F , actually j(F ) = F . This implies that j is a birational involution and F is stable too. Since F is globally generated the map defined by OPE (1) is a morphism uE : PE → P2r−1 := PH 0 (F )∗ . In particular the restriction of uE to any fibre PE,x of PE is a linear embedding uE,x : PE,x → P2r−1 . Definition 11.3.3. The image of uE , (of uE,x ), will be denoted PE ,(PE,x ). For any d ∈ C (2) , Fd := F ⊗ Od can be naturally seen as a rank r vector bundle over d. Note that its projectivization is p∗ d, where p : PE → C is the projection map. In particular the evaluation map ed : H 0 (Fd ) ⊗ Od → Fd defines an embedding id : p∗ d → PH 0 (Fd )∗ . We have PH 0 (Fd )∗ = P2r−1 and moreover id (p∗ d) is the union of two disjoint linear spaces of dimension r − 1 if d is smooth. The next lemma is therefore elementary. Lemma 11.3.4. Let o ∈ PH 0 (Fd )∗ be a point not in id (p∗ d), then there exists exactly one line L containing o and such that Z := i∗d L is a 0dimensional scheme of length two. Moreover with λ the linear projection
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from o, the scheme Z is the unique 0-dimensional scheme of length two on which λ · id is not an embedding. The central arrow in the long exact sequence 0 → H 0 (F (−d)) → H 0 (F ) → H 0 (Fd ) → H 1 (F (−d)) → 0
(11.1)
defines a linear map λd : PH 0 (Fd )∗ → PH 0 (F )∗ = P2r−1 , and from the construction clearly λd ◦ id is the map uE |p∗ d : p∗ d → P2r−1 . Proposition 11.3.5. uE |p∗ d is not an embedding if and only if d ∈ DE . Proof From the previous remarks and lemma 11.3.4 it follows that uE |p∗ d is not an embedding if and only if λd is not an isomorphism. By the long exact sequence (11.1) λd is not an isomorphism if and only if h0 (E(−d)) ≥ 1, that is if d ∈ DE . We want to use the previous results to study the singular locus of PE . Let Hilb2 (PE ) be the Hilbert scheme of 0-dimensional schemes Z ⊂ PE of length two. Consider ∆ = {Z ∈ Hilb2 (PE ) | uE |Z is not an embedding}. Let Z ∈ ∆ then Z ⊂ p∗ d, where d := p∗ Z belongs to DE . So we have a morphism p∗ : ∆ → DE sending Z to d. Proposition 11.3.6. Let E be general then p∗ : ∆ → DE is biregular. Proof Let Z ∈ ∆ and let p∗ Z = d, then Z is embedded in p∗ d. Since d ∈ DE we have h0 (F (−d)) = 1, see prop. 11.2.2. This implies that the linear map λd : PH 0 (Fd )∗ → P2r−1 is the projection from a point onto a hyperplane in P2r−1 . Then, by lemma 11.3.4, Z is the unique element of ∆ which is contained in p∗ d. Hence p∗ is injective. Conversely let d ∈ DE , then λd ◦ id is not an embedding on exactly one 0-dimensional scheme Z ⊂ p∗ d of length two. Since λd ◦ id = uE |p∗ d , it follows that Z is in ∆ and that p∗ is surjective. Since DE is a smooth curve, p∗ is biregular.
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Proposition 11.3.7. Assume r ≥ 2 and E general, then uE : PE → PE is the normalization map and Sing PE is an irreducible curve. ˜ ⊂ PE be the image of the curve Proof Let D ˜ = {(Z, q) ∈ ∆ × PE | q ∈ Supp Z} ∆ ˜ is the locus of points where under the projection ∆ × PE → PE . The set D uE is not an embedding: it is a proper closed set as soon as r ≥ 2. Hence uE : PE → PE is a morphism of degree one if r ≥ 2. Since PE is smooth, uE is the normalization map if each of its fibres is finite. Assume uE contracts an irreducible curve B to a point o. B cannot be in a fibre PE,p . Otherwise DE would contain the curve {z + p, z ∈ C} and would be reducible. Hence o ∈ ∩PE,x , x ∈ C. Let x+y ∈ C (2) with x 6= y, then PE,x ∪PE,y is contained in a hyperplane and hence h0 (E ⊗ωC (−x−y)) ≥ 1. This implies DE = C (2) : a contradiction. It remains to show that Sing PE is an irreducible curve: this ˜ is clear because Sing PE = uE (D) 11.4 The line bundle HE We will keep the generality assumptions and the notation of the previous section. Recall that d ∈ DE uniquely defines a 0-dimensional scheme Zd ⊂ p∗ d of length two such that uE |Zd is not an embedding, in particular uE (Zd ) is a point. Definition 11.4.1. hE : DE → P2r−1 is the morphism sending d to uE (Zd ), moreover HE := h∗E OP2r−1 (1). Remarks 11.4.2. 1) Let F := ωC ⊗ E and let q1 , q2 : C × C → C be the projections. Note that q1∗ F ⊕ q2∗ F descends to a vector bundle F (2) on C (2) via the quotient map C × C → C (2) . Moreover the evaluation H 0 (F ) → Fx ⊕ Fy induces a natural map e : H 0 (F ) ⊗ OC (2) → F (2) . Then DE is the degeneracy locus of e and HE is its cokernel. This implies that the sheaf HE can be defined for every curve DE and that HE is a line bundle if and only if h0 (ωC ⊗ E(−x − y)) = 1 for each x + y ∈ DE . 2) A very simple geometric definition of hE can be given as follows: let d = x + y ∈ DE with x 6= y then hE (d) = PE,x ∩ PE,y = uE (Zd ).
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Proposition 11.4.3. hE : DE → P2r−1 is generically injective if E is general and r ≥ 2. Proof Let U = {x + y ∈ DE | x 6= y}, assume that d1 , d2 ∈ U and T hE (d1 ) = hE (d2 ) = o: then o ∈ i=1...4 PE,xi , where Σ xi = d1 + d2 . We consider the standard exact sequence 0 → F ∗ → H 0 (F )∗ ⊗ OC → F → 0, where F = ωC ⊗ E. The long exact sequence identifies H 0 (F )∗ with a subspace of H 0 (F ). Hence o is a 1-dimensional space generated by some T s ∈ H 0 (F ). It is standard to verify that o ∈ i=1...4 PExi if and only if s is zero on d1 + d2 − d, where d is the gcd of d1 , d2 . By Lemma 11.3.4 F is stable, hence we must have deg d ≥ 2 that is d1 = d2 = d. Proposition 11.4.4. HE has degree r2 + 2r. Proof Set theoretically we have hE (DE ) = Sing PE , hence Sing PE is an irreducible curve. Let o = hE (x + y) be general then x 6= y and moreover u∗E (o) is supported on two closed points o0 and o00 : this follows because hE is generically injective. Claim: The tangent map duE is injective at o0 and o00 . ^ Let D(u E ) be the double point scheme of uE , defined as in [7, p. 166]. ^ D(uE ) is contained in PE^ × PE , where π : PE^ × PE → PE ×PE is the blowing ^ up of the diagonal ∆. A point of D(uE ) is either the inverse image by π of a pair (o0 , o00 ) in PE × PE − ∆ such that uE (o0 ) = uE (o00 ) or it is a point in π −1 (∆) parametrizing a 1 dimensional space of tangent vectors to PE on which duE is zero. In the former case we have also p(o0 ) 6= p(o00 ) because uE is injective on each fibre of p. On the other hand it is clear that, in our situation, ^ q −1 (DE ) = (p × p) · π(D(u E )) where q : C × C → C (2) is the quotient map. Thus duE is not injective at most along fibres PE,z such that 2z ∈ DE . DE cannot be the diagonal of 2 = 2r 2 . Hence D contains finitely many points 2z and we C (2) because DE E can choose the above point o = hE (x + y) so that 2x and 2y are not in DE . This implies our claim. Let T be the tangent space to PE at o and let T 0 , T 00 ⊂ T be the images of 0 00 duE at o0 , o00 . Since duE is injective at o0 , o00 and u−1 E (o) = {o , o }, it follows 0 00 0 00 that T ∪ T spans T and that T ∩ T is the tangent space to Sing PE at o. We have dim T 0 ∩ T 00 ≥ 1 because Sing PE is a curve. On the other hand
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PE,x ∩ PE,y = o implies dim T ≥ 2r − 1. Since dim T 0 = dim T 00 = r, we deduce that dim T 0 ∩ T 00 = 1. Hence, as a scheme defined by the Jacobian ideal of PE , Sing PE is reduced. Finally the degree of Sing PE can be obtained via double point formula, see [7, 9.3], as follows: V · V − cr−1 (NV |P2r−2 ) = 2 deg(Sing PE ) where V ⊂ P2r−2 is a general hyperplane section of PE , corresponding to a global section σ ∈ H 0 (OPPE (1)) ' H 0 (ωC ⊗ E), cr−1 denotes the (r − 1)-st Chern class of the normal bundle NV |P2r−2 . Note that V = PE 0 , with E 0 , vector bundle of rank r − 1 defined by the section σ as follows: 0 → OC → E ⊗ ωC → E 0 ⊗ ωC → 0, and OPE0 (1) = OPE (1)|P 0 . Let H = [OPE0 (1)] and f be the class of a fibre E of PE 0 , then by computing the total Chern class of the normal bundle, we find cr−1 (NV |P2r−2 ) = rH r−1 + f H r−2 [4r2 − 4r] = 7r2 − 4r. Finally, we have deg Sing PE = deg HE = r2 + 2r. The genus of DE is r2 + 1 and HE has degree r2 + 2r, hence h0 (HE ) ≥ 2r. In fact: Proposition 11.4.5. For a general E the line bundle HE is non special that is h0 (HE ) = 2r. Proof By induction on r. Let r = 1 then A := ωC ⊗ E is a general line bundle of degree 3, PE = C and uE : PE → P1 is the triple covering defined by A. Moreover DE is the family of divisors x + y which are contained in a fibre of uE . It is easy to see that DE is a copy of C and that hE = uE . Then HE = A and hence h0 (HE ) = 2. Let r ≥ 2 and let [Er−1 ] ∈ Ur−1 and E1 ∈ Pic1 (C) be general points satisfying the statement, then their corresponding curves Dr−1 and D1 are smooth and transversal. Taking a general semistable extension 0 → Er−1 → E → E1 → 0
(11.2)
we have h0 (E ⊗ωC (−x−y)) ≤ 1 for any x+y, (see 11.2.2 and its proof). Observe also that DE = D1 ∪Dr−1 and that hE is a morphism. The restrictions of hE to D1 and Dr−1 can be described as follows:
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a) Let F := ωC ⊗ E and let A := ωC ⊗ E1 : tensoring (11.2) by ωC and passing to the long exact sequence, we obtain a surjective map H 0 (F ) → H 0 (A). Its dual is a linear embedding i : PH 0 (A)∗ → P2r−1 . On the other hand we already know that hE1 is the triple cover of PH 0 (A)∗ defined by A. It is easy to conclude that hE |D1 = i · hE1 . In particular hE (D1 ) is a line ` in P2r−1 which is triple for hE (DE ). b) Let B := ωC ⊗ Er−1 : tensoring (11.2)) by ωC and passing to the long exact sequence we get an injection H 0 (B) → H 0 (F ). Its dual induces a projection p : P2r−1 → PH 0 (B)∗ of centre `. It is again easy to conclude that p · hE E|Dr−1 = hEr−1 . It follows from the remarks in (b) that HE ⊗ ODr−1 = HEr−1 (a), where a := (hE |Dr−1 )∗ ` = D1 · Dr−1 . On the other hand (a) implies that HE ⊗ OD1 = HE1 = A. Finally, tensoring the Mayer-Vietoris exact sequence 0 → ODE → ODr−1 ⊕ OD1 → Oa → 0, by HE we obtain 0 → HE → HEr−1 (a) ⊕ A → Oa ⊗ HE → 0. By induction h1 (HEr−1 ) = 0, hence h1 (HEr−1 (a)) = 0. Moreover h1 (A) = 0. Passing to the long exact sequence, the vanishing of H 1 (HE ) follows if the restriction ρ : H 0 (HEr−1 (a)) → Oa (a) ⊗ HEr−1 is surjective. Since h1 (HEr−1 ) = 0, this follows from the long exact sequence of 0 → HEr−1 → HEr−1 (a) → Oa (a) ⊗ HEr−1 → 0. The vanishing of h1 (HE ) extends by semicontinuity to a general point of Ur .
11.5 The Brill-Noether curve of E In the following we will set for simplicity: D := DE . D is an abstract curve endowed with an embedding D ⊂ C (2) . These data are in general not sufficient to reconstruct the vector bundle E. As we will see the additional datum of HE makes such a reconstruction possible. The embedding D ⊂ C (2) uniquely defines the family of divisors bx := Cx · D where x ∈ C and Cx := {x + y | y ∈ C}. bx fits in the standard exact sequence 0 → H 0 (ωC ⊗ E(−x)) ⊗ OC → ωC ⊗ E(−x) → Obx → 0
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and its degree is 2r. The determinant of E can be reconstructed from the family {bx | x ∈ C}. Indeed let x + x0 ∈ |ωC |, then the previous exact sequence implies det E ∼ = OC (bx − rx0 ). Let t + ΘE be the translate of ΘE by t ∈ Div0 (C): DE(−t) = a∗ (t + ΘE ). Thus, upon replacing E by E(−t), D is transversal to Cx and bx is smooth for a general x. Mainly we will consider bx as a divisor on D. Let d ∈ D, it is clear that: d ∈ Supp bx ⇐⇒ d = x + y ⇐⇒ hE (d) ∈ PE,x . This implies that bx = hE ∗ PE,x for each x ∈ C. The line bundles HE (−bx ) have degree r2 . Since D has 2 genus r2 + 1 they define a family of points in the theta divisor of Picr (D). We can say more: Proposition 11.5.1. Let E be general, then for every point x ∈ C we have h0 (HE (−bx )) = r. Proof We know from Prop. 11.4.5 that h0 (HE ) = 2r. We also know that bx = hE ∗ (PE,x ). Since the space PE,x has dimension r − 1, it follows h0 (HE (−bx )) ≥ r. Moreover the equality holds if the set hE (Supp bx ) spans PE,x . We prove this by induction on r. Let r = 1 then PE,x is a point: since hE is a morphism hE (Supp bx ) = PE,x . Let r ≥ 2, as in the proof of 11.4.5 we consider a general extension 0 → Er−1 → E → E1 → 0
(11.3)
with [Er−1 ] ∈ Ur−1 and E1 ∈ Pic1 (C) general. We use the same assumptions and notations of the proof of 11.4.5 which is similar to this proof. In particular the curves DEr−1 and DE1 are smooth and transversal, moreover the exact sequence 0→B→F →A→0 just denotes the above exact sequence (11.3) tensored by ωC . Such a sequence induces a linear embedding i : PH 0 (A)∗ → PH 0 (F )∗ . The image of i is the line ` considered in 11.4.5 and it holds the equality proved there: hE |DE1 = i ◦ uE1 . Then it turns out that ` ∩ PE,x = hE (Cx ∩ DE1 ) = one point ox for each x ∈ C. On the other hand let p : PE → PH 0 (B)∗ be the projection of centre `, then the latter exact sequence implies that p(PE,x ) = PEr−1 ,x .
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Moreover we also know from the proof of 11.4.5 that hEr−1 = p ◦ hE . By induction PEr−1 ,x is spanned by hEr−1 (Cx ∩ DEr−1 ) = p(hE (Cx ∩ DEr−1 )). Hence the linear span of hE (Cx ∩ DEr−1 ) is a space L ⊂ PE,x of dimension ≥ r − 2 and such that p(L) = PEr−1 ,x . If ox ∈ PE,x − L then PE,x is spanned by hE (Cx ∩ D). If ox ∈ L then dim L = r − 1 and L = PE,x . In both cases the statement follows. For each ` ∈ Pic1 (C) we consider the curve B` := {x + y ∈ |`(z)|, z ∈ C}. B` is biregular to C unless ` = OC (x) for some x ∈ C. In the latter case B` is Cx ∪ |ωC |. We define b` := D · B` . Note that b` = bx if B` = Cx ∪ |ωC |. The reason is that we are assuming E general, then h0 (E) = 0 and hence D ∩ |ωC | = ∅. Lemma 11.5.2. The morphism b : Pic1 (C) → Pic2r (D) sending ` to b` is an embedding. We will denote its image by JD : JD := {OD (b` ) | ` ∈ Pic1 (C)}. In particular JD contains the canonical theta divisor CD := {OD (bx ) | x ∈ C}.
(11.4)
Proof Up to shifting degrees, b is a morphism between the complex tori Pic0 (C) and Pic0 (D). Hence it is an isogeny up to translations, so b is an embedding if it is injective. Let `1 , `2 ∈ Pic1 (C) and set L : = OD (b`1 − b`2 ). L is defined by the standard exact sequence: 0 → OC (2) (−D + B`1 − B`2 ) → OC (2) (B`1 − B`2 ) → L → 0. D + B`2 − B`1 is the pull-back by the Abel map a : C (2) → Pic0 (C) of a divisor homologous to rΘ, where Θ is a theta divisor in Pic0 (C). Since rΘ is ample, it follows: h0 (−D + B`1 − B`2 ) = h1 (−D + B`1 − B`2 ) = 0. So the associated long exact sequence gives: h0 (L) = h0 (OC (2) (B`1 − B`2 )). Moreover, it is easy to see that if `1 6= `2 , then h0 (OC (2) (B`1 − B`2 )) = 0 hence b`1 and b`2 are not linearly equivalent.
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As an immediate consequence of the lemma, the map b0 : Pic0 (C) → Pic0 (D) sending OC (x − y) → OD (bx − by ) is an embedding too. As we already pointed out, the knowledge of CD is not sufficient to reconstruct E. Instead we need another curve: Definition 11.5.3. The Brill-Noether curve of E is the curve CE := {HE (−bx ), x ∈ C}. 2
CE is a copy of C embedded in Picr (D). Since h0 (HE (−bx )) = r, each point of CE is a point of multiplicity r for the theta divisor 2
ΘD := {L ∈ Picr (D) | h0 (L) ≥ 1}. In particular CE is contained in the Brill-Noether locus 2
Wrr−1 (D) := {L ∈ Picr (D) | h0 (L) ≥ r}. 2 The Brill-Noether number ρ(r − 1, r2 , r2 + 1) yields the expected dimension of Wrr−1 (D). We have ρ(r − 1, r2 , r2 + 1) = 1 for each r, so we expect that 2 CE is an irreducible component of Wrr−1 (D), see [1, ch. V]. Of course D 2 is not a general curve of genus r2 + 1, so the latter property is not a priori granted. Remark 11.5.4. E can uniquely be reconstructed from the pair (CD , HE ) as follows. Consider the correspondence B = { (x, y + z) ∈ C × D | x ∈ {y, z} }, with B · (x × D) = bx . Let p1 : C × D → C and p2 : C × D → D be the projection maps, then we apply the functor p1∗ to the exact sequence 0 → p∗2 HE (−B) → p∗2 HE → p∗2 HE ⊗ OB → 0. This yields the exact sequence ∗
0 → F → H 0 (HE ) ⊗ OC → p1∗ OB ⊗ p∗2 HE → R1 p1∗ p∗2 HE (−B) → 0, ∗
where F := p1∗ p∗2 HE (−B). With F = ωC ⊗ E, we have the natural identities ∗
Fx = H 0 (HE (−bx )) = H 0 (F (−x)). The left one is immediate. Let I be the ideal of PE,x , then we have H 0 (HE (−bx )) = H 0 (I(1)) by prop. 11.5.1. Hence the right equality follows from the identity
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H 0 (I(1)) = H 0 (F (−x)). The above identities, together with H 0 (HE ) = H 0 (F ), imply that ∗
F = H 0 (HE ) ⊗ OC /F . As an immediate consequence of the above construction we have: Proposition 11.5.5. Let [E1 ], [E2 ] be general points of Ur . Assume that θr ([E1 ]) = θr ([E2 ]) = D and HE1 = HE2 . Then [E1 ] = [E2 ]. Remark 11.5.6. The previous construction also defines the vector bundles −1 E : = F ⊗ ωC
,
e : = ω −1 ⊗ (R1 p1 ∗ p∗ HE (−B)). E 2 C
We already know from 11.3.2 that the assignment [E] → [E] defines a bie is semistable for E rational involution j : Ur → Ur . Note also that E general: to prove this it suffices to produce one semistable Eo such that eo is semistable. The existence of Eo follows by induction on r: this is E obvious for r = 1. Let r ≥ 2 and let Eo be defined by a semistable extension e ∈ Ext1 (E1 , Er−1 ) where [U]r−1 ∈ Ur−1 and E1 ∈ U1 . It is easy to eo is defined by some ee ∈ Ext1 (E e1 , E er−1 ): we leave the details to show that E e the reader. Hence Eo is semistable. Due to this property we can define a e rational map κ : Ur → Ur sending [E] to [E]. e is birational: Proposition 11.5.7. The map κ : Ur → Ur sending [E] to [E] its inverse is j ◦ k ◦ j. Proof Let T := OD (bx + bi(x) ), where i : C → C is the hyperelliptic involution. T does not depend on x because the family of divisors {bx + bi(x) , x + i(x) ∈ |ωC |} is rational. Then we define the line bundle of degree r2 + 2r e E := ωD ⊗ T ⊗ H −1 . H E e E ) = 0 for E general. Indeed ωD ⊗ H e −1 is HE (−bx − First, we note that h1 (H E e E ) = h0 (HE (−bx − bi(x) ) by Serre duality. Since bi(x) ) and hence h1 (H eE ) = 0 h0 (HE (−bx − bi(x) ) = h0 (ωC ⊗ E(−x − i(x)) = h0 (E), it follows h1 (H for each [E] ∈ Ur − Θr . Secondly we note that, with the previous notations, Serre duality yields a natural identification e E (−bi(x) ))∗ , ∀x ∈ C. R1 p1∗ p∗2 HE (−B)x = H 0 (H e = R1 p1∗ p∗ HE (−B) ∼ e E (−B)∗ . It is then easy to deduce that ωC ⊗ E = p1∗ p∗2 H 2 e E it is clear that one obtains HE and with the same Starting from H −1 e E is the line bundle H e construction E = ωC ⊗ p1∗ p∗2 HE . Note also that H E
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e This implies that k −1 = j ◦ k ◦ j and hence defined by the vector bundle E. that k is birational.
11.6 The fibres of the theta map We want to see that E can also be uniquely reconstructed from the pair (D, CE ). For this we consider, more generally, any smooth curve D ⊂ C (2) such that a∗ D ∈ Tr : Definition 11.6.1. A Brill-Noether curve of D is a copy 2
C 0 ⊂ Picr (D) of C satisfying the following property: there exists H ∈ Picr that
2 +2r
(D) such
C 0 = {H(−by ), y ∈ C}, moreover H is non special and h0 (H(−bx )) = r for every point x ∈ C. The set of the Brill-Noether curves of D will be denoted by SD . Let D = θr ([E]) then CE is a Brill-Noether curve of D. Let r = 1 then D = C and the canonical theta divisor of Pic1 (C) is the unique Brill-Noether curve of D. Lemma 11.6.2. Let H be as in the previous definition then H is unique. Proof Assume that C 0 = {H 0 (−bx ), x ∈ C} for a second H 0 . Then there exists an automorphism u : C → C which defined by setting u(x) = y precisely if H 0 (−bx ) = H(−by ). Let γ : C × C → Pic0 (D) be the map sending (x, y) to H 0 ⊗ H −1 (bx − by ): we claim that the image of γ is the copy JD ⊂ Pic0 (D) of Pic0 (C) and that γ : C ×C → Pic0 (C) is the difference map. To see this recall that CD = {bx , x ∈ C} is the theta divisor of JD , see (11.4). The map γ˜ : C × C → Pic0 (D) sending (x, y) → OD (bx − by ) factors through the isomorphism t : C ×C → CD ×CD , sending (x, y) → (bx , by ), and the difference map. Moreover, we have the following commutative diagram t
C × C → CD × CD ↓ ↓ b0 Pic0 (C) → Pic0 (D) where the vertical arrows are difference maps and b0 : Pic0 (C) → Pic0 (D) sending OC (x − y) → OD (bx − by ) is an embedding, see 11.5.2. This implies that γ˜ is a difference map and γ too. The graph of u is obviously contracted
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by γ, on the other hand the only curve contracted by the difference map is the diagonal of C × C. Then u is the identity and H(−bx ) = H 0 (−bx ) for each x ∈ C. Hence H = H 0 . Proposition 11.6.3. Let [E1 ], [E2 ] be general points in Ur , assume that CE1 = CE2 and that θr ([E1 ]) = θr ([E2 ]) = D. Then [E1 ] = [E2 ]. Proof By the previous lemma HE1 = HE2 and by 11.5.5 this implies [E1 ] = [E2 ]. Theorem 11.6.4. The theta map θr : Ur → Tr is generically finite. Proof It suffices to show that θr |U : U → θr (U ) is generically finite for a suitable dense open set U . Hence we can assume that D ∈ θr (U ) is a smooth curve and that the points of (θr |U )−1 (D) = θr−1 (D) ∩ U are sufficiently general in Ur . Let i : (θr |U )−1 (D) → SD be the map sending [E] to CE . By 11.6.3 E can uniquely be reconstructed from (D, CE ), hence it follows that i is injective. On the other hand recall that CE is contained in the Brill-Noether locus Wrr−1 (D). Since the Brill2 Noether number ρ(r − 1, r2 , r2 + 1) is one, each irreducible component of Wrr−1 (D) has dimension ≥ 1. This implies that θr |U is finite if CE is an 2 irreducible component of Wrr−1 (D). This property is proved in the next 2 theorem. Hence the statement follows. Lemma 11.6.5. Let D = θr ([E]) for a general [E] ∈ Ur and let a = D · D1 for a general D1 ∈ T1 , then the line bundle HE (a − bx ) is non special. Proof Let D1 = θ1 ([E1 ]) ⊂ C (2) with E1 = OC (x), then we have: D1 = x × C ∪ |ωC | ⊂ C (2) . Note that a = D · D1 = bx if E is general. Hence HE (a − bx ) = HE is non special. By semicontinuity, the same is true for a general D1 . Theorem 11.6.6. For a general [E] ∈ Ur the Brill-Noether curve CE is an irreducible component of Wrr−1 (D), D = θr ([E]). 2 Proof Let H := HE , it is sufficient to show the injectivity of the Petri map µ : H 0 (H(−bx )) ⊗ H 0 (ωD ⊗ H −1 (bx )) → H 0 (ωD ) for a general x ∈ C. This implies that the tangent space to Wrr−1 (D) at its 2 point H(−bx ) is 1-dimensional, see [1, Ch. V]. We proceed by induction on
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r. Let r = 1, then D = C and CE = {OC (x), x ∈ C}. Hence the injectivity of µ is immediate. For r ≥ 2 we borrow once more the notations and the method from the proof of proposition 11.4.5. So we specialize E to the semistable vector bundle defined by the exact sequence 0 → Er−1 → E → E1 → 0. Then D is the transversal union of the curves Dr−1 = θr−1 ([Er−1 ]) and D1 = θ1 ([E1 ]), hE is a morphism and H is the line bundle h∗E OP2r−1 (1). Let a = Dr−1 · D1 : from the proof of 11.4.5 we have D1 = C and HE1 = H1 and moreover H ⊗ ODr−1 = HEr−1 (a) and H ⊗ OD1 = H1 . Since x is general we can assume Supp bx ∩ Sing D = ∅ so that OD (bx ) is a line bundle. Let I be the ideal sheaf of D1 in D: at first we show that µ|I ⊗ W is injective, where I := H 0 (I ⊗ H(−bx )) and W := H 0 (ωD ⊗ H −1 (bx )). Since I ⊗ ODr−1 = ODr−1 (−a) and ωD ⊗ ODr−1 = ωDr−1 (a), we have the restriction maps ρI : I → H 0 (HEr−1 (−bx,r−1 )) and ρW : W → H 0 (ωDr−1 ⊗ HE−1 (bx,r−1 )) with bx,r−1 = bx · Dr−1 . r−1 Claim. ρI is an isomorphism and ρW is surjective. Assuming the claim, we shall complete the proof. First of all, ρ := ρI ⊗ρW is surjective and defines the exact sequence 0 → ker ρ → I ⊗W → H 0 (HEr−1 (−bx,r−1 ))⊗H 0 (ωDr−1 ⊗HE−1 (bx,r−1 )) → 0. r−1 In particular it follows dim ker ρ = r − 1. By induction on r the Petri map on the tensor product at the right side is injective. Therefore µ|I ⊗ W is injective ⇐⇒ µ| ker ρ is injective. But our claim implies dim ker ρW = 1 and ker ρ = I ⊗ hwi, where w generates ker ρW . Hence µ| ker ρ is injective as well as µ|I ⊗ W . Let V := H 0 (H(−bx )) and consider the exact sequence 0 → I ⊗ W → V ⊗ W → (V /I) ⊗ W → 0. The map µ induces a multiplication ν : (V /I) ⊗ W → H 0 (ωD )/µ(I ⊗ W ). The injectivity of µ|I ⊗ W implies that µ is injective if and only if ν is injective. On the other hand, ρI is an isomorphism, hence dim I = r −1 and dim V /I = 1. Let v ∈ V − I. Then ν is injective ⇐⇒ vW ∩ µ(I ⊗ W ) = (0)
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⇐⇒ no w ∈ W − (0) vanishes on D1 . This is equivalent to the injectivity of the restriction map u : H 0 (ωD ⊗ H −1 (bx )) → H 0 (ωC (a − x)); in fact W = H 0 (ωD ⊗ H −1 (bx )) and ωD ⊗ H −1 (bx ) ⊗ OD1 = ωC (a − x). To prove that u is injective consider the Mayer-Vietoris sequence (bx,r−1 )) ⊕ H 0 (ωC (a − x)) → Oa → . . . 0 → W → H 0 (ωDr−1 ⊗ HE−1 r−1 The left non zero arrow followed by the projection onto H 0 (ωC (a − x)) is exactly u. This implies that ker u, via the restriction map, injects in H 0 (ωDr−1 ⊗ HE−1 (bx,r−1 − a)). So u is injective if the latter space is zero r−1 1 that is if H (HEr−1 (a − bx,r−1 )) = 0: this has been shown in lemma 11.6.5. Hence µ is injective. Then, by semicontinuity, the same property is true for a general [E] ∈ Ur and the statement follows. It remains show the above claim. – Let h : D → P2r−1 be the map defined by H. As in 11.4.3 h(D1 ) is a line ` and PE,x ∩ ` is a point. Moreover PE,x is spanned by h(bx ). Hence we have I = H 0 (J(1)) and dim I = r − 1, J being the ideal of PE,x ∪ `. In particular ρ` is the pull-back (h|Dr−1 )∗ restricted to a space of linear forms vanishing on h(D1 ). Since h(D) is non degenerate ρ` is injective. Then, for dimension reasons, ρ` is an isomorphism. – As in 11.4.5 the projection p : PE,x → PEr−1 ,x from PE,x ∩ ` is surjective. Equivalently the restriction H 0 (E ⊗ ωC (−x)) → H 0 (Er−1 ⊗ ωC (−x)) is er , E er−1 be surjective. So this property holds for general E, Er−1 . Let E defined from Er , Er−1 as in Remark 11.5.6. By 11.5.7 they are general. e ⊗ ωC (−x)) → H 0 (E er−1 ⊗ ωC (−x)) is surjective: Hence the restriction H 0 (E this map is just ρW . Our partial geometric description of the theta map can be summarized as follows. Theorem 11.6.7. Let D ∈ Tr be general and smooth, then there exists a natural injective map iD between the fibre of θr at D and the set of the Brill-Noether curves of D. Namely the map iD : θr −1 (D) → SD associates to [E] ∈ θr−1 (D) its Brill-Noether curve CE ∈ SD . Proof Since θr is generically finite, each point [E] ∈ θ−1 (D) is sufficiently general in Ur . The injectivity then follows from corollary 11.6.4.
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Remark 11.6.8. Each Brill-Noether curve C ∈ SD uniquely defines a vector bundle EC of rank r and degree r: to construct EC it suffices to take the line bundle H appearing in the definition 11.6.1 of Brill-Noether curve. Applying the reconstruction given in remark 11.5.4 to the pair (CD , H) we indeed obtain such a vector bundle EC . If EC is semistable it turns out that θr ([EC ]) = D and that iD ([EC ]) = C. In particular iD is bijective if each C ∈ SD defines a semistable EC . This property seems very plausible for a general D, however we do not have a rigorous proof. References [1] Arbarello, E., M. Cornalba, P. A. Griffiths and J. Harris: Geometry of Algebraic Curves, vol. I, (1984), Springer-Verlag, New York, Berlin, Heidelberg, Tokyo. [2] Beauville, A.: Fibr´es de rang 2 sur les courbes, fibre determinant et fonctions thˆeta, Bull. Soc. Math.France 116 (1988), 431–448. [3] Beauville, A.: Vector bundles and theta functions on curves of genus 2 and 3, Preprint math.AG/0406030,(2004), to appear. [4] Beauville, A.: Vector bundles on curves and theta functions, Preprint math. AG/0502179, (2005). [5] Beauville, A., M. S. Narasimhan and S. Ramanan:, Spectral curves and the generalised theta divisor, J.Reine Angew. Math.398 (1989), 169–179. [6] Drezet, J. M. and M. S. Narasimhan: Groupe de Picard des vari´et´es de modules de fibr´es semi-stables sur les courbes alg´ebriques, Invent. math. 97 (1989), 53–94. [7] Fulton, W.: Intersection Theory, (1984), Springer Verlag Berlin. [8] Laszlo, Y.: Un th´eor`eme de Riemann pour le diviseur Thˆeta generalis´e sur les espaces de modules de fibr´es stables sur une courbe, Duke Math. J. 64 (1991), 333–347. [9] Raynaud, M.: Sections des fibr´es vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125 [10] Teixidor, M.: For which Jacobi varieties is Sing Θ reducible? J. f. reine u. angew. Math. 354 (1985), 141–149.
12 On Tannaka duality for vector bundles on p-adic curves Christopher Deninger Mathematisches Institut Einsteinstr. 62 48149 M¨ unster, Germany
[email protected]
Annette Werner Fachbereich Mathematik Pfaffenwaldring 57 70569 Stuttgart, Germany
[email protected]
Dedicated to Jacob Murre
12.1 Introduction In our paper [DW2] we have introduced a certain category Bps of degree zero bundles with “potentially strongly semistable reduction” on a p-adic curve. For these bundles it was possible to establish a partial p-adic analogue of the classical Narasimhan–Seshadri theory for semistable vector bundles of degree zero on compact Riemann surfaces. One of the main open questions of [DW2] is whether our category is abelian. The first main result of the present note, Corollary 12.3.4, asserts that this is indeed the case. It follows ps is that Bps and also the subcategory Bps red of all polystable bundles in B a neutral Tannakian category. In the second main result, theorem 12.4.6, we calculate the group of connected components of the Tannaka dual group of Bps red . This uses a result of A. Weil characterizing vector bundles that become trivial on a finite ´etale covering as the ones satisfying a “polynomial equation” over the integers. Besides, in section 12.3 we give a short review of [DW2], and in section 12.2 we discuss the “strongly semistable reduction” condition. Finally we would like to draw the reader’s attention to the paper of Faltings [F] where a non-abelian p-adic Hodge theory is developped. It is a pleasure for us to thank Uwe Jannsen for a helpful discussion. 399
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12.2 Vector bundles in characteristic p Throughout this paper, we call a purely one-dimensional separated scheme of finite type over a field k a curve over k. Let C be a connected smooth, projective curve over k. For a vector bundle E on C we denote by µ(E) = deg(E) rk (E) the slope of E. Then E is called semistable (respectively stable), if for all proper non-trivial subbundles F of E the inequality µ(F ) ≤ µ(E) (respectively µ(F ) < µ(E)) holds. Lemma 12.2.1. If π : C 0 → C is a finite separable morphism of connected smooth projective k-curves, then semistability of E is equivalent to semistability of π ∗ E. Proof See [Gie2], 1.1. If char(k) = 0, then by lemma 12.2.1 any finite morphism of smooth connected projective curves preserves semistability. However in the case char(k) = p, there exist vector bundles which are destabilized by the Frobenius map, see [Gie1], Theorem 1. Assume that char(k) = p, and let F : C → C be the absolute Frobenius morphism, defined by the p-power map on the structure sheaf. Definition 12.2.2. A vector bundle E on C is called strongly semistable of degree zero if deg(E) = 0 and if F n∗ E is semistable on C for all n ≥ 0. Now we also consider non-smooth curves over k. Let Z be a proper curve over k. By C1 , . . . , Cr we denote the irreducible components of Z endowed with their reduced induced structures. Let C˜i be the normalization of Ci , and write αi : C˜i → Ci → Z for the canonical map. Note that the curves C˜i are smooth irreducible and projective over k. Definition 12.2.3. A vector bundle E on the proper k-curve Z is called strongly semistable of degree zero, if all αi∗ E are strongly semistable of degree zero. A alternative characterization of this property is given by the following well-known result. Proposition 12.2.4. A vector bundle E on Z is strongly semistable of degree zero if and only if for any k-morphism π : C → Z, where C is a smooth connected projective curve over k, the pullback π ∗ E is semistable of degree zero on C. Note that in [DM], (2.34) bundles with this property are called semistable of degree zero.
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Proof Let X be a scheme over k. The absolute Frobenius F sits in a commutative diagram X
spec k
F
/X
F
/ spec k
If we denote for all r ≥ 1 by X (r) the scheme X together with the structure Fr map X → spec k −−→ spec k, then F r : X (r) → X is a morphism over spec k. Assume that the curve Z has the property in the claim. Applying it to the smooth projective curves C˜i and the k-morphisms r
αi (r) F Z, C˜i −−→ C˜i −→
we find that all αi∗ E are strongly semistable of degree zero, i.e. that E is strongly semistable of degree zero in the sense of definition 12.2.3. Conversely, assume that all αi∗ E are strongly semistable of degree zero. Let π : C → Z be a k-morphism from a smooth connected projective curve C to Z. Then π factors through one of the Ci . If π is constant, then π ∗ E is trivial, hence semistable of degree zero. Hence we can assume that π(C) = Ci . Since C is smooth, π also factors through the normalization C˜i , i.e. there is a morphism πi : C → C˜i satisfying αi ◦ πi = π. Since π is dominant, it is finite and hence πi is the composition of a separable map and a power of Frobenius, see e.g. [Ha], IV, 2.5. Hence there exists a smooth projective curve D over k and a finite separable morphism f : D → C˜i such ∼ that C −→ D(r) for some r ≥ 1 and πi factors as r
f ∼ F πi : C −→ D(r) −−→ D − → C˜i .
Write Ei = αi∗ E. Then we have to show that π ∗ E = πi∗ Ei is semistable of degree 0 on C. By assumption, Ei is strongly semistable of degree 0 on C˜i . Using Lemma 12.2.1 and the fact that F commutes with all morphisms in characteristic p, we find that the pullback f ∗ Ei under the finite, separable map f is strongly semistable of degree 0 on D. Hence F r∗ f ∗ Ei is semistable, which implies that πi∗ Ei is semistable of degree zero. Generalizing a result by Lange and Stuhler in [LS], one can show Proposition 12.2.5. If k = Fq is a finite field, then a vector bundle E on the proper k-curve Z is strongly semistable of degree zero, if and only if there exists a finite surjective morphism ϕ:Y →Z
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of proper k-curves such that ϕ∗ E is trivial. In fact, one can take ϕ to be the composition Frsq
π
→Z ϕ : Y −−→ Y − of a power of the k-linear Frobenius morphism Frq (defined by the q-th power map on OY ) and a finite, ´etale and surjective morphism π. Proof See [DW2], Theorem 18. Note that every semistable vector bundle E of degree zero on a smooth geometrically connected projective curve C of genus g ≤ 1 is strongly semistable. Namely, for g = 0, every semistable vector bundle of degree zero is in fact trivial. For g = 1, the claim follows from Atiyah’s classificaL tion [At]: let E = i Ei be the decomposition of E into indecomposable components. Since E is semistable of degree zero, all Ei have degree zero since they are subbundles and quotients. Therefore by [At] Theorem 5, we have Ei ' L⊗G, where L is a line bundle of degree zero and G is an iterated extension of trivial line bundles. The pullback of Ei under some Frobenius power is also of this form. Since the category of semistable vector bundles of degree 0 on C is closed under extensions and contains all line bundles of degree zero, we conclude that Ei is indeed strongly semistable of degree 0. This proves the following fact: Lemma 12.2.6. Let Z be a proper k-curve such that the normalizations C˜i of all irreducible components Ci are geometrically connected of genus g(C˜i ) ≤ 1. Consider a vector bundle E on Z. If all restriction E |C˜i are semistable of degree zero, then E is strongly semistable of degree zero. By [Gie1], for every genus ≥ 2 there are examples of semistable vector bundles of degree zero which are not strongly semistable. On the other hand, there are results indicating that there are “a lot of” strongly semistable vector bundles of degree zero. In [LP], Laszlo and Pauly show that for an ordinary smooth projective curve C of genus two over an algebraically closed field k of characteristic two, the set of strongly semistable rank two bundles is Zariski dense in the coarse moduli space of all semistable rank two bundles with trivial determinant. See [JRXY] for generalizations to higher genus. In [Du], Ducrohet investigates the case of a supersingular smooth projective curve of genus two over an algebraically closed field k with char(k) = 2. It turns out that in this case all equivalence classes of semistable bundles with trivial determinant but one are in fact strongly semistable.
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12.3 Vector bundles on p-adic curves Before discussing the p-adic case, let us recall some results in the complex case, i.e. regarding vector bundles on a compact Riemann surface X. Let ˜ → X the universal covering of x ∈ X be a base point and denote by π : X X. Every representation ρ : π1 (X, x) → GL r (C) gives rise to a flat vector ˜ ×Cr bundle Eρ on X, which is defined as the quotient of the trivial bundle X by the π1 (X, x)-action given by combining the natural action of π1 (X, x) on the first factor with the action induced by ρ on the second factor. It is easily seen that every flat vector bundle on X is isomorphic to some Eρ . Regarding Eρ as a holomorphic bundle on X, a theorem of Weil [W] says that a holomorphic bundle E on X is isomorphic to some Eρ (i.e. E comes from a representation of π1 (X, x)) if and only if in the decomposition L E = ri=1 Ei of E into indecomposable subbundles all Ei have degree zero. A famous result by Narasimhan and Seshadri [NS] says that a holomorphic vector bundle E of degree 0 on X is stable if and only if E is isomorphic to Eρ for some irreducible unitary representation ρ. Hence a holomorphic vector bundle comes from a unitary representation ρ if and only if it is of L the form E = ri=1 Ei for stable (and hence indecomposable) subbundles of degree zero. Now let us turn to the p-adic case. Let X be a connected smooth projective curve over the algebraic closure Qp of Qp and put XCp = X ⊗Qp Cp . We want to look at p-adic representations of the algebraic fundamental group π1 (X, x) where x ∈ X(Cp ) is a base point. It is defined as follows. Denote by Fx the functor from the category of finite ´etale coverings X 0 of X to the category of finite sets which maps X 0 to the set of Cp -valued points of X 0 lying over x. ∼ For x, x0 ∈ X(Cp ) we call any isomorphism Fx −→ Fx0 of fibre functors an ´etale path from x to x0 . (Note that any topological path on a Riemann surface induces naturally such an isomorphism of fibre functors.) Then the ´etale fundamental group π1 (X, x) is defined as π1 (X, x) = Aut(Fx ) . The goal of our papers [DW1] and [DW2] is to associate p-adic representations of the ´etale fundamental group π1 (X, x) to certain vector bundles on XCp . Let us briefly describe the main result. We call any finitely presented, proper and flat scheme X over the integral closure Zp of Zp in Qp with generic fibre X a model of X. By o we denote the ring of integers in Cp , and by k = Fp the residue field of Zp and o. We write Xo = X ⊗Zp o and Xk = X ⊗Zp k.
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Definition 12.3.1. We say that a vector bundle E on XCp has strongly semistable reduction of degree zero if E is isomorphic to the generic fibre of a vector bundle E on Xo for some model X of X, such that the special fibre Ek is a strongly semistable vector bundle of degree zero on the proper k-curve Xk . E has potentially strongly semistable reduction of degree zero if there is a finite ´etale morphism α : Y → X of connected smooth projective curves ∗ E has strongly semistable reduction of degree zero on over Qp such that αC p YCp . By Bs (respectively Bps ) we denote the full subcategory of the category of vector bundles on XCp consisting of all E with strongly semistable (respectively potentially strongly semistable) reduction of degree zero. Besides, for every divisor D on XCp we define BXCp ,D to be the full subcategory of those vector bundles E on XCp which can be extended to a vector bundle E on Xo for some model X of X, such that there exists a finitely presented proper Zp -morphism π : Y −→ X satisfying the following two properties: i) The generic fibre of π is finite and ´etale outside D ii) The pullback πk∗ Ek of the special fibre of E is trivial on Yk (c.f. [DW2], definition 6 and theorem 16). Then we show in [DW2], Theorem 17: [ Bs = BXCp ,D , D
where D runs through all divisors on XCp . By [DW2], Theorem 13, every bundle in BXCp ,D is semistable of degree zero, so that Bs and also Bps are full subcategories of the category Tss of semistable bundles of degree zero on XCp . Line bundles of degree zero lie in Bps by [DW2] Theorem 12 a. The main result in [DW2] is the following (c.f. [DW2], theorem 36): Theorem 12.3.2. Let E be a bundle in Bps . For every ´etale path γ from x to y in X(Cp ) there is an isomorphism ∼
ρE (γ) : Ex −→ Ey of “parallel transport”, which behaves functorially in γ. The association E 7→ ρE (γ) is compatible with tensor products, duals and internal homs of vector bundles in the obvious way. It is also compatible with Gal(Qp /Qp )conjugation. Besides, if α : X → X 0 is a morphism of smooth projective
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, then ρα∗ E 0 (γ) and ρE 0 (α∗ γ) coincurves over Qp and E 0 a bundle in Bps X0 Cp
cide, where α∗ γ is the induced ´etale path on X 0 . For every x ∈ X(Cp ) the fibre functor Bps −→ VecCp , mapping E to the fibre Ex in the category VecCp of Cp -vector spaces, is faithful. In particular one obtains a continuous representation ρE,x : π1 (X, x) → GL r (Ex ). The functor E 7→ ρE,x is compatible with tensor products, duals, internal homs, pullbacks of vector bundles and Gal(Qp /Qp )-conjugation. Let us look at two special cases of this representation: For line bundles on a curve X with good reduction, ρ induces a homomorphism Pic0X (Cp ) −→ Homcont (π1 (X, x), C∗p ) mapping L to ρL,x . As shown in [DW1] this map coincides with the map defined by Tate in [Ta] § 4 on an open subgroup of Pic0X (Cp ). Secondly, applying ρ to bundles E in H 1 (XCp , O) = Ext1XCp (O, O), one recovers the Hodge–Tate map to H 1 (X´et , Qp ) ⊗ Cp = Ext1π1 (X,x) (Cp , Cp ), see [DW1], corollary 8. It follows from [DW2], Proposition 9 and Theorem 11 that the categories s B and Bps are closed under tensor products, duals, internal homs and extensions. We will now prove another important property of those categories. Theorem 12.3.3. If a vector bundle E on XCp is contained in Bs (respectively Bps ), then every quotient bundle of degree zero and every subbundle of degree zero of E is also contained in Bs (respectively Bps ). Proof It suffices to show this property for the category Bs . By duality, it ˜ be a vector bundle with strongly suffices to treat quotient bundles. So let E semistable reduction of degree zero on Xo , where X is a model of X. Denote ˜ and let by E the generic fibre of E, 0 → E 0 → E → E 00 → 0
(12.1)
be an exact sequence of vector bundles XCp , where E 00 has degree zero. By [DW2], Theorem 5, E 0 can be extended to a vector bundle F0 on Yo , where Y is a model of X such that there is a morphism ϕ : Y → X inducing an ˜ o Cp = Hom(E 0 , E), isomorphism on the generic fibres. Since Hom(F0 , ϕ∗o E)⊗ we may assume that the embedding E 0 → E can be extended to a OYo ˜ after changing the morphisms in the module homomorphism F0 → ϕ∗o E diagram (12.1).
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˜ → F00 → 0 Let F00 be the quasi-coherent sheaf on Yo such that F0 → ϕ∗o E 00 is exact. Then F is of finite presentation. Note that the generic fibre of this sequence is isomorphic to the sequence (12.1). Let r be the rank of E 00 . The same argument as in the proof of [DW2], Theorem 5 shows that the blowing-up ψo : Zo → Yo of the r-th Fitting ideal of F00 descends to a finitely presented morphism ψ : Z → Y inducing an isomorphism on the generic fibres. Besides, if A denotes the annihilator of the r-th Fitting ideal of ψo∗ F00 , the sheaf ψo∗ F00 /A is locally free by [RG] (5.4.3). Hence it gives rise to a vector bundle E00 on ˜ Then we have a natural Zo with generic fibre E 00 . Let us write E = ψo∗ ϕ∗o E. surjective homomorphism of vector bundles E → E00 on Zo extending the quotient map E → E 00 on the generic fibre XCp . Let K be a finite extension of Qp such that Z descends to a proper and flat scheme ZoK over the ring of integers oK . We can choose K big enough so that all irreducible components of the special fibre of Z are defined over the residue field of K. Let XK be the generic fibre of ZoK . Then XK ⊗K Qp ' X. The scheme Zo is the projective limit of all ZA = ZoK ⊗oK A, where A runs over the finitely generated oK -subalgebras of o. By [EGAIV], (8.5.2), (8.5.5), (8.5.7), (11.2.6) there exists a finitely generated oK -subalgebra A of o with quotient field Q ⊂ Cp such that E → E00 descends to a surjective homomorphism EA → E00A of vector bundles on ZA . Let x ∈ spec A be the point corresponding to the prime ideal A ∩ m in A, where m ⊂ o is the valuation ideal. If πK is a prime element in oK , we have oK /(πK ) ⊂ A/A ∩ m ⊂ o/m = k, so that A ∩ m is a maximal ideal in A. Hence x is a closed point with residue field κ = κ(x) which is a finite extension of oK /(πK ) in k. ˜ k on the special fibre Xk of X is By assumption, the vector bundle E strongly semistable of degree zero. By Proposition 12.2.4, strong semista˜k bility is preserved under pullbacks via k-morphisms, so that Ek = (ϕ ◦ ψ)∗ E k
is also strongly semistable of degree zero. The bundle Eκ = EA ⊗A κ satisfies Eκ ⊗κ k ' Ek , hence it is strongly semistable of degree zero on Zκ = ZA ⊗A κ. Let C1 , . . . , Cr be the irreducible components of Zκ with normalizations ˜ C1 , . . . , C˜r and denote by αi : C˜i → Ci → Zκ the natural map. Since the Euler characteristics are locally constant in the fibres of the flat and proper 00 = 0. A-scheme ZA , we find deg E00κ = deg EQ By the degree formula in [BLR], 9.1, Proposition 5, deg(E00κ ) is a linear combination of the deg(αi∗ E00κ )’s with positive coefficients. Since αi∗ E00κ is a
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quotient bundle of the semistable degree zero vector bundle αi∗ Eκ on C˜i , it has degree ≥ 0. Hence for all i we find deg(αi∗ E00κ ) = 0. Now let F be a vector bundle on the smooth projective curve C˜i which is a quotient of αi∗ E00κ . Then F is also a quotient of the semistable degree zero bundle αi∗ Eκ , which implies deg(F) ≥ deg αi∗ Eκ = 0. This shows that αi∗ E00κ is semistable of degree 0 on C˜i . The same argument applies to all Frobenius pullbacks of αi∗ E00κ , so that αi∗ E00κ is strongly semistable of degree 0. Hence E00κ is strongly semistable of degree zero on Zκ . By [HL], 1.3.8 the base change E00k = E00κ ⊗κ k is also strongly semistable of degree zero. Since E00 has generic fibre E 00 , it follows that E 00 is indeed contained in Bs . Corollary 12.3.4. Bs and Bps are abelian categories. Proof Recall that Bs and Bps are full subcategories of the abelian category T ss of semistable vector bundles of degree zero on XCp . Since the trivial bundle is contained in Bs and Bps , and both categories are closed under direct sums by [DW1], Proposition 9, they are additive. By the theorem, Bs and Bps are also closed under kernels and cokernels, hence they are abelian categories.
12.4 Tannakian categories of vector bundles In this section we look at several categories of semistable vector bundles from a Tannakian point of view. Useful references in this context are [DM] and [S] for example. As before let X be a smooth projective curve over Qp with a base point x ∈ X(Cp ). We call a vector bundle on XCp polystable of degree zero if it is isomorphic to the direct sum of stable vector bundles of degree zero. Let ss be the strictly full subcategory of vector bundles on X Tred Cp consisting of ps ∩ T ss . Then we have polystable bundles of degree zero and set Bps = B red red the following diagram of fully faithful embeddings ps Bps red ⊂ B ∩ ∩ ss Tred ⊂ T ss
(12.2)
Note that because of theorem 12.3.3, every vector bundle E in Bps red is the ps direct sum of stable vector bundles of degree zero contained in B . ss and Bps are closed under taking subLemma 12.4.1. The categories Tred red quotients in T ss .
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Proof Since Bps is closed under subquotients in T ss by theorem 12.3.3, it L ss . A bundle E in T ss can be written as E = suffices to consider Tred i Ei red 00 with Ei stable of degree zero. Let ϕ : E → E be a surjective map in T ss . Then the images ϕ(Ei ) lie in T ss and the surjective map ϕ|Ei : Ei → ϕ(Ei ) is an isomorphism or the zero map since Ei is stable. Hence we have E 00 = P 00 00 j Ej for stable degree zero vector bundles Ej (the nonzero ϕ(Ei )). For every j we have Ej00 ∩
X
Ek00 = Ej00
or = 0
k6=j
since the bundle Ej00 being stable is a simple object of Tss . It follows that E 00 ss . The case of is the direct sum of suitably chosen Ej00 ’s and hence lies in Tred subobjects follows by duality.
Consider the fibre functor ωx on T ss defined by ωx (E) = Ex and ωx (f ) = fx . It induces fibre functors on the other categories as well. ss with the fibre ps ss Theorem 12.4.2. a The categories Bps red , B , Tred and T functor ωx are neutral Tannakian categories over Cp . ps ss b The categories Bps red and Tred are semisimple. Every object in B (resp. ps ss ). T ss ) is a successive extension of objects of Bred (resp. Tred ps ss c The natural inclusion B ⊂ T is an equivalence of categories if and ss only if Bps red ⊂ Tred is an equivalence of categories. ss this is well known, see e.g. [Si], p. 29. The Proof a For T ss and Tred ps ps categories B and Bred are abelian by corollary 12.3.4 and lemma 12.4.1. It was shown in [DW2] that Bps is closed under tensor products and duals. ps ps ps ss The same follows for Bps red = B ∩ Tred . Faithfulness of ωx on B and Bred follows because ωx is faithful on T ss . Alternatively a direct proof was given in [DW2] Theorem 36. ss b Every object in Bps red and Tred is the direct sum of simple objects since stable bundles are simple. It is well known that objects of T ss are successive extensions of stable bundles of degree zero. Since subquotients in T ss of objects in Bps lie in Bps by theorem 12.3.3, the corresponding assertion for Bps follows. c This is a consequence of b because both T ss and Bps are closed under extensions, c.f. [DW2].
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Let o Gps red O
Gps O (12.3)
o Gss red
Gss
be the diagram of affine group schemes over Cp corresponding to diagram (12.2) by Tannakian duality. Proposition 12.4.3. All morphisms in (12.3) are faithfully flat. The conss nected components of Gps red and Gred are pro-reductive. Proof The following is known [DM] Proposition 2.21: A fully faithful ⊗-functor F : C → D of neutral Tannakian categories over a field k of characteristic zero induces a faithfully flat morphism F ∗ : GD → GC of the Tannakian duals if and only if we have: Every subobject in D of an object F (C) for some C in C is isomorphic to F (C 0 ) for a subobject C 0 of C. This criterion can be verified immediately for the functors in (12.2) by using either theorem 12.3.3 or lemma 12.4.1. The second assertion of the proposition follows from [DM] Proposition 2.23 and Remark 2.28. Let Tfin be the category of vector bundles on XCp which are trivialized by a ∼ finite ´etale covering of XCp . Since π1 (XCp , x) −→ π1 (X, x) is an isomorphism it follows that Tfin is equivalent to the category of representations of π1 (X, x) with open kernels on finite dimensional Cp -vector spaces V , c.f. [LS], 1.2. Such a representation factors over a finite quotient G of π1 (X, x) and the corresponding bundle in Tfin is E = XC0 p ×G V. Here α : X 0 → X is the Galois covering corresponding to the quotient π1 (X, x) → G and V is the affine space over Cp corresponding to V . With the fibre functor ωx the category Tfin is neutral Tannakian over Cp with Tannaka dual π1 (X, x)/Cp = lim(π1 (X, x)/N )/Cp . ←− N
Here N runs over the open normal subgroups of π1 (X, x) and for a finite (abstract) group H we denote by H/Cp the corresponding constant group scheme. Using Maschke’s theorem it follows that Tfin is semisimple. Proposition 12.4.4. The category Tfin is a full subcategory of Bps red . The ps induced morphism Gred π1 (X, x)/Cp is faithfully flat. Proof Obviously, Tfin is a full subcategory of Bps . Let E be a vector bundle
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in Tfin . Since a finite ´etale pullback of E is trivial and hence polystable, E ss . It is also polystable by [HL], Lemma 3.2.3. Hence E is an object of Tred ps ss . The next assertion follows that Tfin is a full subcategory of Bred = Bps ∩Tred ps follows from fully faithfulness of Tfin ,→ Bred since Bps red is semisimple, c.f. [DM], Remark 2.29. Consider a Galois covering α : X 0 → X with group Gal(X 0 /X) of smooth projective curves over Qp and choose a point x0 ∈ X 0 (Cp ) above x ∈ X(Cp ). ps ss ss of vector Let us write CX for any of the categories Bps red , B , Tred and T ∗ bundles on XCp . The pullback functor α : CX → CX 0 is a morphism of neutral Tannakian categories over Cp commuting with the fibre functors ωx and ωx0 . Let i : GX 0 → GX be the morphism of Tannaka duals induced by α∗ . We also need the faithfully flat homomorphism obtained by composition: q : GX π1 (X, x)/Cp Gal(X 0 /X)/Cp . Here the second arrow is determined by our choice of x0 . Note that every σ in Gal(X 0 /X) induces an automorphism σ ∗ of CX 0 and hence an automorphism σ : GX 0 → GX 0 of group schemes over Cp . Lemma 12.4.5. There is a natural exact sequence of affine group schemes over Cp i
q
1 → GX 0 − → GX − → Gal(X 0 /X)/Cp → 1 . Proof Every bundle E 0 in CX 0 is isomorphic to a subquotient of α∗ (E) for some bundle E in CX . Namely, thinking of E 0 as a locally free sheaf, the L sheaf E = α∗ E 0 is locally free again and we have α∗ E ∼ where σ = σ σ∗ E 0L 0 ∗ ∼ runs over Gal(X /X). Incidentally, E lies in CX because α E = σ σ ∗ E 0 ss and hence for Bps it lies in CX 0 . This is clear for C = T ss or Bps . For Tred red follows from [HL] Lemma 3.2.3. It follows from [DM] Proposition 2.21 (b) that i is a closed immersion. By descent the category CX is equivalent to the category of bundles in CX 0 equipped with a Gal(X 0 /X)-operation covering the one on X 0 . In other words, the category of representations of GX is equivalent to the category of representations of GX 0 together with a Gal(X 0 /X)-action, i.e. a transitive system of isomorphisms σ ∗ ρ = ρ ◦ σ → ρ for all σ in Gal(X 0 /X). Hence, GX is an extension of GX 0 by Gal(X 0 /X)/Cp inducing the above Gal(X 0 /X)action on GX 0 . (Because such an extension has the same ⊗-category of representations.) In particular, the sequence in the lemma is exact.
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For a commutative diagram of Galois coverings X 00 B
BB BB BB B!
x0
X
/ X0 | | || || | |~
00 00 and the choice of points ∈ p ) and x ∈ X (Cp ) over x we get a commutative diagram of affine group schemes over Cp :
1
1
X 0 (C
/ GX 00 _
/ GX
/ Gal(X 00 /X)/C p
/0
/ GX 0
/ GX
/ Gal(X 0 /X)/C p
/0.
Passing to the projective limit, we get an exact sequence 1 −→ lim GX 0 −→ GX −→ π1 (X, x)/Cp −→ 1 . ←−0
(12.4)
X
Right exactness follows from propositions 12.4.3 and 12.4.4. If G is a group scheme, we denote by G0 its connected component of identity. Theorem 12.4.6. We have a commutative diagram 1
/ (Gss )0 red
/ Gss red
/ π1 (X, x)/C p
/1
1
/ (Gps )0 red
/ Gps red
/ π1 (X, x)/C p
/1
In particular π1 (X, x)/Cp is the common group scheme of connected compops nents of both Gss red and Gred . Moreover we have: 0 lim Gss (Gss red ) = ← −0 red,X 0 X
Here
X 0 /X
and
0 . (Gps lim Gps red ) = ← −0 red,X 0 X
runs over a cofinal system of pointed Galois covers of (X, x).
ss Proof Let CX denote either Bps red,X or Tred,X and let GX be its Tannaka dual. The exact sequence (12.4) implies that G0X ⊂ limX 0 GX 0 . Hence it suffices ←− to show that limX 0 GX 0 is connected. The category of finite dimensional ←− representations of limX 0 GX 0 on Cp -vector spaces is limX 0 CX 0 . In order to ←− −→ show that limX 0 GX 0 is connected, by [DM] Corollary 2.22 we have to prove ←− the following: Claim Let A be an object of limX 0 CX 0 . Then the strictly full subcategory −→
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[[A]] of limX 0 CX 0 whose objects are isomorphic to subquotients of AN , N ≥ 0 −→ is not stable under ⊗ unless A is isomorphic to a trivial bundle. Proof Let [[A]] be stable under ⊗. The category limX 0 CX 0 is semisimple −→ since (limX 0 GX 0 )0 = limX 0 G0X 0 is pro-reductive. Hence we may decompose ←− ←− A into simple objects A = A1 ⊕ . . . ⊕ As . By assumption, for every j ≥ 1 N for some N = N (j). the object A⊗j 1 is isomorphic to a subquotient of A The same argument as in the proof of Lemma 12.4.1 shows that up to isomorphism the subquotients of N A := AN have the form m1 A1 ⊕. . .⊕ms As P for integers mi ≥ 0. Hence we get isomorphisms where means “direct sum”: s X ∼ A⊗j mij Ai for 1 ≤ j ≤ r . 1 = i=1
Here M = (mij ) is an s × r-matrix over Z. Fixing some r > s there is a relation with integers cj , not all zero: r X
cj (m1j , . . . , msj )t = 0 .
j=1
This gives the relation r X
t c+ j (m1j , . . . , msj )
=
j=1
r X
t c− j (m1j , . . . , msj )
j=1
− where c+ j = max{cj , 0} and cj = − min{cj , 0}. “Left multiplication” with (A1 , . . . , As ) gives isomorphisms r X j=1
c+ j
s X
mij Ai ∼ =
i=1
r X j=1
c− j
s X
mij Ai ,
i=1
and hence r X
⊗j ∼ c+ j A1 =
j=1
For the polynomials P ± (T ) = P + 6= P − and:
r X
⊗j c− j A1 .
j=1 ± j j=1 cj T
Pr
with coefficients in Z≥0 we have
P + (A1 ) ∼ = P − (A1 ) . Let E1 be a bundle in CX 0 representing A1 in limX 0 CX 0 . Then we have an −→ isomorphism P + (β ∗ E1 ) ∼ = P − (β ∗ E1 )
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of vector bundles on a suitable Galois cover β : X 00 → X 0 . A theorem of Weil, c.f. [W] Ch. III or [N], now implies that β ∗ E1 and hence E1 is trivialized by a finite ´etale covering of X 0 . Hence A1 , the class of E1 is isomorphic in limX 0 CX 0 to a trivial bundle. The same argument applies to −→ A2 , A3 , . . . Hence A is isomorphic to a trivial bundle as well. This proves the claim and hence the theorem. ss We now determine the structure of Gab for G = Gps red and G = Gred . In either case, this group is pro-reductive and abelian, hence diagonalizable and therefore determined by its character group
X(Gab ) = MorCp (Gab , Gm ) . The characters of Gab correspond to isomorphism classes of one-dimensional representations of G i.e. to isomorphism classes of degree zero line bundles ss in Bps red resp. Tred . Since both categories contain all degree zero line bundles we get X(Gab ) = Pic0X (Cp ) and hence Gab = Hom(Pic0X (Cp ), Gm,Cp ) = lim Hom(A, Gm,Cp ) . ←− A
Here A runs over the finitely generated subgroups of Pic0X (Cp ). A similar 0 ss 0 argument using the fact that (Gps red ) resp. (Gred ) is the Tannaka dual of ps ss limX 0 Bred,X 0 resp. limX 0 Tred,X 0 shows the following: For G as above, the −→ −→ group (G0 )ab is diagonalizable with character group X((G0 )ab ) = lim Pic0X 0 (Cp ) . −→0 X
Note that the right hand group is torsionfree because line bundles of finite order become trivial in suitable finite ´etale coverings. This corresponds to the fact that (G0 )ab is connected. We can therefore write as well: X((G0 )ab ) = lim(Pic0X 0 (Cp )/tors) −→0 X
and hence (G0 )ab = lim Hom(Pic0X 0 (Cp )/tors, Gm,Cp ) . ←−0 X
This is a pro-torus over Cp . 0 ss 0 In particular we have seen that (Gps red ) and (Gred ) have the same maximal
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abelian quotient. Incidentally we may compare (G, G0 ) with (G0 , G0 ): There is an exact sequence: 1 −→ (G, G0 )/(G0 , G0 ) −→ (G0 )ab −→ (Gab )0 −→ 1 . The sequence of character groups is 1 −→ Pic0X (Cp )/tors −→ lim(Pic0X 0 (Cp )/tors) −→ X((G, G0 )/(G0 , G0 )) −→ 1 . −→0 X
If X is a curve of genus g(X) ≥ 2 then by the Riemann Hurwitz formula, the middle group is infinite dimensional and in particular (G, G0 )/(G0 , G0 ) contains a non-trivial pro-torus. If X is an elliptic curve, it follows from Atiyah’s classification in [At] that every stable bundle of degree 0 on X is in fact a line bundle. Hence ss = Bps is the full subcategory of all vector bundles which can be deTred red composed as a direct sum of line bundles of degree 0. This implies that the corresponding Tannaka group G is abelian and diagonalizable with character group Pic0X (Cp ), which fits in the picture above. We proceed with some remarks on the structure of G0 which follow from the general theory of reductive groups. Let C be the neutral component of the center of G0 . Then we have G0 = C · (G0 , G0 ) . Here C is a pro-torus and (G0 , G0 ) is pro-semisimple. The projection C → (G0 )ab is faithfully flat and its kernel is a commutative pro-finite group scheme H. In the exact sequence: 1 −→ X((G0 )ab ) −→ X(C) −→ X(H) −→ 1 the group X((G0 )ab ) is divisible because Pic0X 0 (Cp ) is divisible. Since X((G0 )ab ) is also torsionfree it is a Q-vector space. The group X(H) being torsion it follows that X(C) ⊗ Q = X((G0 )ab ) canonically and X(C) ∼ = X((G0 )ab ) ⊕ X(H) non-canonically. It would be interesting to determine X(H) for both ss G = Gps red and Gred . We end with a remark on the Tannaka dual GE of the Tannaka subcatps egory of Bps red generated by a vector bundle E in Bred . The group GE is a subgroup of GLEx the linear group over Cp of the Cp -vector space Ex . It can be characterized as follows: The group GE (Cp ) consists of all g in GL (Ex ) with g(sx ) = sx for all n, m ≥ 0 and all sections s in Γ(XCp , (E ∗ )⊗n ⊗ E ⊗m ) = HomXCp (E ⊗n , E ⊗m ) .
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Here g(sx ) means the extension of g to an automorphism g of (Ex∗ )⊗n ⊗ Ex⊗m applied to sx . Consider the representation attached to E by theorem 12.3.2: ρE,x : π1 (X, x) −→ GL (Ex ) . Its image is contained in GE (Cp ) because the functor F 7→ ρF,x on Bps red is compatible with tensor products and duals and maps the trivial line bundle to the trivial representation. Hence GE contains the Zariski closure of Im ρE,x in GLEx . It follows from a result by Faltings [F] that the faithful functor F 7→ ρF,x is in fact fully faithful. If ρE,x is also semisimple then a standard argument shows that GE is actually equal to the Zariski closure of Im ρE,x .
References [At] Atiyah, M.F.: Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452 ¨tkebohmert and M. Raynaud: N´eron models. Springer [BLR] Bosch,S., W. Lu 1990 [DM] Deligne, P. and J.S. Milne: Tannakian categories. LNM 900. Springer 1982 [DW1] Deninger, C. and A. Werner: Line bundles and p-adic characters. In: G. van der Geer, B. Moonen, R. Schoof (eds.): Number Fields and Function Fields - Two Parallel Worlds. Birkh¨auser 2005, 101-131. [DW2] Deninger, C. and A. Werner: Vector bundles on p-adic curves and par´ Norm. Sup. 38 (2005), 553-597 allel transport. Ann. Scient. Ec. [Du] Ducrohet, L.: The action of the Frobenius map on rank 2 vector bundles over a supersingular genus 2 curve in characteristic 2. Preprint 2005. http://www.arXiv.math.AG/0504500 ´ ements de G´eom´etrie ´: El´ [EGAIV] Grothendieck, A. and J. Dieudonne Alg´ebrique IV, Publ. Math. IHES 20 (1964), 24 (1965), 28 (1966), 32 (1967) [F] Faltings, G.: A p-adic Simpson correspondence. Preprint 2003 [Gie1] Giesecker, D.: Stable vector bundles and the Frobenius morphism. Ann. ´ Norm. Sup. 6 (1973), 96–101 Scient. Ec. [Gie2] Giesecker, D.: On a theorem of Bogomolov on Chern classes of stable bundles., Am. J. Math. 101 (1979), 79–85 [Ha] Hartshorne, R.: Algebraic Geometry, Springer 1977 [HL] Huybrechts, D. and M. Lehn: The geometry of moduli spaces of sheaves Viehweg 1997 [JRXY] Joshi, K., S. Ramanan, E. Z. Xia and J.-K. Yu: On vector bundles destabilized by Frobenius pull-back, preprint 2002. http://www.arXiv.math.AG/0208096 [LP] Laszlo,Y. and C. Pauly: The action of the Frobenius map on rank 2 vector bundles in characteristic 2, preprint 2004 http://www.arXiv.math.AG/0005044 [LS] Lange, H. and U. Stuhler: Vektorb¨ undel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1977), 73–83
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[Liu] Q. Liu: Algebraic Geometry and Arithmetic Curves. Oxford University Press 2002 [N] Nori, M.V.: On the representations of the fundamental group, Composition Math. 33 (1976), 29–41 [NS] Narasimhan, M.S. andC.S. Seshadri: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 82 (1965), 540–567 [RG] Raynaud, M. and L. Gruson: Cri`eres de platitude et de projectivit´e. Invent. Math. 13 (1971), 1–89 [S] Serre, J.-P: Propri´et´es conjecturales des groupes de Galois motiviques et des repr´esentations l-adiques. In: U. Jannsen et al. (eds.): Motives. Proc. Symp. Pure Math. 55 vol. 1. AMS 1994, 377-400. [Si] Simpson, C.: Higgs bundles and local systems. Publ. Math. IHES 75 (1992), 5–92 [Ta] Tate, J.: p-divisible groups, in Proceedings of a Conference on local fields, Driebergen 1966, 158–183 [W] Weil, A.: G´en´eralisation des fonctions ab´eliennes, J. de Math. P. et App., (IX) 17 (1938), 47–87
13 On finite-dimensional motives and Murre’s conjecture Uwe Jannsen NWF I - Mathematik Universit¨ at Regensburg 93040 Regensburg GERMANY
[email protected]
To Jacob Murre
13.1 Introduction The conjectures of Bloch, Beilinson, and Murre predict the existence of a certain functorial filtration on the Chow groups (with Q-coefficients) of all smooth projective varieties, whose graded quotients only depends on cycles modulo homological equivalence. This filtration would offer a rather good understanding of these Chow groups, and would allow to prove several other conjectures, like Bloch’s conjecture on surfaces of geometric genus 0. In Murre’s formulation (cf. 13.5.1 below) one can check the validity of the conjecture for particular smooth projective varieties, and in fact, a slightly weaker form of the conjecture has been proved for several cases, e.g., for surfaces [Mu1] and several threefolds [GM] (proving parts (A), (B) and (D) of the conjecture, and giving evidence for (C)). But to my knowledge, there are few results for higher-dimensional varieties, and the strongest form of Murre’s conjecture (including part (C)) is only known for curves, rational surfaces, and, trivially, for Brauer-Severi varieties. The first aim of this paper is to exhibit some cases, where the full Murre conjecture can be shown. The positive aspect is that we get this for some non-trivial cases of varieties of higher (in fact arbitrarily high) dimension, the negative aspect is that we get this just for some special varieties and special ground fields. In particular, not over some universal domain. As a sample, we get the following:
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Theorem 13.1.1. Let k be a rational or elliptic function field (in one variable) over a finite field F. Let X0 be an arbitrary product of rational and elliptic curves over F, and let X = X0 ×F k. Then Murre’s conjecture holds for X. One ingredient is the notion of “finite-dimensionality” of motives, as introduced independently by Kimura [Ki] and O’Sullivan [OSu]. Up to now it is only known that the Chow motive of a smooth projective variety is finite-dimensional, if it lies in the tensor category generated by the motives of abelian varieties. But I believe that this notion will be fundamental for further progress on Chow groups, and motivic cohomology in general, in view of the nilpotence properties it implies. Therefore a second aim of this paper is to investigate this property in several directions. We add some observations to the existing results (cf. [An1] for a survey) which may be interesting in their own right, but also bear on our investigation of Murre’s conjecture. In particular, we are interested in Chow endomorphisms and nilpotence results: For a smooth projective variety X of pure dimension d let CHd (X × X)Q be the ring of Chow selfcorrespondences, i.e., the endomorphism ring End(hrat (X)) of the Chow motive hrat (X) associated to X, and let J(X) = CHd (X × X)Q, hom be the ideal of homologically trivial correspondences. Then we get the following (a similar result appeared in [DP]): Theorem 13.1.2. Let X be a smooth projective variety over a field k, and X be the projector onto the even degree part of the cohomology. Then let π+ the following properties are equivalent. X is algebraic and J(X N ) is nilpotent for all N > 0. i) π+ X is algebraic and J(X N ) is a nil ideal for all N > 0. ii) π+ iii) X is finite-dimensional (i.e., h(X), the motive of X, is finitedimensional).
The implication from (iii) to (ii) is Kimura’s nilpotence theorem. The other implications give a certain converse. The following again sharpens results of Kimura. Theorem 13.1.3. Let M be a motive modulo rational equivalence over a field k, and assume that M is either oddly or evenly finite-dimensional. For an endomorphism f ∈ End(M ) let P (t) = det( f | H ∗ (M, Q` ) ) be the characteristic polynomial of f acting on the `-adic cohomology of M (` 6= char(k)). Then P (f ) = 0 in End(M ).
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This fact (which in the even case was also proved by O’Sullivan) is somewhat surprising, because a priori the equality P (f ) = 0 only holds modulo homological equivalence. Coming back to Murre’s conjecture, it is well-known by now that [GP] that part (A) (the Chow-K¨ unneth decomposition) follows from the standard conjecture on algebraicity of the Kuenneth components, and finitedimensionality. In particular, this applies to abelian varieties over arbitrary ground fields. In this paper, we explore which additional ingredients can give the remaining part of Murre’s conjecture. We start with the case of a finite ground field F. Here it is known by work of Geisser [Gei] and Kahn [Ka] that the conjunction of finite-dimensionality and Tate’s conjecture “implies everything”. In particular it implies that rational and numerical equivalence agrees (with Q-coefficients). Evidently, the latter also implies Murre’s conjecture. However, we are interested in getting some unconditional theorems, and hence we take some pain to single out the minimal conditions to get such results. In particular, we don’t want to argue with Tate’s conjecture for all varieties, but want to get by with conditions just on the given variety X (For this, we have to rectify some statements in the literature.) As a sample, we get: Theorem 13.1.4. Let X be a smooth projective variety over the finite field F. Assume that the ideal J(X) is nilpotent (e.g., assume that h(X) is finitedimensional). Fix an integer j ≥ 0 and assume that the Tate conjecture holds for H 2j (X ×F F¯ , Q` (j)), and that the Frobenius eigenvalue 1 is semi-simple on H 2j (X ×F F¯ , Q` (j)). Then the cycle map induces an isomorphism ∼
CHj (X)Q ⊗Q Q` −→ H 2j (X ×F F , Q` (j))Gal(F /F ) , i (X, Z(j)) is of finite exponent for all i 6= 2j. and the motivic cohomology HM
Corollary 13.1.5. Let X be a smooth projective variety of pure dimension d over the finite field F , and assume that h(X) is finite-dimensional. Then i (X, Z(d)) has finite exponent for all i 6= 2d. HM This generalizes results of Soul´e [So1]. The theorems on function fields over F (like Theorem 13.1.1) are obtained by considering arbitrary (not necessarily smooth or projective) varieties over F and passing to certain limits of `-adic cohomology, as in [Ja1]. The results in this paper were mainly obtained during a stay at the University of Tokyo during the academic year 2003/2004, and it is my pleasure to thank the department and my host Takeshi Saito for the invitation and
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the hospitality. I thank the referee for suggesting a more elegant version and proof of Theorem 13.4.3(a).
13.2 Weil cohomology theories, motives, tensor categories In this section, we recall some notions and properties needed later. We fix a base field k and consider the category SPk of smooth projective varieties over k. For X in SPk , we denote by Z j (X) the group of algebraic cycles of codimension j on X, with Q-rational coefficients. The following definition is equivalent to the one in [Kl]. Definition 13.2.1. Let E be a field of characteristic 0. An E-linear Weil cohomology theory H is a contravariant functor X 7→ H(X) , (f : X → Y ) 7→ f ∗ := H(f ) : H(Y ) → H(X) from SPk to the category of graded commutative E-algebras ( a.b = (−1)i+j b.a for a ∈ H i (X), b ∈ H j (X) ), together with the following data and properties: i) dimE H(X) < ∞, and H i (X) = 0 for i < 0 or i > 2 · dim X. ii) To each morphism f : X → Y there is functorially associated an E-linear map f∗ : H(X) → H(Y ) which is of degree dim Y – dim X if X and Y are irreducible. iii) (projection formula) f∗ (f ∗ y · x) = y · f∗ x for f : X → Y, x ∈ H(X) and y ∈ H(Y ). ∗ a · pr ∗ b iv) (K¨ unneth formula) The association a ⊗ b 7→ a × b := prX Y gives an isomorphism H(X) ⊗E H(Y ) → H(X × Y ). v) (Poincar´e duality) H(Spec k) = H 0 (Spec k) ∼ = E, and the bilinf∗ · ear pairing H(X) × H(X) −→ H(X) −→ H(Speck) ∼ = E , (a, b) 7→ ha, bi := f∗ (a.b), is non-degenerate. vi) (cycle map) There are cycle class maps c`j : Z j (X) → H 2j (X) compatible with products, pull-backs f ∗ and push-forwards f∗ , whenever these operations are defined on the algebraic cycles. Remarks 13.2.2. a) It follows that f∗ is the transpose of f ∗ under Poincar´e duality. b) Let X be a smooth projective variety of pure dimension d, and denote, as usual, by ∆X the cycle in Z d (X × X) corresponding to the diagonal X ,→ X × X, and also the associated cycles class in H 2d (X × X) for a given Weil cohomology theory H. The K¨ unneth components of the diagonal, πi ∈ H 2d−i (X) ⊗ H i (X) (i = 1, . . . , 2d) P are defined by decomposing ∆X = 2d unneth i=0 πi according to the K¨
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isomorphism (13.2.1)(iv). The standard conjecture C(X) predicts that the πi are algebraic, i.e., again classes of algebraic cycles. c) Let X and Y ∈ SPk . Any correspondence from X to Y , i.e., any element α ∈ Z ∗ (X × Y ), induces an E-linear map, again denoted α, from H(Y ) to H(X) by defining α(x) = (pY )∗ ((pX )∗ (x).c`(α)) for x ∈ H(Y ), where pX : X × Y → X and pY : X × Y → Y are the projections. The same already holds for any cohomology class α0 ∈ H ∗ (X × Y ) in place of c`(α). Via this interpretation, the element πi , as cohomological correspondence from X to X, is the identity on H i (X), and zero on H j (X) for j 6= i. Examples 13.2.3. The following are examples of Weil cohomology theories: a) If k = C: singular cohomology H(X) = H ∗ (X(C), Q) (E = Q). ∗ (X × k, Q ), for ` 6= char k b) For arbitrary k: `-adic cohomology Het k ` (E = Q` ). ∗ (X/k), (E = k). c) If char k = 0: de Rham cohomology HdR d) If k is a perfect field of characteristic p > 0: crystalline cohomol∗ (X/B(k)) := H ∗ (X/W (k)) ⊗ ogy Hcrys W (k) B(k) (E = B(k) := crys Frac(W (k)). 13.2.4. Let ∼ be an adequate equivalence relation on algebraic cycles, i.e., an equivalence relation on all cycle groups Z j (X) for all X in SPk such that product, push-forward and pull-back of cycles is well-defined on the cycle groups Aj∼ (X) := Z j (X)/ ∼ [Ja3]. We recall that we have the adequate equivalence relations rational, algebraic, homological and numerical equivalence with the relationship α ∼rat 0 ⇒ α ∼alg 0 ⇒ α ∼hom 0 ⇒ α ∼num 0 The category M∼ (k) of (Q-rational) motives modulo ∼ over k can be defined as follows. For X, Y ∈ SPk the group of correspondences (modulo ∼) dim(Xi )+n of degree n from X to Y is defined as Corrn∼ (X, Y ) = ⊕i A∼ (Xi × Y ), where the Xi are the irreducible components of X. The composition of corm respondences f ∈ Corrm ∼ (X, Y ) and g ∈ Corr∼ (Y, Z) is defined as g ◦ f = (pXZ )∗ (p∗XY (f ).p∗Y Z (g)) ∈ Corrm+n (X, Z), where pXZ , pXY and pY Z are the projections from X × Y × Z to X × Z, X × Y and Y × Z, respectively. Then the objects of M∼ (k) can be described as triples (X, p, m), with X ∈ SPk , p ∈ Corr0∼ (X, X) an idempotent and m ∈ Z, and one has Hom((X, p, m), (Y, q, n)) = q Corrn−m (X, Y )p, with composition given by
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the above composition of correspondences. The Tate objects are defined by 1(n) = (Spec k, id, n) for n ∈ Z. 13.2.5. The precise definition of a tensor category can be found in [DM]. Let us just recall that it is a category with a bifunctor (A, B) 7→ A ⊗ B together with asssociativity constraints ψA,B,C : A ⊗ (B ⊗ C) ∼ = (A ⊗ B) ⊗ C, commutativity constraints φA,B : A ⊗ B ∼ = B ⊗ A and an unity constraint l : 1⊗A ∼ = A, satisfying certain compatibilities modelled after the situation of the tensor product of vector spaces. A tensor category is called rigid, if it has internal Homs Hom(A, B) (characterized by Hom(A ⊗ B, C) = Hom(A, Hom(B, C))) satisfying some reasonable properties [DM, 1.7]. In this case, the dual of an object is defined as A∨ = Hom(A, 1). Examples 13.2.6. a) In particular, let E be a field. Then the category VecE of finite-dimensional E-vector spaces is a rigid E-linear tensor category, with the usual tensor product and the obvious constraints. b) The category GrVecE of finite-dimensional (Z-) graded E-vector spaces V ∗ is a rigid E-linear tensor category by defining (V ⊗ W )r = ⊕i+j=r V i ⊗E W j , taking the associativity constraints from VecE , defining 1 = E placed in degree 0, and defining φA,B (a ⊗ b) = (−1)i+j b ⊗ a for a ∈ Ai and b ∈ B j . The relationship between the objects introduced in 13.2.2, 13.2.3 and 13.2.4 is as follows. 13.2.7. For any adequate equivalence relation ∼, the category M∼ (k) of motives modulo ∼ becomes a rigid Q-rational tensor category by defining (X, p, m) ⊗ (Y, q, n) = (X × Y, p × q, m + n), and taking the obvious associativity constraint, the unit object 1 = (Spec(k), id, 0), and the commutativity constraints induced by the transpositions τX,Y : X × Y ∼ = Y × X. Recall that a tensor functor Φ : A → B between tensor categories is a functor together with functorial isomorphisms αA,B : Φ(A) ⊗ Φ(B) ∼ = Φ(A ⊗ B), satisfying some obvious compatibilities with respect to the constraints [DM, 1.8] . Then one has: Lemma 13.2.8. Let E be a field. Giving an E-linear Weil cohomology theory H is the same as giving a tensor functor Φ : Mrat (k) −→ GrVecfF with Φ(1(−1)) of degree 2. This is well-known, and the proof is straightforward (cf. [An2, 4.1.8.1]): Given a Weil cohomology theory H we can extend it to a (covariant) functor
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on Mrat (k) by defining H ∗ (X, p, m) = pH ∗+m (X). Here we have used the fact that correspondences act on the cohomology, cf. 13.2.2 (ii), and this also gives the functoriality of this association. Conversely, given a tensor functor Φ, we can compose it with the functor SPk → Mrat (k) to obtain a Weil cohomology theory. Here, for a morphism f : X → Y , f∗ is induced by Φ(Γf ), where Γf is the graph of f , and f ∗ is induced by Φ(Γtf ), where Γtf is the transpose of Γf .
13.3 Finite-dimensional motives For any object M in a tensor category C, and every natural number N , the symmetric group SN acts on the N -fold tensor product M ⊗N as follows. For an elementary transposition (i, i + 1), 1 ≤ i < N , the induced isomorphism (i, i + 1)∗ of M ⊗N is induced by applying the commutativity constraint between the i-th and (i + 1)-st place, i.e., we have (i, i + 1)∗ = idM ⊗(i−1) ⊗ φM,M ⊗ idM ⊗(N −i−1) . One can check that one obtains a well defined action of Q SN by decomposing each element σ as a product σ = τν of such elementary Q transpositions, and defining σ∗ = (τν )∗ . Now let C be a Q-linear pseudoabelian tensor category. Then, by linearity, the group ring Q[SN ] acts on M ⊗N , and we can define the symmetric product as SymN M = esym M ⊗N V P and the exterior product as N M = e alt M ⊗N , where esym = N1 ! σ and P ealt = N1 ! sign(σ)σ, with the sum taken over all σ ∈ SN . Note that these are idempotents in Q[SN ], and that by the very definition of pseudo-abelian categories, every projector has an image in C. The following definition goes back to Kimura [Ki] and, independently, to O’Sullivan [OSu] (with a different terminology). Definition 13.3.1. An object M in a Q-linear pseudo-abelian tensor category C is called V i) evenly finite-dimensional, if there is some N > 0 with N M = 0, ii) oddly finite-dimensional, if there is some N < 0 with SymN M = 0, iii) finite-dimensional, if M = M+ ⊕M− with M+ evenly and M− oddly finite-dimensional, respectively. For such objects one has the following notion of dimension. Definition 13.3.2. If M is finite-dimensional, and if M+ and M− are as in definition 13.3.1 (iii), define the (Kimura-) dimension of M as dim M = V dim+ M +dim− M , where dim+ M := max{r | r M+ 6= 0} and dim− M := max{s | Syms M− 6= 0}.
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This is well-defined, because M+ and M− are unique up to (non unique) isomorphism (loc.cit.). Examples 13.3.3. a) If E is a field of characteristic zero, then any V object V in VecE is evenly finite-dimensional. In fact, n V is the Vn usual alternating power E V , and this is zero for n > dimE V . One has dim V = dimE V . b) With E as above, every object in the tensor category GrVecE (cf. 13.2.6 (b) for our conventions) is finite-dimensional: For V = ⊕i∈Z V i V let V+ = ⊕i even V i and V− = ⊕i odd V i . Then one has N Gr V+ = VN VN N E V+ and SymGr V− = E V− , by the sign rule for our commutation constraints. Thus dim V = dimE V is the usual dimension of V as an E-vector space (This should not be confused with the rank of V (cf. [DM, p.113]) with respect to the structure of GrVecE as a rigid tensor category, which would be rankV = dimE V+ − dimE V− ). Fix a Weil cohomology theory H ∗ . For a smooth projective variety X let πi = πiX be the K¨ unneth components of the diagonal, and let X π+ = π+ = π0 + π2 + π4 + . . .
,
X π− = π− = π1 + π3 + π5 + . . .
be the projectors onto the even and odd degree part of the cohomology, respectively. Then we have the ’sign conjecture’ X and π X are algebraic. Conjecture S(X) The projectors π+ −
It is implied by standard conjecture C(X) (cf. 13.2.2 (b)), and hence known for curves, surfaces and abelian varieties, and in general over finite fields. + + × πY+ + = πX Moreover S(X) and S(Y ) imply S(X × Y ) (because πX×Y πY− × πY− ). If S(X) holds (for the given H), then the motive hhom (X) modulo homological equivalence (for the given H) is finite-dimensional. In fact, decompose hhom (X) = M+ ⊕ M− , with M± = (hhom (X), π± ), and let b± (X) = dim H ∗ (M± ) = dim H ∗ (X)± , where H ∗ (X)± is defined as in 13.3.3(b). Then b+ (X)+1
^
M+ = 0 = Symb− (X)+1 M− .
because H ∗ : Mhom (k) → GrVecF is a faithful tensor functor). In particular, conjecture S(X) implies that also the motive hnum (X) modulo numerical equivalence is finite-dimensional. However, the following conjecture is much deeper:
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Conjecture 13.3.4. (Kimura, O’Sullivan) Every motive modulo rational equivalence is finite-dimensional. Remark 13.3.5. If M = (X, p, m) is a motive modulo the equivalence relation ∼, then Symn M = (X n , esym ◦ p⊗n , n · m). So this is 0 if and only if esym ◦ p⊗n ∼ 0. Here p⊗n = p × . . . × p on (X × X)n ∼ = X n × X n , and ⊗n ⊗n esym ◦ p = p ◦ esym , so this is again an idempotent. In fact, for every endomorphism f of h(X) and every σ ∈ Sn obviously σ ◦ f ⊗n = f ⊗n ◦ σ. V Similarly for n and ealt . Now let C be a smooth projective curve over k, and let x ∈ C be a closed point of degree m. Then we have a decomposition hrat (C) = 1 ⊕ h1rat (C) ⊕ 1 1 1(−1), with h1rat (C) := (C, ∆C − m x×C − m C × x, 0). As for the notation, 1 1 note that π ˜o := m x×C and π ˜2 := m C ×x are orthogonal idempotents lifting the K¨ unneth components π0 and π2 , respectively. Hence ∆C − π ˜0 − π ˜2 is an idempotent lifting the K¨ unneth component π1 = ∆C − π0 − π2 . Now 1 and 1(r), for every r ∈ Z, are evenly finite-dimensional, since S2 acts trivially on 1 ⊗ 1 and 1(r) ⊗ 1(r). The following results thus show that h(C) is finite-dimensional. Theorem 13.3.6. ([Ki, 4.2]) The motive h1rat (C) is oddly finite-dimensional. More precisely, one has Sym2g+1 h1rat (C) = 0 where g is the genus of C. Proposition 13.3.7. ([Ki]) Let M and N be objects in a Q-linear pseudoabelian tensor category. a) If M and N are finite-dimensional, then M ⊕N is finite-dimensional, with dim M ⊕ N ≤ dim M + dim N . b) If M and N are finite-dimensional, then M ⊗N is finite-dimensional, with dim M ⊗ N ≤ dim M. dim N . c) If M is finite-dimensional, then also every direct factor of M . d) M = 0 if and only if M is finite-dimensional with dim M = 0.
13.4 Nilpotence and finite-dimensionality For each smooth projective variety X over k, let J(X) ⊆ Corr0rat (X, X) be the ideal of correspondences which are numerically equivalent to zero. recall the following conjecture. Conjecture N(X) J(X) is a nilpotent ideal. A remarkable consequence of this conjecture would be that there is no phantom motive, i.e., no non-trivial motive which becomes zero after passing to numerical equivalence, and that every idempotent modulo numerical or
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homological equivalence can be lifted to an idempotent modulo rational equivalence. In fact, for any motive M modulo rational equivalence let J(M ) ⊆ End(M ) be the ideal of numerically trivial endomorphisms (so that J(X) = J(h(X)) for X in SPk ). Then we have Lemma 13.4.1. Assume that J(M ) is a nil ideal. Let Mnum and Mhom be the images of M in Mnum (k) and Mhom (k) (with respect to a given Weil cohomology), respectively. Then the following holds. i) If Mnum = 0 (e.g., if H ∗ (M ) = 0 for a Weil cohomology theory), then M = 0. ii) Any idempotent in End(Mnum ) or End(Mhom ) can be lifted to an idempotent in End(M ), and any two such liftings are conjugate by a unit of End(Mhom ) lying above the identity of End(Mnum ). iii) If the image of f ∈ End(M ) in End(Mnum ) is invertible, then so is f. Proof (i): If idM maps to zero in End(Mnum ), it is nilpotent, hence zero. (ii) and (iii): These properties holds for any surjection A A = A/I where A is a (not necessarily commutative) ring with unit, and I is a (two-sided) nil ideal. For (iii), it suffices to assume that the element a ∈ A maps to 1 ∈ A. But then a is unipotent, hence invertible. As for (ii), if e is idempotent in A and a is any lift in A, then (a − a2 )N = 0 for some N > 0, and it follows easily that e˜ = (1 − (1 − a)N )N is an idempotent lifting e ([Ki, 7.8]). If e and e0 are idempotents of A lying above e, then u = e0 e + (1 − e0 )(1 − e) lies above 1 ∈ A. Thus u is invertible, and the equality e0 u = e0 e = ue shows that e0 = ueu−1 . Conjecture N (X) would follow from the existence of the Bloch-Beilinson filtration [Ja2], or Murre’s conjecture [Mu1], or the following conjecture of Voevodsky: Conjecture 13.4.2. ([Voe]) If an algebraic cycle z is numerically trivial, it is smash nilpotent, i.e., there is an n > 0 such that z ×n = 0. In fact, as is observed in loc. cit., a smash nilpotent correspondence from X to X is nilpotent; more precisely, z ×n = 0 implies z n = 0 in Corr(X, X). The following result gives (in part (b)) another criterion for nilpotence. Here we may consider motives modulo any (fixed) adequate equivalence relation ∼. Recall that, for a motive M = (X, p, m) and an endomorphism f of M , the trace of f is defined as tr(f ) = hf.pt i, where hα.βi is the intersection number of two cycles α and β. This coincides with the trace coming from the rigid tensor category structure of M∼ (k)..
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Theorem 13.4.3. Let f : M → M be an endomorphism of a motive. P a) If ∧d+1 M = 0 (resp. Symd+1 M = 0), then di=0 (−1)d−i tr(∧d−i f )f i = Pd d−i f )f i = 0). 0 (resp. i=0 tr(Sym b) In particular, if M is either evenly finite-dimensional or oddly finitedimensional, and d = dim M , then there is a monic polynomial G(t) ∈ Q[t] of degree d with G(f ) = 0. If f is numerically equivalent to 0, then f is nilpotent, viz., f d = 0. This was originally proved by Kimura [Ki] in a slightly weaker form giving (b) with d + 1 instead of d, and not giving the description (a) of G(t). The following corollaries already follow from this original form, except for 13.4.7. Corollary 13.4.4. If M is a finite dimensional motive, then the ideal JM ⊆ End(M ) of numerically trivial endomorphisms is nilpotent. In fact, by decomposing M = M+ ⊕ M− , it is shown in [Ki] that J(M ) is a nil ideal, with degree of nilpotence bounded by n = (dim+ M. dim− M + 1).max(dim+ M, dim− M ) + 1. By a result of Nagata-Higman (cf. [AK, 7.2.8)] it follows that J(M ) is in fact a nilpotent ideal, of nilpotence degree ≤ 2n − 1 (since we assume Q-coefficients). Corollary 13.4.5. If M is a finite-dimensional motive, and H is any F P rational Weil cohomology theory, then dim M = i∈Z dimF H i (M ). In particular, the right hand side is independent of H. Proof (cf. [Ki, 3.9 and 7.4]) We may assume that M is either evenly or oddly finite-dimensional. Obviously, the dimension decreases under any tensor functor, so dim M ≥ dimF H(M ). On the other hand, by the nilpotence reV V V sult (together with 13.3.7 and 13.4.1), r M = 0 if r H(M ) = H( r M ) = 0, similarly for Symr . Thus dim M ≤ dimF H(M ). Corollary 13.4.6. (compare 13.3.7) If M and N are finite-dimensional motives, then dim(M ⊕ N ) = dim M + dim N and dim M ⊗ N = dim M · dim N . P In fact, this holds for dimF H i (−). For d = 1 Theorem 13.4.3 implies: i∈Z
Corollary 13.4.7. If M is a finite-dimensional motive with dim M = 1, then J(M ) = 0, i.e., on End(M ) numerical and rational equivalence coincide. Moreover, End(M ) = Q.
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Examples 13.4.8. a) If M is an evenly (resp. oddly) finite-dimensional V motive of dimension d, then d M (resp. Symd M ) is one-dimensional. In fact, by 13.4.5 it suffices to show this after applying some Weil cohomology theory, and then it holds, again by 13.4.5. b) If C is a curve of genus g, then h1rat (C) is oddly finite-dimensional with dim h1rat (C) = 2g: This follows from 13.4.6, 13.4.5, and the fact that for the `-adic cohomology, ` 6= char k, one has dimQ` H 1 (C ×k k, Q` ) = 2g. Moreover Sym2g h1rat (c) ∼ = 1(−g) : First of all, Sym2g h1rat (C) is one-dimensional, by (a). Then, by 13.4.4 and 13.4.1, we only have to show this isomorphism modulo (some) homological equivalence. But one knows that h1hom (C) ∼ h1 (Jac(C)), where Jac(C) is the JacoV2g =1 hom 2g bian of C, and that hrat (Jac(C)) ∼ = hrat (Jac(C)) ∼ = 1(−g). Here we have used the fact that Jac(C) is an abelian variety of dimension g, that for an abelian variety A the K¨ unneth components πi of the i diagonal are algebraic, and that for hhom (A) := hhom (A, πi ) one has a Vi 1 canonical isomorphism hihom (A) ∼ hhom (A). = We can deduce a certain converse of Theorem 13.4.3. Consider the following, a priori weaker variant of conjecture N (X) (for a smooth projective variety X). Conjecture (N0 (X)). J(X) is a nil ideal. Corollary 13.4.9. Let X be a smooth projective variety X, and let H ∗ be any Weil cohomology theory. Then the following statements are equivalent, where S(X) is meant with respect to H ∗ : a) h(X) is finite-dimensional. b) S(X) holds, and N (X n ) holds for all n ≥ 1. c) S(X) holds, and N 0 (X n ) holds for all n ≥ 1. Proof If M = h(X) = M+ ⊕ M− , where M+ = h(X, p+ ) (resp. M− = h(X, p− )) is evenly (resp. oddly) finite-dimensional, then H ∗ (M+ ) (resp. H ∗ (M− )) is the even (resp. odd) degree part of H ∗ (M )), because H ∗ : Mrat (k) → GrV ecE is a tensor functor. Therefore, modulo homological X . Therefore (a) implies S(X), and by 13.4.4, it also imequivalence, p± = π± plies N (X). Since (a) also implies finite-dimensionality of h(X n ) = h(X)⊗n , for all n ≥ 1 (by 2.7), (a) implies (b). (b) ⇒ (c) is trivial. (c) ⇒ (a): If S(X) holds, the π± are algebraic projectors modulo homological equivalence, and if N 0 (X) holds, these lift to orthogonal projectors π ˜+ and π ˜− modulo rational equivalence with π ˜+ + π ˜− = id by 3.1 (lift π+ to a
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projector π ˜+ and let π ˜− = id − π ˜+ ). Let M± = (X, π ˜± , 0) modulo rational equivalence. Then M = M+ ⊕ M− , and for b± = dim H ∗ (M± ) one has ∧b+ +1 M+ = 0 =Symb− +1 M− modulo homological equivalence. By 3.1 and N 0 (X n ), for n = b+ + 1 and n = b− + 1, one concludes that this vanishing also holds modulo rational equivalence, i.e., we obtain (a). Corollary 13.4.10. Voevodsky’s nilpotence conjecture (cf. 13.4.2) implies the conjecture of Kimura and O’Sullivan (cf. 13.3.4). Proof It implies the standard conjecture D(X) (postulating ∼hom = ∼num ), hence B(X), hence C(X), hence S(X). Moreover, it implies N 0 (X) (cf. the lines after 13.4.2). Remark 13.4.11. O’Sullivan (cf. [An1, Th. 3.33]) has proved: Let C be a rigid tensor subcategory of M(k). If every motive in C is finite-dimensional, and if every tensor functor ω : C −→ sVecF (where F is a field of characteristic 0 and sVecF is the category of finitedimensional super (i.e., Z/2-graded) vector spaces over K) factors through numerical equivalence, then Voevodsky’s conjecture holds for C. So far we have only applied the nilpotence result in 13.4.3 (b). Theorem 13.4.3 (a) gives the following Cayley-Hamilton theorem. Theorem 13.4.12. Let f be an endomorphism of a motive M , and assume that M is either evenly finite-dimensional or oddly dimensional. Let H ∗ be a Weil cohomology theory, and let P (t) = det (t − f | H ∗ (M ) ) be the characteristic polynomial of f on H ∗ (M ). Then P (t) is independent of the chosen Weil cohomology theory, and one has P (g) = 0. Proof If M is an evenly finite-dimensional motive, its cohomology is even, and by the trace formula [Kl, 1.3.6 c] one has tr(f ) = tr(f, H ∗ (M )). Therefore one has d X i=0
d−i
(−1)
d−i
tr(∧
i
f )t =
d X
d−i
(−1)
n−i
tr(∧
f|
n−i ^
H ∗ (M ))ti .
i=0
On the other hand, it is known that the right hand side is the charcteristic polynomial P (t). For an oddly finite-dimensional motive its cohomology is
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odd and one has tr(f ) = −tr(f, H ∗ (M )), so that d X
tr(Sym
d−i
i
f )t =
i=0
d X
d−i
(−1)
n−i
tr(∧
f|
n−i ^
H ∗ (M ))ti
i=0
is again equal to P (t). Therefore the claim follows with 13.4.3 (a). We now come to the proof of Theorem 13.4.3. It is straightforward to prove the following two lemmas. (Note that for n = 3, Lemma 13.4.13 is just the definition of composition of correspondences.) Lemma 13.4.13. Let pij : X n → X × X be the projection onto the i-th and j-th factor ((x1 , . . . , xn ) 7→ (xi , xj )). Consider algebraic cycles f1 , . . . , fn−1 on X × X, regarded as correspondence from X to X. Then one has (p1,n )∗ (p∗1,2 f1 . p∗2,3 f2 . . . p∗n−2,n−1 fn−2 . p∗n−1,n fn−1 ) = fn−1 ◦ fn−2 ◦ . . .
◦ f2 ◦ f1
(composition of correspondences on the right hand side). Lemma 13.4.14. Consider morphisms f : V → M , g : W → N of smooth, projective varieties, and the diagram
V
V × WH HH p HH W HH HH $
vv vv v vv zv v pV
f ×g
W
f
M ×N HH v
v vv vv v v zv pM
M
g
HH pN HH HH H$
N
where pV , pW , pM and pN are the projections. Then, for algebraic cycles α on V and β on W one has (f × g)∗ (p∗V α . p∗W β) = p∗M f∗ α . p∗N g∗ β , i.e., (f × g)∗ (α × β) = f∗ α × g∗ β for the exterior products. Proof of Theorem 13.4.3 Let us consider the case where M is evenly finiteV dimensional, with n M = 0 (The odd case is similar). If M = (X, p, m), then f is a cycle on X × X such that pf = f = f p. Then, by assumption, we have P sgn(σ) σ ◦ p × . . . × p = 0 σ∈Sn
On finite-dimensional motives and Murre’s conjecture
(where we have n factors p) since this endomorphism factors through This means P sign(σ) p∗1,n+σ(1) p . p∗2,n+σ(2) p . . . p∗n,n+σ(n) p = 0
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Vn
M.
σ∈Sn
since p × . . . × p = p∗1,n+1 p . p∗2,n+2 p . . . p∗n,2n p on X 2n , where pij : X 2n → X × X is the projection onto the i-th and j-th factor as in 13.4.13. In particular, we have P sign(σ) (p1,n+1 )∗ (p∗1,n+σ(1) p . · · · . p∗n,n+σ(n) p . p∗2,n+2 f t . p∗3,n+3 f t . · · · . p∗n,2n f t ) = 0 σ∈Sn
Let o1 (σ) = {1, σ(1), σ 2 (1), . . . , σ s−1 (1)} (with σ s (1) = 1) be the orbit of 1 ∈ {1, . . . , n} under σ ∈ Sn . Then σ is the product σ = (1 σ(1) σ 2 (1) . . . σ s−1 (1)) · σ 0 of the s-cycle σ1 : 1 7→ σ(1) 7→ σ 2 (1) 7→ . . . 7→ σ s−1 (1) 7→ 1 and a product σ 0 of cycles which are disjoint from σ1 . Thus, in the above sum, the summand corresponding to σ is sgn(σ) (p1,n+1 )∗ (p∗1,n+σ(1) p . p∗n+σ(1),σ(1) g . pσ(1),n+σ2 (1) p . pn+σ2 (1),σ2 (1) g . . . p∗n+σs−1 (1),σs−1 (1) g . p∗σs−1 ,n+1 p . β) where β is the product of the 2(n − s) factors p∗i,n+σ(i) p and p∗i,n+i f t with i ∈ o1 (σ) = {1, . . . , n} r o1 (σ). Writing X 2n = V × W , with V being the product of the 2s factors at the places i or n + i for i ∈ o1 (σ), and W the product over the 2(n − s) factors at the other places, it follows from 13.4.11 (applied to p1,n+1 : V → X ×X and the structural morphism W → Spec(k)) and 13.4.10 that the summand is sign(σ) · (f )s−1 deg(β 0 ) where β 0 is a zero cycle on W and (f )s−1 is the (s − 1)-fold self product in End(M ). If σ is an n-cycle, then sign(σ) = (−1)n−1 , s = n, β = β 0 = 1 and deg(β 0 ) = 1, so that the summand is (−1)n−1 (f )n−1 . If σ is not an n-cycle, then s < n. This shows that we get a polynomial equation for f with leading term (−1)n−1 (n − 1)! (f )n−1 . If f is numerically equivalent to zero, then so is β 0 for s < n, so that deg(β 0 ) = 0 unless σ is an n-cycle. This proves 13.4.3 (a). For 3.3 (b) we note that, by choosing a bijection ρ : {1, . . . , n − s} → o1 (σ), we may identify W with X n−s × X n−s and β 0 with β 00 = p∗1,n−s+σ00 (1) p . · · · . p∗n−s,n−s+σ00 (n−s) p . p∗1,n−s+1 f t . · · · . p∗n−s,2(n−s) f t = (σ 00 ◦ p × · · · × p).(f × · · · × f )t ,
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where σ 00 = ρ−1 σ 0 ρ ∈ Sn−s . Thus deg(β 0 ) = tr(σ 00 ◦ p × · · · × p ◦ f × · · · × f ) = tr(σ 00 ◦ f × · · · × f ) . Summing over all σ ∈ Sn with fixed σ1 , and keeping the bijection ρ, we thus get X sign(τ )tr(τ ◦ f × . . . × f ) = (−1)s−1 (n − s)!tr(∧n−s f )f s−1 . (−1)s−1 τ ∈Sn−s
After summing over all σ ∈ Sn we then see that the coefficient of f s−1 is (−1)n−1 (n − 1)! (−1)n−s tr(∧n−s f ) , because there are (n − 1)!/(n − s)! cycles σ1 containing 1 of length s. This proves 3.3 (b).
13.5 Finite fields Let us recall Murre’s conjecture. Let X be a purely d-dimensional smooth projective variety over a field k, fix a Weil cohomology theory, and assume the standard conjecture C(X), i.e., that the K¨ unneth components πi = πiX ∈ 2d H (X × X) of the diagonal ∆X are algebraic. Conjecture 13.5.1 (Murre, [Mu1]). A) X has a Chow-K¨ unneth decomposition, i.e., the πiX lift to an orthogonal set of idempotents {˜ πi } with P d ˜i = ∆X in CH (X × X). i π B) The correspondences π ˜2j+1 , . . . , π ˜2d act as zero on CHj (X). C) Let F ν CHj (X) = Ker π ˜2j ∩ Ker π ˜2j−1 ∩ . . . ∩ Ker π ˜2j−ν+1 ⊆ CHj (X). · Then the descending filtration F is independent of the choice of the π ˜i . D) F 1 CHj (X) = CHj (X)hom := {z ∈ CHj (X)|z ∼hom 0}. It is known [Ja2] that this conjecture, taken for all smooth projective varieties, is equivalent to the conjecture of Bloch-Beilinson on a certain functorial filtration on Chow groups, and that this Bloch-Beilinson filtration would be equal to the filtration F · defined above. The advantage of Murre’s conjecture is that it can be formulated and proved for specific varieties, and that will be used below. Remarks 13.5.2. a) The condition in 13.5.1(A) is called the ChowK¨ unneth decomposition, because it amounts to saying that the K¨ unneth 2d i i decomposition hhom (X) = ⊕i=0 h (X) in Mhom (k), with h (X) = ˜i (X, πiX ), can be lifted to a decomposition hrat (X) = ⊕2d i=0 h (X) in ˜ i (X) = (X, π the category Mrat (k) of Chow motives, via h ˜i ).
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b) Conjecture 13.5.1 (A) would follow from the nilpotence of J(X), i.e., from conjecture N (X) (cf. 13.4.1), and hence from finite-dimensionality of hrat (X). On the other hand it is known that the Bloch-BeilinsonMurre conjecture would imply the conjecture (13.3.4) of Kimura-O’Sullivan ([AK], [An1]). Let us note here that, more precisely, Murre’s conjecture for X, X × X and X × X × X implies N (X) ([Ja2, pp. 294, 295]), so that Murre’s conjecture for all sufficient high powers X N implies finite-dimensionality of X (13.4.9; note that we have assumed C(X), hence S(X)). Now let k be a finite field. Then the standard conjecture C(X) holds for every smooth projective variety X over k [KM]. It is furthermore known (cf. [Ja3, 4.17]) that the conjecture of Bloch-Beilinson-Murre over k is equivalent to the equality ∼rat = ∼hom where homological equivalence is taken with respect to any Weil cohomology theory H satisfying weak Lefschetz (cf. [Kl, p. 368 ] or [KM, p. 74]) (e.g., the `-adic cohomology (13.2.3 (b)) for any fixed ` 6= char(k)). Again we want to make this more precise. Theorem 13.5.3. Let k be a finite field. The equality ∼rat = ∼hom on X × X implies Murre’s conjecture for X. Conversely, Murre’s conjecture for X and X × X implies the equality ∼rat = ∼hom for X. Proof The first claim is trivial. For the second claim we use a result of Soul´e: Proposition 13.5.4 (Prop. 2, [So1]). Let X be smooth projective over k. The k-linear Frobenius F : X → X acts on CHj (X) as the multiplication by q = cardinality of k. Given this, assume Murre’s conjecture for X and X × X. We may assume that X is irreducible of dimension d. Let π˜0 , . . . , π ˜2d be orthogonal X X idempotents lifting the K¨ unneth components π0 , . . . , π2d of the diagonal, ˜ i (X) = (X, π and define h ˜i ) in the category Mrat (k) of Chow motives. By Murre’s conjecture for X × X, X×X CHd (X × X)hom = ⊕ π^ CHd (X × X) r r<2d
P X×X where π^ = µ+ν=r (˜ π2d−µ )t × π ˜ν lifts the K¨ unneth component πrX×X = r P X X X t X µ+ν=r πµ × πν of X × X (note that (π2d−µ ) = πµ ). But ((˜ π2d−µ )t × π ˜ν ) CHd (X × X) = π ˜ν ) ◦ CHd (X × X) ◦ π ˜2d−µ ,
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and for α ∈ CHd (X × X) we have π ˜ν α˜ π2d−µ π ˜i CHj (X) = 0 for i 6= 2d − µ . On the other hand, for i = 2d − µ and µ + ν < 2d we have π ˜ν α˜ π2d−µ π ˜i CHj (X) ⊆ π ˜ν CHj (X) with ν < i. This shows that CHd (X × X)hom acts trivially on GriF CHj (X) for Murre’s filtration, because X π ˜m CHj (X) F i CHj (X) = m≤2j−i
by 13.5.1 (B) and (C). In other words, the correspondences in CHd (X × X) act on GriF CHj (X) modulo homological equivalence, and then this quotient just depends on the motive modulo homological equivalence h2j−i (X) = X ). Let P (t) = det ( t − F ∗ | H ∗ (X) ) be the characteristic polyno(X, π2j−i i mial of the k-linear Frobenius F : X → X acting on the cohomology. It is known from [KM] that Pi (t) = det( t − F ∗ | H i (X ×k k, Q` )) for any ` 6= char(k) and hence, by Deligne’s proof of the Weil conjectures, that Pi (t) is in Z[t], and has zeros with complex absolute values q i/2 . By the Cayley-Hamilton theorem, Pi (F ) acts as zero on H i (X), hence P2j−ν (F ) acts as zero on GrνF CHj (X). Since F = q j on CHj (X) by Soul´e’s result, and P2j−ν (q i ) 6= for ν 6= 0, we deduce GrνF CHj (X) = 0 for ν ≥ 1. q.e.d. One can prove part of Murre’s conjecture from finite-dimensionality, by applying ideas of Soul´e [So1], Geisser [Gei], and Kahn [Ka]. Theorem 13.5.5. Let k be a finite field, and let X/k be a smooth projective variety such that J(X) is a nil ideal (e.g., assume that hrat (X) is finite-dimensional). Then there is a unique Chow-K¨ unneth decomposition 2d i e hrat (X) = ⊕i=0 h (X), and one has CH j (e hi (X)) = 0 for i 6= 2j . In particular, parts (A), (B) and (C) of Murre’s conjecture hold for X and, moreover, π ˜i acts as zero on CH j (X) for all i 6= 2j, so that F ν = 0 for all ν ≥ 1.
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Proof The existence of the Chow-K¨ unneth decomposition was noted in 13.5.2. Let Pi (t) = det(t − F | H i (X)) be as above. By Cayley-Hamilton we have Pi (F ) = 0 in End(hihom (X)), so that Pi (F )r = 0 in End(e hi (X)) for some r ≥ 1, by assumption. Therefore 0 = Pi (F )r · CHj (e hi (X)) = Pi (q j )r · CHj (e hi (X)) , but Pi (q j ) 6= 0 for j 6= 2i, by Deligne’s proof of the Weil conjecture. The claimed consequences for Murre’s conjecture are now immediate. Finally, the uniqueness of the Chow-K¨ unneth decomposition is seen as follows. Let Q P (t) = Pi (t). Then P (F ) is homologically trivial, so that P (F )r = 0 for some r ≥ 0 in End(hrat (X)). Again by Deligne, the polynomials Pν (t) are also pairwise coprime, so that, for each i ∈ {0, . . . , 2d} there are polynomials ai (t) and bi (t) in Q(t) with ai (t)Pi (t)r + bi (t)Qi (t)r = 1 , Q ˜i = bi (F )Qi (F )r are pairwise where Qi (t) = j6=i Pj (t). Then the elements π orthogonal idempotents in End(hrat (X)) summing up to 1, and π ˜i is a lift of the i-th K¨ unneth projector πiX , as follows from Cayley-Hamilton. Finally, by 13.4.1 every other idempotent lifting πiX is of the form (1 + a)˜ πi (1 + a)−1 with a ∈ J(X), cf. [Ja2, 5.4]. But every endomorphism of hrat (X) commutes with F (cf. [So1, Prop. 2 ii]), hence with π ˜i , so that we obtain π ˜i again. This shows the uniqueness of the Chow-K¨ unneth decomposition. Remarks 13.5.6. a) The above proof, together with the fact that the Frobenius F commutes with all morphisms of Chow motives ([So1, in Prop. 2 ii)]), shows that the full (tensor) subcategory Mfrat (k) ⊂ Mrat (k) consisting of the finite-dimensional motives, possesses a unique weight grading in the sense of [Ja3, 4.11], i.e., a grading lifting the weight grading of Mhom (k): Every motive M has a unique decomposition M = ⊕i M i with H i (M ) = H i (M i ) and H j (M i ) = 0 for j 6= i, and this filtration is respected by any morphism. in b) By 2.6 and 2.7, the category Mfrat (k) contains Mav rat (k), the full rigid pseudo-abelian tensor subcategory of Mrat (k) generated by the motives of abelian varieties. Hence the assumptions of Theorem 13.5.5 hold for (products of) curves, abelian varieties, Fermat hypersurfaces of degree m invertible in k, and Kummer or Enriques surfaces. As in loc. cit., one can extend the above result to higher algebraic Ktheory, or rather motivic cohomology with rational coefficients. Recall that, for any smooth variety V over a field L its Q-rational motivic cohomology
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i (V, Q(j)) = K (j) can be defined by HM 2j−i (V ) , where
Km (V )(j) := { x ∈ Km (V )Q | ψn (x) = nj x for all n ≥ 1 } . Here ψn is the n-th Adams operator, acting on the algebraic K-group 2j Km (V ). One has HM (V, Q(j)) = K0 (V )(j) ∼ = CHj (V ) by the RiemannRoch theorem for Chow groups. It is known (and follows from the fact just recalled) that algebraic correspondences modulo rational coefficients act on motivic cohomology, so that motivic cohomology extends to a covariant functor on Mrat (k) by defining i+2n i HM (M, Q(j)) = pHM (X, Q(j + n))
for M = (X, p, n). Theorem 13.5.7. Under the assumptions of theorem 13.5.5 , one has i ˜ν HM (h (X), Q(j)) = 0 for ν 6= 2j . i (X, Q(j)) = 0 for j > d = dim(X). In particular, HM
Proof This follows as above, by using that F acts on H i (X, Q(j)) as q j , because F = ψq on Km (X) [So2, 6.1], while Pν (q j ) 6= 0 for ν 6= 2j. It remains to investigate part (D) of Murre’s conjecture, i.e., the equality F 1 CHj (X) = CHj (X)hom . Recall that the Tate conjecture for H 2j (X, Q` ) states the surjectivity of the cycle map CHj (X) ⊗Q Q` −→ H 2j (X, Q` (j))Γ , where Γ = Gal(k sep /k) is the absolute Galois group of k. Theorem 13.5.8. Let X be smooth, projective of pure dimension d. Assume that i) J(X) is a nil ideal (e.g., assume that X is finite-dimensional), ii) the Tate conjecture holds for H 2j (X, Q` ) and H 2(d−j) (X, Q` ), and iii) the eigenvalue 1 of F is semi-simple on H 2j (X, Q` (j)). Then the following holds. a) ∼rat = ∼num on CHν (X) (i.e., CHν (X)num = 0), for ν = j, d − j. i (X, Q(ν)) = 0 for all i 6= 2ν, for ν = j, d − j. b) HM Proof This follows from results of Geisser [Gei] and Kahn [Ka]. Let us give a brief argument, for avoiding a little problem with the arguments given in [Ka], and for getting a statement used below.
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By Poincar´e duality, (iii) also holds for d − j, so it suffices to consider ˜ 2j (X) ν = j. Then, by theorems 13.5.5 and 13.5.7, it suffices to consider h instead of X in the statements. Now it is well-known (cf. [Ta, (2.9)]) that the assumptions on the Tate conjecture and the semi-simplicity of F imply that ∼num = ∼hom on CHν (X) for ν = j and d − j, and that ∼
Aνnum (X) ⊗Q Q` → H 2j (X, Q` (j))Γ for ν = j, d − j via the cycle map. Let P2j (t) = det(t − F | H 2j (X)) where H is the Weil cohomology theory given by `-adic cohomology, ` 6= p = char(k). Write P2j (t) = Q(t)(t − q j )ρ with (t − q j ) - Q(t) and some ρ ≥ 0. By assumption, ˜ 2j (X)). Now we have there is an integer r > 0 with P2j (F )r = 0 in End(h 1 = q(t)Q(t)r +r(t)(t−q j )ρr with polynomials q(t) and r(t) in Q. This shows that Q0 (F ) = q(F )Q(F )r and P 0 (F ) = r(F )P (F )r , with P (t) = (t − q j )ρ , ˜ 2j (X)) with Q0 (F ) + P 0 (F ) = 1. Let are orthogonal idempotents in End(h 0 2j 0 e M1 = P (F )h (X) and M2 = Q (F )e h2j (X). Then M = M1 ⊕ M2 and j CH (M1 ) = 0 as in the proof of 13.5.5, because Q(F )r M1 = 0 and Q(q j ) 6= 0, i (M , Q(j)) = 0. and similarly HM 1 We now claim that M2 ∼ = 1(−j)ρ . Then the claims follow, because it is clear that ∼ rat = ∼hom on 1(−j), and well-known (by work of Quillen) that i (1(−j), Q(j)) = H i−2j (Spec k, Q(0)) = 0 for i 6= 2j if k is a finite field. HM The characteristic polynomial of F on P 0 (F )H 2j (X, Q` (j)) is (t − q j )ρ . Hence ρ H(M2 (j)) = Q0 (F )H 2j (X, Q` (j)) ∼ = Q` = H 2j (X, Q` (j))F =1 ∼
as a Galois module, by semi-simplicity (iii). By Tate’s conjecture (ii), this cohomology has a basis given by algebraic cycles. Using the equality Ajhom (X) = Hom(1, hhom (X)(j)) = Hom(1, (M2 )hom (j)), and the identification of the composition map Hom(1, hhom (X)(j)) × Hom(hhom (X)(j), 1) → Hom(1, 1) = Q sending (α, β) to β ◦ α with the intersection number pairing Ajhom (X) × Ad−j hom (X) → Q ϕ
ψ
sending (α, β) to hα . βi we now get two maps 1ρ −→ M2 (−j) −→ 1ρ whose composition is the identity. (Note that the above intersection number pairing is non-degenerate, because Aνhom (X) = Aνnum (X) for ν = j and d − j, as remarked above.) Therefore 1ρ becomes a direct factor of (M2 )hom , and we conclude that ϕ : 1ρ ∼ = (M2 )hom is an isomorphism with inverse ψ, because ρ ∼ H(M2 (j)) = Q` as was shown above. But this implies that one also has an
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isomorphism 1ρ ∼ = M2 (j) in the category of Chow motives, because J(M2 ) is a nil ideal, and J( 1 ) = 0. Corollary 13.5.9. If J(X) is a nil ideal (e.g., if X is finite-dimensional), i (X, Q(d)) = 0 for i 6= 2d. then HM Proof In fact, it is clear that the Tate conjecture holds in degrees 0 and 2d, and that the corresponding cohomology groups are semi-simple Galois representations. We remark that the bijectivity of c`d : CHd (X) ⊗ Q` → H 2d (X, Q` (d)) is known without the assumption on X, by higher class field theory (note that we have Q-coefficients). Corollary 13.5.10. Assume that J(X) is a nil ideal, the Tate conjecture holds for X (i.e., for all cohomology groups of X), and the eigenvalue 1 is semi-simple on all groups H 2j (X, Q` (j). Then ∼
a) CHj (X)num = 0, and CHj (X) ⊗Q Q` −→ H 2j (X, Q` (j)Γ via the cycle map (strong Tate conjecture) for all j ≥ 0, b) Km (X) ⊗ Q = 0 for all m 6= 0 (Parshin conjecture). Remarks 13.5.11. a) The problems with the arguments in [Ka] concern the meaning of the statement that rational and numerical equivalence agree on X. In this paper, the meaning is that ∼rat = ∼num on CHj (X) for all j, and this would also fit with the assumptions in [Ka]. It does not imply that one can identify hrat (X) and hnum (X) as written in the parenthesis following loc.cit. Cor. 2.2, because that would rather mean that rational and numerical equivalence agree on X × X. Similarly, the reference in [Ka, 2.2] to [Gei, th. 3.3] has to be completed, because in the latter reference the argument is by assuming ∼rat = ∼num for all varieties, and deducing an action of End(hnum (X)) on Ka (X)(j) , which again requires ∼rat = ∼num on X × X. Finally, in the proof of [Ka, Th´eorem 1.10], the reference to [Mi, th. 2.6] has to be taken with similar care, because again, in that reference the (strong) Tate conjecture is assumed for all varieties, and in principle used for a product of two varieties when deducing semi-simplicity of the category Mhom (k) and considering the question of isomorphy of two motives. The final conclusion is that the stated results in [Ka] remain correct, while the proofs have to be modified - basically by noting that in the considered cases it suffices to consider morphisms between Tate objects 1(j) and h(X) instead of endomorphisms of h(X). b) In principle, the proof given in [An1, 4.2] is correct, but the short formulation might disguise the fact that, to my knowledge, it does not
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suffice to assume the Tate conjecture and 1-semi-simplicity just for H 2j (X, Q` (j)) if one wants to get the results for CHj (X). c) In view of 13.5.6 (b), the assumptions of Corollary 13.5.10 hold, e.g., for arbitrary products of elliptic curves [Sp], for abelian varieties of dimension ≤ 3 [So1, Th. 4], Fermat surfaces of degree m invertible in k and dimension ≤ 3 (loc. cit.), for rational, Enriques or Kummer surfaces, and for many abelian varieties. In particular, for the sub tensor category of Mrat (k) generated by elliptic curves one gets ∼ rat = ∼hom , and hence the validity of Murre’s conjecture. Theorem 13.5.12. Under the assumptions of Corollary 13.5.10 , the regulator map ∼
i (X, Q(j)) ⊗ Q` → H i (X, Q` (j))Γ HM
is an isomorphism for all i, j ∈ Z, where Γ = Gal(k/k) and ` 6= char k. Proof This is clear from 13.5.10 and the fact that H i (X, Q` (j))Γ = 0 for i 6= 2j by Deligne’s proof of the Weil conjectures. For the definition and properties of these regulator maps we refer to [Ja1, ch. 8], where they are deduced from Chern characters on higher algebraic K-theory constructed by Gillet. They exist for any smooth variety U over k instead of X, and coincide with the cycle maps for i = 2j, via the isomorphisms K0 (U )(j) ∼ = CHj (U ). The following application will be used in the next section. Corollary 13.5.13. If C is an elliptic curve or a rational curve, and X is a product of elliptic curves, then, for every open U ⊆ C, and all i, j ∈ Z, the regulator map i (X × U, Q(j)) ⊗Q Q` → H i (X × U , Q` (j))Γ HM
is an isomorphism, and the eigenvalue 1 of F on H i (X × U , Q` (j)) is semisimple. This is a special case of the following conjecture (in which k is still a finite field). Conjecture 13.5.14. ([Ja1, 12.4]) For any separated scheme of finite type Z over k, and all a, b ∈ Z, the regulator map HaM (Z, Q(b)) ⊗Q Q` → Ha´et (Z, Q` (b))Γ is an isomorphism, and the eigenvalue 1 of Frobenius on Ha´et (Z, Q` (b))is semi-simple.
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Again we refer to [Ja1, ch. 8] for the definition and properties of these homological versions of the regulator maps. Corollary 13.5.13 now follows from the following lemma, because the assertion of 13.5.12 holds for X × C and for X × Spec(k(x)), for any closed point x ∈ C − U . Lemma 13.5.15. a) ([Ja1, 8.4]) If Z is smooth and of pure dimension d, then the regulator map in 13.5.14 coincides with the regulator map 2d−a HM (Z, Q(d − b)) → H´e2d−a (Z, Q` (d − b))Γ . t
b) ([Ja1, Th. 12.7 b])) If Z 0 ⊆ Z is closed, U = Z − Z 0 , and Conjecture 13.5.14 holds for two of the three schemes Z, Z 0 , U, then it also holds for the third one. Although in this paper, we always used Chow groups and motivic cohomology groups with Q-coefficients, we note that we can also get consequence for groups with Z-coefficients, as in Soul´e’s paper [So1]. Recall that, for a smooth variety X over a field L one has motivic cohomology with Zi (X, Z(j)) = CHj (X, 2j− coefficients, which can for example be defined as HM i), where the latter groups are the higher Chow groups as defined by Bloch [Bl]. By definition, these groups vanish for j < 0 or i > 2j or i > d + j, where d = dim(X), and it is known that CHa (X, b) ⊗Z Q ∼ = Kb (X)(a) i i 2j so that HM (X, Z(j)) ⊗Z Q = HM (X, Q(j)). Moreover H (X, Z(j)) = CHj (X, 0) = CHj (X)Z , the usual Chow groups with integral coefficients. Finally, for an irreducible smooth projective variety X of dimension d, the group CHd (X × X, Z) of integral correspondences is a ring and acts on the i (X, Z(j)). The additive category of integral motives motivic cohomology HM (modulo rational equivalence) is defined by the same formalism as recalled in section 1. We denote the objects as (X, p, m)Z where p is now an integral idempotent correspondence, and define h(X)Z = (X, id, 0)Z , the integral motive corresponding to X and 1(j)Z = (Spec(k), id, j)Z , the j-fold Tate twist of the trivial motive 1. Corollary 13.5.16. Under the assumptions of Theorem 13.5.8 , the groups i (X, Z(j)) have finite exponent for i 6= 2j. For H 2j (X, Z(j)) = CHj (X) , HM Z M the subgroup CHj (X)Z,num of numerically trivial cycles has finite exponent, and the quotient group Aj (X)Z,num = CHj (X)Z /CHj (X)Z,num is isomorphic to Zρ , where ρ = dimQ` H 2j (Xk , Q` (j))Gal(F /F ) . Proof In the proof of Theorem 13.5.8 it was shown that there is an isomorphism of Q-linear motives h(X) ∼ = M1 ⊕ 1(−j)ρ ,
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with Q(F )M1 = 0 for a polynomial Q(t) ∈ Z[t] with Q(q j ) 6= 0. Then there are morphisms α
β
h(X) −→ 1(−j)ρ −→ h(X) with αβ = id, and for the idempotent βα one has Q(F )(id − βα) = 0 (id − βα is the idempotent corresponding to M1 ). Thus there exist integers N1 , N2 , N3 > 0 such that N1 α and N2 β lift to integral correspondences α ˜ d ˜ ˜ and β, and N3 Q(F )(N1 N2 − β α ˜ ) = 0 in CH (X × X)Z . Now the argument of Soul´e for Chow groups ([So1, Prop. 2 i)]) immediately extends to higher i (X, Z(j)) as multiplication by q j . Chow groups to show that F acts on HM We conclude i i i (X, Z(j)) (X, Z(j)) = N HM (X, Z(j)) = N1 N2 N3 Q(F )HM N3 β˜α ˜ HM
with the non-zero integer N = N1 N2 N3 Q(q j ). But the composition α ˜
β˜∗
∗ i i i (X, Z(j)) HM (X, Z(j)) −→ HM (1(−j)ρ , Z(j)) −→ HM
i (1(−j), Z(j)) = H i−2j (Spec(k), Z), which is zero for i 6= 2j, because HM M is known to be zero for i 6= 2j if k is a finite field. For i = 2j we have i−2j 2j HM (Spec(k), Z) = Z, and thus N HM (X, Z(j)) is isomorphic to Zρ (note that α ˜ ∗ β˜∗ = N1 N2 , which can be checked after tensoring with Q, where it holds by definition). We deduce that the torsion group of CHj (X)Z is killed by N , and coincides with CHj (X)Z,num : it is contained in the latter group, and the quotient embeds into the group CHj (X)Q = Anum (X)Q .
Corollary 13.5.17. Let X be a smooth projective variety over the finite field k such that the associated motive (with Q-coefficients) is finite-dimensional i (X, Z(j)) has finite expo(or that J(X) is a nil ideal). Then the group HM i nent for j > d = dim(X), and HM (X, Z(d)) has finite exponent for i < 2d. Proof The last statement follows from 13.5.16, because the Tate conjecture and the semi-simplicity hold for H 0 and H 2d . Formally, the first statement follows as well, since the condition on the Tate conjecture is empty here, but we give a simpler direct proof: Let the integral polynomial P (t) = Q2d i=0 Pi (t) be as in the proof of Theorem 13.5.5. Then the assumption implies that P (F )r = 0 in CHd (X × X, Q), for some integer r ≥ 1. Therefore N P (F )r = 0 in CHd (X × X, Z) for some integer N ≥ 1. Because F acts i (X, Z(j)), the integer N P (q j ) annihilates this group, but one as q j on HM j has P (q ) 6= 0 for j ∈ / {0, . . . , 2d} by Deligne’s proof of the Weil conjectures.
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13.6 Global function fields Using the last section, we will deduce some results for global function fields. The most complete results are obtained for certain isotrivial varieties. Let F be a finite field, let C be a smooth projective geometrically irreducible curve over F , and let k = F (C) be its function field. Theorem 13.6.1. Let W be a smooth projective variety over k, and assume that, after possibly passing to a finite extension F 0 /F , W is isomorphic to Y ×F k, where Y /F is a smooth projective variety such that the assumptions of Corollary 13.5.10 hold for X = Y ×F C. Then the strong Tate conjecture holds for W , i.e., the cycle maps induce isomorphisms ∼
Ajhom (W ) ⊗Q Q` −→ H´e2jt (Wk , Q` (j))Gal(k/k) , for all j ≥ 0, and the Abel-Jacobi map ∼
1 (Wk , Q` (j))) CHj (W )hom ⊗Q Q` −→ Hcont (Gk , H´e2j−1 t
is an isomorphism for all j ≥ 0. Furthermore Murre’s conjecture holds for W , with the filtration F 1 CHj (W ) = CHj (W )hom and F 2 CHj (W ) = 0, and numerical and homological equivalence agree on W (i.e., on all Chow i (W, Q(j)) = 0 for 2j − i 6= 0, 1, i.e., groups of W ). Finally one has HM Km (W )Q = 0 for m ≥ 2. Remarks 13.6.2. The assumptions of the theorem hold, if C is a rational or elliptic curve and X is a product of rational or elliptic curves (or the motive of X is contained in the rigid tensor subcategory generated by elliptc curves and Artin motives). We will prove a somewhat more general result. For any smooth variety W over k = F (C) define the arithmetic ´etale cohomology as i Har (W, Q` (j)) := lim H i (XU 0 ×F F , Q` (j))Gal(F /F ) , −→
where U ⊆ C is some non-empty open, X → U is a smooth model for W (W ∼ = X ×U k), and the limit is over all non-empty open subschemes U 0 ⊆ U , with XU 0 = X ×U U 0 . By standard limit theorems this cohomology does not depend on the choice of U and X. Moreover, this cohomology is functorial in W , receives a cycle class and allows an action of Chow correspondences if W is smooth and proper. In fact, there are regulator maps i i HM (W, Q(j)) −→ Har (W, Q` (j))
by taking the limit of the regulator maps i HM (XU0 , Q(j)) −→ H i (XU0 ×F F , Q` (j))Gal(F /F )
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discussed at the end of the previous section and noting that motivic cohomology commutes with filtered inductive limits, so that the limit on the left i (W, Q(j)). hand side is HM Let V be an `-adic representation of Gk (i.e., a finite-dimensional Q` vector space with continuous action of Gk ). Call V arithmetic, if it comes from a representation of the fundamental group π1 (U, η), where U ⊆ C is a non-empty open and η = Spec(k) the geometric generic point of U . Then we define the arithmetic Galois cohomology of V as i Har (Gk , V ) := lim H 1 (π1 (UF0 , η), V )Gal(F /F ) , −→
where the limit is again over the non-empty open U 0 ⊆ U . (This is i (Spec(k), F) for the `-adic sheaf F on Spec(k) corresponding to V , if one Har defines arithmetic ´etale cohomology more generally for an arithmetic `-adic sheaf G on W , i.e., one that extends to some model X over some U ⊆ C as above, cf. [Ja1, 11.7, 12.15].) This definition is functorial in V . With these notations we have the following. Theorem 13.6.3. Let W be a smooth projective variety over k. Assume the condition (∗) There is a scheme X of finite type over C with generic fiber W = X ×C k such that for some non-empty open U ⊆ C, Conjecture 13.5.14 holds for XU = X ×C U and the fibres Xt = X ×C t for all closed points t ∈ U . Then the regulator maps induce isomorphisms ∼
i i HM (W, Q(j)) ⊗Q Q` −→ Har (W, Q` (j)) ,
(13.1)
i (W, Q(j)) = 0 for i − 2j 6= 0, 1, i.e., for all i, j ∈ Z. Moreover, HM i (W, Q (j)) ∼ Km (W )Q = 0 for m > 1. For i−2j = 1 one has isomorphisms Har = ` 1 (G , H i (W , Q (j))). For i−2j = 0, the cycle maps induce isomorphisms Har k ` k for all j ≥ 0
∼
H´e2jt (Wk , Q` (j))Gal(k/k)
∼
1 Hcont (Gk ,
Ajhom (W ) ⊗Q Q` −→
(13.2)
(strong Tate conjecture) j
CH (W )hom ⊗Q Q` −→
H´e2j−1 (Wk , Q` (j))) t
(13.3)
(Abel-Jacobi map). If W has a Chow-K¨ unneth decomposition (e.g., if standard conjecture C(W ) holds and condition (∗) also hold for W ×k W ), then Murre’s conjecture holds for W , with F 1 CHj (W ) = CHj (W )hom and F 2 CHj (W ) = 0.
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Proof The first isomorphism is clear from the above, since the maps i HM (XU 0 , Q(j)) −→ H i (XU 0 ×F F , Q` (j))Gal(F /F )
are isomorphisms for all sufficiently small U 0 ⊆ U , by Lemma refjan:4.15 (a) and (b). The next three claims follow from [Ja1, Thm. 12.16, diagram (12.16.3) and Rem. 12.17 b)]. Now assume that W is of pure dimension d and has a Chow-K¨ unneth decomposition, i.e., the K¨ unneth projectors πi are algebraic, and lift to an orthogonal set of idempotents π ˜i in CHd (W × W ). Since the cycle maps (6) and (7) are functorial with respect to correspondences, it follows that, for the filtration F ν CHj (W ) defined in the theorem, the action of correspondences on GrνF CHj (W ) factors through homological equivalence, and that πi = δi,2j−ν id (Kronecker symbol) on GrνF CHj (W ). From this the remaining parts of Murre’s conjecture follow easily, including the given description of the filtration. Finally assume that the K¨ unneth components πi are algebraic and that condition (∗) holds for W × W . For any smooth projective variety V over k let F`ν be the descending filtration on the continuous ´etale cohomology i Hcont (V, Q` (j)) coming from the Hochschild-Serre spectral sequence p p+q (Gk , H q (Vk , Q` (j))) ⇒ Hcont E2p,q = Hcont (V, Q` (j)) .
Then, by (7) for W × W , the cycle map CHd (W × W ) → H 2d ((W × W )k , Q` (d)) (which is compatible with the cycle map (5) for W × W and (i, j) = (2d, d)) induces an injection 2d (W × W, Q (d)) CHd (W × W )hom → Gr1F` Hcont ` 1 ∼ = H (Gk , H 2d−1 ((W × W )k , Q` (d))) .
On the other hand, the filtration F`ν is respected under the action of correspondences, and F ν .F µ ⊆ F ν+µ under cup product. This shows that J(W ) = CHd (W × W ) is an ideal of square zero. Hence W has a ChowK¨ unneth decomposition. Proof of Theorem 13.6.1: We may assume that Y is of pure dimension d. Next we observe that a finite constant field extension does not matter, because we have Galois descent for ´etale cohomology with Q` -coefficients and motivic cohomology with Q-coefficients. Thus we may assume that W = Y ×F k. Then it is clear that Theorem 13.6.1 follows from 5.3, except possibly for the statement on Murre’s conjecture. But, in the situation of 13.6.1, the pull-back via the morphism W = Yk → Y induces isomorphism ∼ H i (YF , Q` ) → H i (Wk , Q` ) by proper and smooth base change. This shows
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that the projectors π ˜iY of a Chow-K¨ unneth decomposition for Y (which exist by the assumptions on Y ) map to idempotents lifting the K¨ unneth compod d nents of W under the pull back CH (X × X) → CH (W × W ). Therefore W has a Chow-K¨ unneth decomposition, and we can apply 13.6.3. While the emphasis of this paper was to investigate conjectures, results and conditions for fixed varieties, we conclude with statements on all varieties over a given field. From Theorem 13.6.3 we get: Corollary 13.6.4. If conjecture refjan:4.14 holds for all (smooth) varieties over F , then the results of theorem 13.6.3 hold for all smooth projective varieties W over function fields k in one variable over F . In particular, the strong Tate conjecture and the Murre’s conjecture hold over such k. The reduction to the smooth case is done by lemma 13.5.15 and induction on dimension. Moreover, we note: Proposition 13.6.5. Conjecture 13.5.14 holds for all varieties over F if and only if the following holds for all smooth projective varieties X over F : i) Tate’s conjecture (surjectivity of (5)), ii) the eigenvalue 1 is semi-simple on H 2j (XF , Q` (j)) for all j, iii) the Chow motive hrat (X) is finite-dimensional. Proof First we note that the properties (i) - (iii) for all X ∈ SPF are equivalent to conjecture 13.5.14 for all X ∈ SPF . This follows from theorems 13.5.12 and 13.4.9 , and the fact that S(X) holds for all X ∈ SPF . Secondly, conjecture 13.5.14 holds for all varieties if it holds for smooth projective varieties. The proof goes like in [Ja1, 12.7], but instead of assuming resolution of singularities, one may use de Jong’s version: Let Z be any reduced separated algebraic F -scheme. By [dJ] there is a smooth projective variety X and a morphism f : X → Z which is generically ´etale. Choose a dense smooth open U ⊆ Z such that the restriction g : V = f −1 (U ) → U is finite ´etale. By induction on dimension and lemma 13.5.15 (b) it suffices to prove conjecture 13.5.14 for U , and we may assume that it holds for V . But g induces degree-respecting pull-backs g ∗ and push-forwards g∗ in motivic and ´etale cohomology making the diagrams i (V, Q(j)) ⊗ Q i Gal(F /F ) HM ` −→ H´ et (VF , Q` (j)) ↓ g∗ ↓ g∗ i i HM (U, Q(j)) ⊗ Q` −→ H´et (UF , Q` (j))Gal(F /F )
commutative; similarly with g ∗ . On the other hand, one has g∗ g ∗ = m on
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both sides, where m is the degree of g. This implies that the bottom line is a retract of the top line, and hence that conjecture 13.5.14 for V implies conjecture 13.5.14 for U . Finally we indicate that the above results can easily be generalized to function fields k of arbitrary transcendence degree over F , by replacing C by any variety over F and using the same definitions of arithmetic ´etale and Galois cohomology, and a corresponding Hochschild-Serre spectral sequence p p+q E2p,q = Har (Gk , H q (Wk , Q` (j))) ⇒ Har (W, Q` (j)).
This gives the following result. Proposition 13.6.6. If conjecture 13.5.14 holds for all (smooth projective) varieties over Fp , then the strong Tate conjecture, the equality of numerical and homological equivalence and the conjecture of Bloch-Beilinson-Murre hold over all fields of characteristic p.
References ´, Y.: Motifs de dimension finie, d’apr´es S.-I. Kimura, P.O’Sullivan..., [An1] Andre Sem. Bourbaki 929, March 2004. ´, Y.: Une introduction aux motifs, Panoramas et synth`eses 17. Soci´et´e [An2] Andre de Math´ematique de France, Paris, 2004. ´, Y., B. Kahn and P. O’Sullivan: Nilpotence, radicaux et structures [AK] Andre mono¨ıdales., Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291. [Bl] Bloch, S.: Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), no. 3, 267–304. ´ [dJ] de Jong, A. J.: Smoothness, sem-stability and alterations, Inst. Hautes Etudes Sci. Publ. Math. 83 (1996), 51–93. [DM] Deligne, P. and J.S. Milne: Tannakian categories., in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, 900. SpringerVarlag, Berlin, 1982, 101–228. [DP] Del Padrone, A and C. Mazza: Schur finiteness and nilpotency. C.R.Acad.Sci. Paris, Ser. I 341 (2005), 283–286. [Gei] Geisser, Th.:Tate’s conjecture, algebraic cycles, and rational K-theory in characteristic p, K-Theory 13 (1998), 109-122. [GM] Gordon, B. B. and J.P. Murre: Chow motives of elliptic modular threefolds, J. Reine Angew. Math. 514 (1999), 145–164. [GP] Guletskii, V. and Pedrini, C.: Finite-dimensional motives and the Conjectures of Beilinson and Murre, K-Theory 30, No.3 (2003), 243–263. [Ja1] Jannsen, U.:Mixed Motives and Algebraic K-Theory, Lecture Notes in Mathematics, 1400. Springer-Verlag, Berlin, 1990. xiv + 246pp. [Ja2] Jannsen, U.: Motivic sheaves and filtrations on Chow groups, in Motives (Seattle, WA, 1991), 245–302, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
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[Ja3] Jannsen, U.: Equivalence relations on algebraic cycles, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 225–260, NATO Sci. Ser. C Math. Phys. Sci., 548, Kluwer Acad. Publ., Dordrecht, 2000. ´ [Ka] Kahn, B.: Equivalences rationnelle et num´erique sur certaines vari´et´es de ´ type ab´elien sur un corps fini, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 6, 977–1002 (2004). [KM] Katz, N. and W. Messing: Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. math. 23, (1974), 73–77. [Ki] Kimura, S.-I.: Motives are finite-dimensional, in some sense, Math. Ann. 331 (2005), 173–201. [Kl] Kleiman, S.: Algebraic cycles and the Weil conjectures in Dix espos´es sur la cohomologie des sch´emas, 359–386. North-Holland, Amsterdam; Masson, Paris, 1968. [Mi] Milne, J. S.: Motives over finite fields, in Motives (Seattle, WA, 1991), 401– 459, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. [Mu1] Murre, J. P. On a conjectural filtration on the Chow groups of an algebraic variety I, The general conjecture and some examples. Indag. Math. (N.S.) 4 (1993)2, 177–188. [Mu2] Murre, J. P.: On a conjectural filtration on the Chow groups of an algebraic variety II, Verification of the conjectures for threefolds which are the product on a surface and a curve, Indag. Math. (N.S.) 4 (1993), 189–201. [OSu] O’Sullivan, P.: Letters to Y. Andr´e and B. Kahn, 29/4/02 and 12/5/02. ´, C.: Groupes de Chow et K-th´eorie de vari´et´es sur un corps fini, Math. [So1] Soule Ann. 268 (1984), 317–345. ´, C.: Op´erations en K-th´eorie alg´ebrique,Canad. J. Math. 37 (1985), [So2] Soule no. 3, 488–550. [Sp] Spiess, M.: Proof of the Tate conjecture for products of elliptic curves over finite fields, Math. Ann. 314 (1999), 285–290. [Ta] Tate, J.: Conjectures on algebraic cycles in `-adic cohomology, in Motives (Seattle, WA, 1991), 71–83, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994. [Voe] Voevodsky, V.: A nilpotence theorem for cycles algebraically equivalent to zero, Internat. Math. Res. Notices 1995,4, 187–198
14 On the Transcendental Part of the Motive of a Surface Bruno Kahn † Institut de Math´ematiques de Jussieu, 175–179 rue du Chevaleret, 75013 Paris, France,
[email protected]
Jacob P. Murre Universiteit Leiden, Mathematical Institute, P.O. Box 9512, 2300 RA Leiden, the Netherlands,
[email protected]
Claudio Pedrini ‡ Universit´ a di Genova, Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy,
[email protected]
Introduction Bloch’s conjecture on surfaces [B1], which predicts the converse to Mumford’s famous necessary condition for finite-dimensionality of the Chow group of 0-cycles [Mum1], has been a source of inspiration in the theory of algebraic cycles ever since its formulation in 1975. It is known for surfaces not of general type by [B-K-L] (see also [G-P1]), for certain generalised Godeaux surfaces [Voi] and in a few other scattered cases. Thirty years later, it remains open. As was seen at least implicitely by Bloch himself early on, his conjecture is of motivic nature (see [B2, 1.11]). This was made explicit independently by Beilinson and the second author [Mu1]: we refer to Jannsen’s article [J2] for an excellent overview. In particular, the Chow-K¨ unneth decomposition for a surface S constructed in [Mu1] easily shows that the information necessary to study Bloch’s conjecture is concentrated in the summand h2 (S) § of the Chow motive of S. The main purpose of this article is to introduce and study a finer invariant of S: the transcendental part t2 (S) of h2 (S). Let us immediately clarify to the reader that we will not give a proof of Bloch’s conjecture! Instead, we study the endomorphism ring of t2 (S) for a general S and prove the following † supported by RTN Network HPRN-CT-2002-00287 ‡ supported by RTN Network HPRN-CT-2002-00287 and partially by the Italian MIUR § In this article we adopt a covariant convention for motives, hence write hi (S) rather than hi (S).
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two formulas (Theorems 14.4.3 and 14.4.8): EndMrat (t2 (S)) '
T (Sk(S) ) A2 (S × S) . ' J(S, S) T (Sk(S) ) ∩ H≤0
(14.1)
Here k is the base field, Mrat is the category of Chow motives over k with rational coefficients, A∗ denotes Chow groups tensored with Q, and • J(S, S) is the subgroup of A2 (S × S) generated by those correspondences which are not dominant over S via either the first or the second projection; • T (S) is the Albanese kernel; • H≤1 is the subgroup of A2 (Sk(S) ) generated by the images of the A2 (SL ), where L runs through the subextensions of k(S)/k of transcendence degree ≤ 1. The first formula is a higher-dimensional analogue of a classical result of Weil concerning divisorial correspondences. The second formula provides a variant of Bloch’s proof of Mumford’s theorem in [B2, App. to Lecture 1] (with actually a slightly more precise result), cf. Corollary 14.4.9. A conjecture generalising Bloch’s conjecture: ?
EndMrat (t2 (S)) ' EndMhom (thom (S)) 2 (here hom stands for homological equivalence) may therefore be reformulated as saying that the cycles homologically equivalent to 0 should be contained in J(S, S). This conjecture, in turn, appears in a wider generality in Beilinson’s article [Bei]; a link with this point of view is outlined in the last section, via the theory of birational motives ([K-S], see §14.5 here; in the context of Bloch’s conjecture this point of view goes right back to Bloch, Colliot-Th´el`ene and Sansuc, see [B2, App. to Lect. 1]). In particular, it is proven that t2 (S) does not depend on the choice of the refined ChowK¨ unneth decomposition of Propositions 14.2.1 and 14.2.3, and is functorial in S for the action of correspondences (Corollary 14.8.10). We now describe the contents of this paper in more detail. Section 14.1 fixes notation and reviews motives. Section 14.2 reviews the Chow-K¨ unneth decomposition of a surface S (Proposition 14.2.1) and introduces the hero of our story, t2 (S) (Proposition 14.2.3). Section 14.3 reviews the conjectures of [Mu2] on the Chow-K¨ unneth decomposition as well as some of their consequences established by Jannsen in [J2], and proves a part of these consequences for the case of a product of two surfaces (Theorem 14.3.10). In Section 14.4 the isomorphisms (14.1) are established; they are reinterpreted in the next section in terms of birational motives. Section 14.6 studies the relationship of the previous results with Kimura’s notion of finite-dimensional
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motives [Ki, G-P1, A-K]. The next section discusses some (conditional) higher-dimensional generalizations. Finally, Section 14.8 reproves and generalizes some of the previous results from a categorical viewpoint. This article may be seen as a convergence point of the ways its 3 authors understand the theory of motives. The styles of the various sections largely reflect the styles of the various authors: we didn’t attempt (too much) to homogenize them.
Acknowledgements This collaboration developed over a series of conferences: one on algebraic cycles in Morelia in 2003 and three on motives and homotopy theory of schemes in Oberwolfach, on K-theory and algebraic cycles in Sestri-Levante and on cycles and algebraic geometry in Leiden in 2004. We would like to thank their organisers for these opportunities and also for providing excellent mathematical environments. To add more acknowledgements, the first author also gratefully acknowledges the hospitality of TIFR, Mumbai, where he stayed in 2005 during the completion of this work. The second author thanks Uwe Jannsen for a helpful discussion.
14.1 Definitions and Notation 14.1.1 Categories Throughout this paper we shall use interchangeably the notations HomC (X, Y ) and C(X, Y ) for Hom sets between objects of a category C. In particular, the notation C(−, −) is more convenient when the symbol designating C is long, but the notation EndC (X) may be more evocative than C(X, X). 14.1.2 Pure motives Let k be a field and let V = Vk be the category of smooth projective varieties over k. We shall sometimes write X = Xd to say that X ∈ V is irreducible (or equidimensional) of dimension d. We denote by Ai (X) = Ad−i (X) the group CH i (X) ⊗ Q of cycles of codimension i (or dimension d − i) on X, modulo rational equivalence, with Q coefficients. We shall assume that the reader is familiar with the definition of pure motives and will only give minimal recollections on it, except for one thing. In [Mu1, Mu2, J1] and [Sch], pure motives are defined in the Grothendieck tradition so that the natural functor sending a variety to its motive
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is contravariant. On the contrary, here we are going to consider covariant motives, in order to be compatible with Voevodsky’s convention that his triangulated motives are covariant. Moreover, we change the sign of the weight. Since the translation thoroughly confused all three authors, we prefer to give a dictionary of how to pass from one convention to the other, for clarity and the benefit of the reader: To start with, for X, Y ∈ V we introduce as in [Sch, p. 165] the groups of contravariant Chow correspondences M Corri (X, Y ) = Adα +i (Xα × Y ) α
`
if X = α Xα with Xα equidimensional of dimension dα ; composition of correspondences is given by the usual formula (ibid.): g ◦contr f = (p13 )∗ (p∗12 f · p∗23 g). Let us denote by CHM (k) = CHM the category of Chow motives considered in [Mu1, Mu2, J1] or [Sch]. Thus an object of CHM is a triple M = (X, p, m) where X ∈ V, p is an idempotent in Corr0 (X, X) and m ∈ Z, while morphisms are given by CHM ((X, p, m), (Y, q, n)) = q Corrn−m (X, Y )p. To X ∈ V we associate ch(X) = (X, 1X , 0) ∈ CHM and to a morphism f : X → Y we associate ch(f ) = [Γf ]t , where Γf is the graph of f and γ t denotes the transpose of a correspondence γ†: this defines a contravariant functor ch : V → CHM . We could merely define Mrat as the opposite category to CHM . However it is much more comfortable to have an explicit description of it: Definition 14.1.1. i) The groups of covariant Chow correspondences are defined as follows: for X, Y ∈ V Corri (X, Y ) = Corr−i (Y, X). Composition of covariant correspondences is given by the formula g ◦cov f = (f t ◦contr g t )t . (From now on, we drop the index cov for the composition sign.) ii) The category of covariant Chow motives Mrat (k) = Mrat has objects triples M = (X, p, m) as above, while morphisms are given by Mrat ((X, p, m), (Y, q, n)) = q Corrm−n (X, Y )p. † On [Sch, p. 166], Scholl writes ch(f ) = [Γf ], which is slightly misleading.
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iii) The “covariant motive” functor h : V → Mrat is given by the formulas h(X) = (X, 1X , 0) h(f ) = [Γf ]. Note that, by definition Corri (X, Y ) =
M
Adα +i (Xα × Y )
α
if X =
`
Xα with dim Xα = dα . The reader will easily check the following
Lemma 14.1.2. There is an anti-isomorphism of categories F : CHM → Mrat defined by F (X, p, m) = (X, pt , −m) F (γ) = γ t . One has the formula F ◦ ch = h. Except at the beginning of Section 14.2, we shall never mention the category CHM again and will work only with Mrat . Let us review a few features of this category: 14.1.2.1 Effective motives Let Meff rat be the full subcategory of Mrat consisting of the (X, 1X , 0) for X ∈ V (see [Sch]): this is the category of effective Chow motives. 14.1.2.2 Tensor structure The product of varieties and of correspondences defines on Mrat a tensor structure (= a symmetric monoidal structure which is distributive with respect to direct sums); Meff rat is stable under this tensor structure. Then Mrat is an additive, Q-linear, pseudoabelian tensor category. It is also rigid, in the sense that there exist internal Homs and dual objects M ∨ satisfying suitable axioms. Namely one has (X, p, m)∨ = (X, pt , −d − m) if dim X = d, and γ∨ = γt if γ ∈ Corrn (X, Y ) = Mrat (X, 1X , n), (Y, 1Y , 0)).
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14.1.2.3 The unit motive and the Lefschetz motive The unit motive is 1 = (Spec(k), 1, 0): it is a unit for the tensor structure. The Lefschetz motive L is defined via the motive of the projective line over k: h(P1k ) = 1 ⊕ L. We then have an isomorphism L ' (Spec(k), 1, 1). 14.1.2.4 Tate twists For every motive M = (X, p, m) we define the Tate twist M (r) to be the motive (X, p, m + r). Note that, with our conventions, M (r) ' M ⊗ L⊗r for r ≥ 0. 14.1.2.5 Inverse image morphisms For a morphism f : X → Y in V, one often writes f∗ instead of h(f ). One may also consider the map in Mrat : f ∗ = [Γtf ] ∈ Ad (Y × X) : h(Y ) → h(X)(e − d) where d = dim X, e = dim Y . 14.1.2.6 Action of correspondences on Chow groups Observe that, by definition Ai (X) = Mrat (h(X), Li ) Ai (X) = Mrat (Li , h(X)) for any X ∈ V. This gives us a way to let correspondences act on Chow groups: • On the left: if α ∈ Ai (X) and γ ∈ Corrn (X, Y ), then γ∗ α = γ ◦ α ∈ Ai+n (Y ). (In terms of cycles: γ∗ (α) = (p2 )∗ (p∗1 (α) · γ).) • On the right: if γ ∈ Corrn (X, Y ) and α ∈ Ai (Y ), then γ ∗ α = α ◦ γ ∈ Ai−n (X). Remark 14.1.3. Suppose that dim X = d. If α ∈ Ai (X) is interpreted as a morphism from Li to h(X), then the dual morphism α∨ : h(X)(−d) → L−i is nothing else than α. The same applies to γ∗ α and γ ∗ α, with notation as above. If we view α = α∨ in Ad−i (X), we thus get a formula comparing left and right actions: γ ∗ α = (γ ∗ α)∨ = (α ◦ γ)∨ = γ ∨ ◦ α∨ = γ t ◦ α = (γ t )∗ α.
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In the same vein, note the formula Ai (M ) = A−i (M ∨ ). 14.1.2.7 Chow groups of motives We now extend the functors Ai : V → VecQ (contravariant) and Ai : V → VecQ (covariant) to functors on Mrat : Ai (X, p, m) = Mrat ((X, p, m), Li ) = p∗ Ai−m (X) Ai (X, p, m) = Mrat (Li , (X, p, m)) = p∗ Ai−m (X).
14.1.3 Weil cohomology theories If ∼ is any adequate equivalence relation on cycles (see [J3]), then a similar definition yields the category M∼ . In particular we will consider the cases where ∼ equals homological equivalence or numerical equivalence. We give ourselves a Weil cohomology theory H ∗ on V, as defined in [Kl] or [An, 3.3]; we shall denote its field of coefficients by K (by convention it is of characteristic 0). We also define M Hi (X) = H 2dα −i (Xα ) α
`
if X = α Xα with dim Xα = dα . For an element α ∈ Ai (X) we denote by cli (α) its image under the cycle map in Ai (X) → H 2i (X); we write Ai (X)hom = Ker(cli ) Aihom (X) = Ai (X)/Ai (X)hom = Coim(cli ) A¯i (X) = Im(cli ) ' Aihom (X). Equivalently, we have “homological” cycle maps cli : Ai (X) → H2i (X) and vector spaces Ai (X)hom , Ahom (X) and A¯i (X). One easily checks that i the K¨ unneth formula and Poincar´e duality carry over to homology without any change. We denote by Mhom the (covariant) category of homological motives, which is is defined as above by considering correspondences modulo homological equivalence, and by hhom the functor which associates to every X ∈ Vk its motive in Mhom .
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14.1.3.1 Action of correspondences on cohomology Let X, Y ∈ V, equidimensional of dimensions d and e for simplicity. The cycle class map gives us a homomorphism cle−i
Corri (X, Y ) = Ad+i (X × Y ) = Ae−i (X × Y ) −→ H 2e−2i (X × Y ). Since we have H 2e−2i (X × Y ) '
M
M
H j (X) ⊗ H 2e−2i−j (Y ) '
j
H j (X) ⊗ H 2i+j (Y )∗
j
'
Y
Hom(H 2i+j (Y ), H j (X))
j
by the K¨ unneth formula and Poincar´e duality, we get a contravariant action of correspondences γ ∗ : H ∗ (Y ) → H ∗−2i (X) for γ ∈ Corri (X, Y ), extending the action of morphisms. Similarly, using the homological cycle map, we get a covariant action γ∗ : H∗ (X) → H∗+2i (Y ) by means of the composition cld+i
Corri (X, Y ) = Ad+i (X × Y ) −→ H2d+2i (X × Y ) M M ' H2d+2i−j (X) ⊗ Hj (Y ) ' Hj−2i (X)∗ ⊗ Hj (Y ) j
j
'
Y
Hom(Hj−2i (X), Hj (Y )).
j
14.1.3.2 Homology and cohomology of motives As in 14.1.2.7, we may use 14.1.3.1 to extend H i , Hi to tensor functors H ∗ : Mhom → Vecgr K (contravariant) H∗ : Mhom → Vecgr K (covariant) with values in graded vector spaces: the first functor corresponds to [An, 4.2.5.1]. Explicitly, H i (X, p, m) = p∗ H i−2m (X), Hi (X, p, m) = p∗ Hi−2m (X) and H ∗ (γ) = γ ∗ , H∗ (γ) = γ∗ . We also have Hi (M ) = H −i (M ∨ ). Definition 14.1.4 (cf. [An, 3.4]). A Weil cohomology theory H is classical if:
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(i) chark = 0 and H is algebraic de Rham cohomology, l-adic cohomology for some prime number l or Betti cohomology relative to a complex embedding of k, or (ii) chark = p > 0 and H is crystalline cohomology (if k is perfect) or l-adic cohomology for some prime number l 6= p. (Recall that, if the prime number l is different from chark, l-adic cohoi (X ⊗ k , Q ), where k is some separable mology is defined by Hli (X) = Het s k s l closure of k.) Lemma 14.1.5. If chark = 0, homological equivalence does not depend on the choice of a classical Weil cohomology theory. Moreover, for any smooth projective X, Y , the Hom groups Mhom (hhom (X), hhom (Y )) are finite-dimensional Q-vector spaces. Proof Suppose first that k admits a complex embedding. Then the first statement follows from the comparison theorems between classical cohomology theories [An, 3.4.2]; the second one follows from taking Betti cohomology, which has rational coefficients. In general, X and Y are defined over some finitely generated subfield k0 of k, and k0 has a complex embedding. If we pick two classical Weil cohomology theories over k, then the base change comparison theorems show that we may compare them with the corresponding ones over k0 ; in turn we may compare the latter two with the help of the complex embedding. Remark 14.1.6. In arbitrary characteristic, the dimension of H i (M ) for M ∈ Mrat is independent of the choice of the (classical) Weil cohomology H [An, P 4.2.5.2], and the Euler characteristic (−1)i dim H i (M ) is independent of the choice of the (arbitrary) Weil cohomology H because it equals the trace tr(1M ) computed in the rigid category Mrat . For example, for a curve C of genus g one always has dim H 1 (C) = 2g. Unless otherwise specified, all Weil cohomology theories considered in this paper will be classical.
14.1.4 Chow-K¨ unneth decompositions Let X ∈ V, X = Xd . We say that X has a Chow-K¨ unneth decomposition in Mrat (C-K for short) if there exist orthogonal projectors πi = πi (X) ∈
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Corr0 (X, X) = Ad (X ×X), for 0 ≤ i ≤ 2d, such that cld (πi ) is the (i, 2d−i)component of ∆X in H 2d (X × X) and X [∆X ] = πi . 0≤i≤2d
This implies that in Mrat the motive h(X) decomposes as follows: M h(X) = hi (X) (14.2) 0≤i≤2d
where hi (X) = (X, πi , 0). Moreover H ∗ (hi (X)) = H i (X),
H∗ (hi (X)) = Hi (X)
(see 14.1.3.2). t for all i, we say that the C-K decomposition is If we have πi = π2d−i self-dual.
14.1.5 Triangulated motives eff (k) DMgm
Let be the triangulated category of effective geometrical motives constructed by Voevodsky [Voev2]: there is a covariant functor M : Sm/k → eff (k) where Sm/k is the category of smooth schemes of finite type over DMgm eff (k, Q) for the pseudo-abelian hull of the category k. We shall write DMgm eff obtained from DMgm (k) by tensoring morphisms with Q, and usually abbreeff . By abuse of notation, we shall denote by Q(1) the image viate it into DMgm eff eff (k) → DM eff (k, Q). of Z(1) in DMgm (k, Q) under the natural functor DMgm gm eff By [Voev2, p. 197], M induces a covariant functor Φ : Meff → DMgm rat which is a full embedding by [Voev3] and sends L to Q(1)[2] (this functor is already defined and fully faithful on the level of Chow motives with integral coefficients). As in [Voev2], we denote by DMgm := DMgm (k, Q) the eff by inverting Q(1). category obtained from DMgm eff admits a natural full embedding, as a tensor trianThe category DMgm gulated category, into the category DM−eff := DM−eff (k) ⊗ Q of (bounded above) motivic complexes [Voev2, 3.2]. If the motive h(X) of a smooth projective variety X has a Chow-K¨ unneth decomposition in Mrat as in (14.2), eff we have the we write Mi (X) = Φ(hi (X)), so that in the category DMgm following decomposition M M (X) = Mi (X). 0≤i≤2d
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14.1.6 Abelian varieties We denote by Ab(k) or Ab the category whose objects are abelian k-varieties and, for A, B ∈ Ab, Ab(A, B) = Hom(A, B) ⊗ Q. Recall that these are finite-dimensional Q-vector spaces (see [Mum2, p. 176]). 14.2 Chow-K¨ unneth decomposition for surfaces In this section we adapt in Proposition 14.2.1 the construction of a suitable Chow-K¨ unneth decomposition (see [Mu1] and [Sch]) for a smooth projective surface S to our covariant setting for Mrat . Then we refine this decomposition in Proposition 14.2.3.
14.2.1 The covariant Chow-K¨ unneth decomposition for surfaces For a smooth projective variety X = Xd , we denote by AlbX and by Pic0X the Albanese variety and the Picard variety of X; T (X) denotes the Albanese kernel of X, i.e the kernel of the map Ad (X)0 → AlbX (k)Q , where Ad (X)0 is the group of 0-cycles of degree 0. From [Mu1], [Mu2] and [Sch, §4] it follows that, in CHM , there exist projectors p0 , p1 , p2d−1 , p2d in EndCHM (ch(X)) = Ad (X × X) with the following properties: i) p0 = (1/n)[P × X] and p2d = (1/n)[X × P ], P a closed point on X of degree n with separable residue field. ii) p2d operates as 0 on Ai (X) for i 6= d; on Ad (X) we have F 1 Ad (X) = Ker p2d = Ad (X)num . iii) p1 , the “Picard projector”, operates as 0 on all Ai (X) with i 6= 1; its image on A1 (X) is A1 (X)hom , hence A1 (ch1 (X)) = Pic0X (k)Q where ch1 (X) = (X, p1 , 0). Moreover, on A1 (X)hom p1 operates as the identity. iv) p2d−1 , the “Albanese projector”, operates as 0 on all Ai (X) with i 6= d; on Ad (X) its image lies in Ad (X)hom and on Ad (X)hom its kernel is T (X); hence Ad (ch2d−1 (X)) = AlbX (k)Q where ch2d−1 (X) = (X, p2d−1 , 0). v) F 2 Ad (X) = Ker(p2d−1 ) ∩ F 1 = T (X). vi) p0 , p1 , p2d−1 , p2d are mutually orthogonal. vii) p2d−1 = pt1 . Note that P exists by [EGA4, 17.15.10 (iii)]. Also, the motive ch0 (X) is in general not isomorphic to 1, but rather to ch(Spec k 0 ) where k 0 is the field of constants of X. If X is equidimensional but reducible, to get the “right”
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p0 and p2d we need to take the sum of the corresponding projectors for all irreducible components Xα of X; then we get M M a ch0 (X) = ch0 (Xα ) ' ch(Spec kα0 ) = ch( Spec kα0 ) where kα0 is the field of constants of Xα : thus ch0 (X) is an Artin motive (cf. [An, 4.1.6.1]). Note finally that the existence of p0 , p1 , p2d−1 , p2d with properties (i) – (vii) is part of a conjectural Chow-K¨ unneth decomposition in CHM for any variety X (see [Mu2] and §14.3). In the case of a smooth projective surface S a Chow-K¨ unneth decomposition of the motive ch(S) always exists by [Mu1] and [Sch, §4]. The following proposition is just a translation of these results in Mrat . Proposition 14.2.1. Let S be a smooth projective connected surface over k and let P ∈ S be a separable closed point. There exists a Chow-K¨ unneth deL composition of h(S) in Mrat : h(S) = 0≤i≤4 hi (S), with hi (S) = (S, πi , 0), πi = πi (S) ∈ A2 (S × S), with the following properties: t i) πi = π4−i for 0 ≤ i ≤ 4. ii) π0 = (1/ deg(P ))[S × P ], π4 = (1/ deg(P ))[P × S]. iii) There exists a curve C ⊂ S of the form C1 ∪C2 , where C1 = S ·H is a general (smooth) hyperplane section of S, such that π1 is supported on S × C (and hence π3 is supported on C × S). iv) Let i1 : C1 → S be the inclusion and ξ = (i1 )∗ i∗1 : h(S) → h(S)(1). Then the compositions ξ
h3 (S) → h(S) −→ h(S)(1) → h1 (S)(1) ξ2
h4 (S) → h(S) −→ h(S)(2) → h0 (S)(2) are isomorphisms. We set π2 := ∆(S) − π0 − π1 − π3 − π4 and h2 (S) = (S, π2 , 0). We have the following tables: M=
h0 (S)
h1 (S)
h2 (S)
h3 (S)
h4 (S)
A0 (M ) = A1 (M ) = A2 (M ) =
A0 (S) 0 0
0 Pic0S (k)Q
0 NS(S)Q T (S)
0 0 AlbS (k)Q
0 0
0
A2num (S)
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M=
h0 (S)
h1 (S)
h2 (S)
h3 (S)
h4 (S)
A0 (M ) = A1 (M ) = A2 (M ) =
Anum (S) 0 0 0
AlbS (k)Q 0 0
T (S) NS(S)Q 0
0 Pic0S (k)Q
0 0 A2 (S)
0
and EndMrat (h1 (S)) = EndAb (AlbS ) EndMrat (h3 (S)) = EndAb (PicS ). Proof In conformity with Lemma 14.1.2, we take πi = pti , where pi are the projectors defined in [Sch, §4]. (The first table is copied from [Sch, p.178]; by Remark 14.1.3, we have the general formula Ai (hj (X)) = Ad−i (h2d−j (X)) for a self-dual C-K decomposition.) For later use we add some precisions on the construction of the pi , hence of the πi , especially concerning rationality issues, making [Sch, 4.2] more specific. Contrary to the case of P , the curve C1 may always be chosen as defined over k and geometrically connected: this is clear if k is infinite by Bertini’s theorem [Ha, p. 179, Th. 8.18 and p. 245, Rem. 7.9.1], and in case k is finite this can also be achieved up to enlarging the projective embedding of S as in [Del-I, 5.7]. In Proposition 14.2.1 (ii) the curve C2 enters the picture because, in the case of a surface, in order to get mutually orthogonal idempotents π1 and π3 , one has to introduce a “correction term” (see [Mu1] and [Sch, p. 177]) as follows: in the contravariant setting one first takes projectors p?1 , p?3 defined in [Mu1] verifying p?3 = (p?1 )t and p?3 p?1 = 0, and then one corrects them by p1 = p?1 − 12 (p?1 ◦ p?3 ) and p3 = p?3 − 21 (p?1 ◦ p?3 ).† The correction term 12 (p?1 ◦ p?3 ) P is supported on C2 × C2 and is of the form Aλ × A0λ , where each Aλ , A0λ is a divisor on S, homologically equivalent to 0. Supposing k perfect, the projector π1 may be described more precisely as follows. Let i : C → S be the closed embedding. Replace C by its ˜ which is smooth. There is a divisor class D ∈ A1 (S × C) ˜ normalization C, such that π1 = (1S טı)(D), where ˜ı : C˜ → S is the proper morphism induced by the projection C˜ → C and ˜ı∗ is the correspondence given by the graph Γ˜ı of. With this description we then have π3 = Dt ◦ ˜ı∗ , where ˜ı∗ = Γ˜tı . Note that Dt (R) is a divisor homologically equivalent to 0 on S for every divisor R on C. † This correction is the one from [Sch] which is different from the one in [Mu1]: its advantage is that p3 = pt1 .
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14.2.2 The refined Chow-K¨ unneth decomposition We now introduce the motive t2 (S), whose construction had been outlined in [An, 11.1.3] in a special case. We start with a well-known lemma (cf. [Bo, §4, no 2, Formula (12) p. 77]): Lemma 14.2.2. Let µ : V × W → Q be a perfect pairing between two finite dimensional Q-vector spaces V, W . Let (ei )1≤i≤n be a basis of V and let (e∗i ) be the dual basis of W with respect to this pairing (µ(ei , e∗j ) = δij ). Then, in Hom(V ⊗ W, Q) ' Hom(V, W ∗ ), we have the identity X µ−1 = ei ⊗ e∗i where we have viewed µ as an isomorphism in Hom(W ∗ , V ). In particular, the right hand side is independent of the choice of the basis (ei ).
Proposition 14.2.3. Let S be a surface provided with a C-K decomposition as in Proposition 14.2.1. Let ks be a separable closure of k, Gk = Gal(ks /k) and NSS = NS(S ⊗k ks )Q be the (Q-linear, geometric) N´eron-Severi group of S viewed as a Gk -module. Then there is a unique splitting π2 = π2alg + π2tr which induces a decomposition h2 (S) ' halg 2 (S) ⊕ t2 (S) alg tr where halg 2 (S) = (S, π2 , 0) ' h(NSS )(1) and t2 (S) = (S, π2 , 0). Here h(NSS ) is the Artin motive associated to NSS . Moreover the tables of Proposition 14.3.6 refine as follows:
M=
halg 2 (S)
t2 (S)
A0 (M ) = A1 (M ) = A2 (M ) =
0 NS(S)Q 0
0 0 T (S)
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M=
halg 2 (S)
t2 (S)
A0 (M ) = A1 (M ) = A2 (M ) =
0 NS(S)Q 0
T (S) 0 0
Finally 2 2 H 2 (S) = Halg (S) ⊕ Htr (S) = π2alg H 2 (S) ⊕ π2tr H 2 (S)
= (NS(S) ⊗ K) ⊕ H 2 (t2 (S)) 2 (S) is (by definition) the “transcendental cohomology”. where Htr
Proof Choose a finite Galois extension E/k such that the action of Gk on NSS factors through G = Gal(E/k). Let [Di ] be an orthogonal basis of NSS = NS(SE )Q . It follows from Lemma 14.2.2 that X 1 [Di ] ⊗ [Di ] ∈ NS(SE )Q ⊗ NS(SE )Q h[Di ], [Di ]i i
is G-invariant, where h[Di ], [Di ]i are the intersection numbers. By Proposition 14.2.1, the k-rational projector π2 defines a G-equivariant section σ of the projection A1 (SE ) → NS(SE )Q . The composition σ⊗σ
∩
λ : NS(SE )Q ⊗ NS(SE )Q −→ A1 (SE ) ⊗ A1 (SE ) −→ A2 ((S × S)E ) is also G-equivariant. It follows that X 1 π2alg = λ( [Di ] ⊗ [Di ]) ∈ A2 ((S × S)E ) h[Di ], [Di ]i i
is a G-invariant cycle, hence descends uniquely to a correspondence π2alg ∈ A2 (S × S). Over E, the correspondences αi =
1 1 [Di × Di ] = λ( [Di ] ⊗ [Di ]) h[Di ], [Di ]i h[Di ], [Di ]i
are mutually orthogonal idempotents, are orthogonal to πj for j 6= 2, and verify π2 ◦ αi = αi ◦ π2 = αi . It follows that their sum π2alg is an idempotent orthogonal to πj for j 6= 2, and that π2 ◦ π2alg = π2alg ◦ π2 = π2alg . We define π2tr := π2 − π2alg .
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In order to prove the isomorphism halg 2 (S) ' h(NSS )(1), it is enough to show that Mi := (SE , αi , 0) ' L for 1 ≤ i ≤ ρ, where ρ is the Picard number. Define fi : L → Mi by fi = αi ◦[Di ]◦1Spec E = [Di ]. The transpose fit is a morphism Mi → L and, by taking gi = h[Di1,Di ]i fit , we get gi ◦ fi = 1L and fi ◦ gi = αi , hence the required isomorphism. 1 From the construction above we also get A∗ (halg 2 (S)) = A (h2 (S)) = NS(S)Q , A∗ (t2 (S)) = T (S). This shows that
Mrat (L, t2 (SE )) = Mrat (t2 (SE ), L) = 0 for any extension E/k. Taking E as above, we get that alg Mrat (halg 2 (SE ), t2 (SE )) = Mrat (t2 (SE ), h2 (SE )) = 0
hence by descent that alg Mrat (halg 2 (S), t2 (S)) = Mrat (t2 (S), h2 (S)) = 0.
Therefore EndMrat (h2 (S)) = EndMrat (halg 2 (S)) × EndMrat (t2 (S)) which implies the uniqueness of the decomposition π2 = π2alg + π2tr . The assertions on cohomology immediately follow from the definition of and π2tr .
π2alg
2 (S) = 0. If chark = 0, this implies Corollary 14.2.4. t2 (S) = 0 ⇒ Htr pg = 0.
Proof The first assertion is obvious from Proposition 14.2.3. The second one is classical [B2]. Definition 14.2.5. For a surface S, we call the set of projectors {π0 , π1 , π2alg , π2tr , π3 , π4 } the refined Chow-K¨ unneth decomposition associated to the C-K decomposition {π0 , π1 , π2 , π3 , π4 }.
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14.3 On some conjectures In this section, we show in Theorem 14.3.10 that part of the results proved by U. Jannsen in [J2, Prop 5.8] hold unconditionally for the Chow motives coming from surfaces. We first recall the conjectures about the existence of a C-K decomposition as formulated in [Mu2] (see also [J2]). Conjecture 14.3.1 (see Conj. A in [Mu2]). Every smooth projective variety X has a Chow-K¨ unneth decomposition. The conjecture is true in particular for curves, surfaces, abelian varieties, uniruled 3-folds and Calabi-Yau 3-folds. If X and Y have a C-K decomposition, with projectors πi (X) and πj (Y ) (0 ≤ i ≤ 2d, d = dim X and 0 ≤ j ≤ 2e, e = dim Y ) then Z = X × Y also has a C-K decomposition with projectors πm (Z) given by πm (Z) = P r+s=m πr (X) × πs (Y ) whith 0 ≤ m ≤ 2(d + e). If the motive h(X) is finite-dimensional in the sense of Kimura [Ki] and the K¨ unneth components of the diagonal are algebraic (i.e., are classes of L algebraic cycles), then h(X) = 0≤i≤2d hi (X) and the motives hi (X) are unique, up to isomorphism as follows from the results of [Ki] (see §14.6). Now let X have a C-K decomposition and consider the action of the correspondence πi (X) on the Chow groups Aj (X). Then Conjecture B in [Mu2] translates as follows in our covariant setting (identical statement): Conjecture 14.3.2 (Vanishing Conjecture). The correspondences πi (X) act as 0 on Aj (X) for i < j and for i > 2j. Assuming that Conjectures 14.3.1 and 14.3.2 hold, one may define a decreasing filtration F • on Aj (X) as follows: F 1 Aj (X) = Ker π2j , F 2 Aj (X) = Ker π2j ∩ Ker π2j−1 , . . . F ν Aj (X) = Ker π2j ∩ Ker π2j−1 , ∩ · · · ∩ Ker π2j−ν+1 . Note that, with the above definitions, F j+1 Aj (X) = 0. Also it easily follows from the definition of F • (see [Mu2, 1.4.4]) that: F 1 Aj (X) ⊂ Aj (X)hom . Conjecture 14.3.3 (see Conj. D in [Mu2]). F 1 Aj (X) = Aj (X)hom , for all j. Finally we mention: Conjecture 14.3.4 (see Conj. C in [Mu2]). The filtration F • is independent of the choice of the πi (X). Remark 14.3.5. Jannsen [J2] has shown that if the Conjectures 14.3.1 ... 14.3.4 hold for every smooth projective variety over k, then the filtration F • satisfies Beilinson’s Conjecture. The converse also holds [J2, 5.2].
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Now let X and Y be two smooth projective varieties: the following result, due to U. Jannsen, relates Conjectures 14.3.2 and 14.3.3 for Z = X × Y , with the groups Mrat (hi (X), hj (Y )). Proposition 14.3.6 ([J2, Prop 5.8]). Let X and Y be smooth projective varieties of dimensions respectively d and e, provided with C-K decompositions, and let Z = X × Y be provided with the product C-K decomposition. i) If Z satisfies the vanishing Conjecture 14.3.2, then: Mrat (hi (X), hj (Y )) = 0 if j < i; 0 ≤ i ≤ 2d; 0 ≤ j ≤ 2e. ii) If Z satisfies Conjecture 14.3.3 then Mrat (hi (X), hi (Y )) ' Mhom (hhom (X), hhom (Y )). i i In particular, if X × X satifies Conjecture 14.3.3, then b) implies that the Q-vector space EndMrat (hi (X)) has finite dimension for 0 ≤ i ≤ 2d, at least if chark = 0 and the Weil cohomology is classical (cf. Lemma 14.1.5). Remark 14.3.7. Note that, because of our covariant definition of the functor h : V → Mrat , in a) we have j < i, while in the contravariant setting (as in [J2, 5.8]) one has i < j. Corollary 14.3.8. Let S be a smooth projective surface and C a smooth P P projective curve. Let ∆S = 0≤i≤4 πi (S) and ∆C = 0≤i≤2 πj (C) be C-K decompositions respectively for S and for C. Then ( i > j and Γ ∈ A1 (S × C) (i) πj (C) · Γ · πi (S) = 0 if i = j and Γ ∈ A1 (S × C)hom ( i > j and Γ ∈ A2 (C × S) (ii) πj (S) · Γ · πi (C) = 0 if i = j and Γ ∈ A2 (C × S)hom ( r < 2 + s and Γ ∈ A1 (S × C) (iii) πr (S) · Γt · πs (C) = 0 if r = 2 + s and Γ ∈ A1 (S × C)hom ( r + 2 < s and Γ ∈ A2 (C × S) (iv) πr (C) · Γt · πs (S) = 0 if r + 2 = s and Γ ∈ A2 (C × S)hom . Proof By the results in [Mu2, Prop. 4.1], Conjectures 14.3.1, 14.3.2 and 14.3.3 hold for the product Z = S × C. Therefore Proposition 14.3.6 applies to S × C (and C × S). Then (1) follows from the equality: πj (C) · Γ · πi (S) ∈ A2 (S × C) = Mrat (hi (S), hj (C)) and similarly for (2), (3) and (4).
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Corollary 14.3.9. Let S and C be as in Corollary 14.3.8. Then: i) Mrat (h1 (C)(1), h2 (S)) = 0; ii) Mrat (h2 (S)), h1 (C)) = 0. Proof The first assertion follows from the equality : Mrat (h1 (C)(1), h2 (S)) = π2 (S) ◦ A1 (C × S) ◦ π1 (C) by applying (3) of Cor. 14.3.8 to Γt for any Γ ∈ A1 (S × C). b) follows from Mrat (h2 (S), h1 (C)) = π1 (C) ◦ A1 (S × C) ◦ π2 (S) and from (1). The next result shows that, in the case of two surfaces S and S 0 , part of Proposition 14.3.6 holds without assuming any conjecture for S × S 0 .† Theorem 14.3.10. Let S and S 0 be smooth projective surfaces over the field k. Then for any C-K decompositions as in Proposition 14.2.1 M M h(S) = hi (S); h(S 0 ) = hj (S 0 ) 0≤i≤4
0≤j≤4
where hi (S) = (S, πi (S), 0) and hj (S 0 ) = (S 0 , πj0 (S 0 ), 0), we have i) Mrat (hi (S), hj (S 0 )) = 0 for all j < i and 0 ≤ i ≤ 4 ii) Mrat (hi (S), hi (S 0 )) ' Mhom (hhom (S), hhom (S 0 )) for i 6= 2. i i Proof Let πi = πi (S) and πj0 = πj (S 0 ). Then S ×S 0 has a C-K decomposition P 0 defined by the projectors r+s=m πr × πs . For any correspondence Z ∈ A2 (S × S 0 ) let us define αji (Z) = πj0 ◦ Z ◦ πi for 0 ≤ i, j ≤ 4. Then, in order to prove part (i) it is enough to show that αji (Z) = 0 for j < i. We will show that α12 (Z) = α23 (Z) = 0: the other cases are easier and follow from the same type of arguments. Let α12 = π10 ◦ Z ◦ π2 : from the construction of the projectors {πi } and {πj0 } in Proposition 1, it follows that π10 = j∗ ◦ D where j : C 0 → S 0 is the closed embedding of the curve C 0 in S 0 and D ∈ A1 (S 0 × C 0 ). By possibly taking a desingularization for each irreducible component of C 0 we get a † Kenichiro Kimura recently informed the second author that he had also found a proof, for the case of the product of two surfaces, of conjecture 14.3.2 and of part of conjecture 14.3.3.
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morphism Y 0 → S 0 where Y 0 is a smooth projective curve. Also, by arguing componentwise, we may as well assume that Y 0 is irreducible and we may replace C 0 by such a Y 0 . Then α12 = j∗ ◦ D ◦ Z ◦ π2 = j∗ ◦ D1 ◦ π2 , with D1 = D ◦ Z ∈ A1 (S × Y 0 ) and D ◦ Z ◦ π2 ∈ HomMrat (h2 (S), h(Y 0 )). P Let us take a C-K decomposition h(Y 0 ) = 0≤j≤2 hj (Y 0 ), where hj (Y 0 ) = (Y 0 , πj (Y 0 ), 0). By applying Corollary 14.3.8 to S × Y 0 , we get πj (Y 0 ) ◦ D1 ◦ π2 (S) = 0 for j = 0, 1 so that D ◦ Z ◦ π2 (S) = π2 (Y 0 ) ◦ D ◦ Z ◦ π2 (S). If π2 (Y 0 ) = [R0 × Y 0 ], with R0 a chosen rational point on Y 0 , then π2 (Y 0 ) ◦ D1 = D1t (R0 ) × Y 0 and D1t (R0 ) = Z t (Dt (R0 )). From the chosen normalization in the construction of the projectors {πi (S)} and {πj (S 0 )} (see the proof of Proposition 14.2.1) it follows that Dt (R0 ) ∈ A1 (S 0 )hom and D1t (R0 ) ∈ A1 (S)hom . Therefore we get: D ◦ Z ◦ π2 (S) = π2 (Y 0 ) ◦ D ◦ Z ◦ π2 (S) = π2 (Y 0 ) ◦ D1 ◦ π2 (S) = (D1t (R0 ) × Y 0 ) ◦ π2 (S) = π2 (S)(D1t (R0 )) × Y 0 = 0 since π2 (S)(D1t (R0 )) = 0 because π2 (S)(A1 (S)hom ) = 0. Therefore α12 (Z) = 0. To show that α23 (Z) = π2 (S 0 ) ◦ Z ◦ π3 (S) = 0 it is enough to look at t . Then αt (Z) = π (S) ◦ Z t ◦ π (S 0 ). By the transpose correspondence α23 1 2 23 0 t (Z) = 0, hence α (Z) = 0. applying the previous case to S × S we get α23 23 We now prove part (ii): for Z homologically equivalent to 0, α11 (Z) = α33 (Z) = 0 follows from the definition of {πi (S)} and {πj (S 0 )} in Proposition 14.2.1 and from the following result in [Sch, 4.5] (by interchanging π1 and π3 because of our covariant set-up): Mrat (h1 (S), h1 (S 0 )) = Ab(AlbS , AlbS 0 ) Mrat (h3 (S), h3 (S 0 )) = Ab(Pic0S , Pic0S 0 ). Both equalities hold also with Mrat replaced by Mhom and therefore we get (ii). The equalities α00 (Z) = α44 (Z) = 0 are trivial because Mrat (hj (S), hj (S 0 )) = Mhom (hhom (S), hhom (S 0 )) ' Q j j for j = 0, 4. Summarizing what we have done so far with Proposition 14.3.6 in mind, let us display our information on the Hom groups Mrat (hi (S), hj (S 0 )) in matrix
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form (r = h means “rational equivalence = homological equivalence”): r=h 0 0 0 0 ∗ r=h 0 0 0 ∗ ∗ ? 0 0 ∗ ∗ ∗ r=h 0 ∗ ∗ ∗ ∗ r=h In the next section we study the remaining group on the diagonal: the one marked with a ‘?’.
14.4 The group Mrat (h2 (S), h2 (S 0 )) Let S, S 0 be smooth projective surfaces over the field k: from Proposition 14.2.1 and from Theorem 14.3.10 it follows that for any C-K decompositions L L 0 0 h(S) = 0≤i≤4 hi (S ) as in Proposition 14.2.1 0≤i≤4 hi (S) and h(S ) = 0 in Mrat , the Q-vector spaces Mrat (hi (S), hi (S )) are finite dimensional for i 6= 2. In fact we have Mrat (h0 (S), h0 (S 0 )) ' Mrat (h4 (S), h4 (S 0 )) ' Q (if S and S 0 are geometrically connected), and Mrat (h1 (S), h1 (S 0 )) ' Ab(AlbS , AlbS 0 ) Mrat (h3 (S), h3 (S 0 )) ' Ab(PicS , PicS 0 ). Moreover, from Proposition 14.3.6 (ii) it follows that, if S × S 0 satisfies Conjecture 14.3.3, then Mrat (h2 (S), h2 (S 0 )) is also a finite dimensional Qvector space, at least in characteristic 0 and for a classical Weil cohomology. In the case k = C, if the surface S has geometric genus 0 then the isomorphism Mrat (h2 (S), h2 (S)) ' Mhom (hhom (S), hhom (S)) in Proposition 14.3.6 2 2 (ii) holds if and only if Bloch’s conjecture holds for S i.e. if and only if the Albanese kernel T (S) vanishes (see §14.6). It is therefore natural to ask how the group Mrat (h2 (S), h2 (S 0 )) may be computed. We have Lemma 14.4.1. There is a canonical isomorphism alg 0 0 Mrat (h2 (S), h2 (S 0 )) ' Mrat (halg 2 (S), h2 (S )) ⊕ Mrat (t2 (S), t2 (S ))
where t2 (S) and t2 (S 0 ) are defined in Proposition 14.2.3.
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Proof It suffices to see that alg 0 0 Mrat (t2 (S), halg 2 (S )) = Mrat (h2 (S), t2 (S )) = 0
which follows immediately from Proposition 14.2.3 (see its proof). 0
alg 0 ρρ Since Mrat (halg 2 (S), h2 (S )) ' Q , this lemma reduces the study of Mrat (h2 (S), h2 (S 0 )) to that of Mrat (t2 (S), t2 (S 0 )). In this section we give two descriptions of this group: one as a quotient of A2 (S × S 0 ) (Theorem 14.4.3) and the other in terms of Albanese kernels (theorem 14.4.8). Then, in §14.5, we will relate these results with the birational motives of S and S 0 i.e. with the images of h(S) and h2 (S 0 ) in the category Morat (k) of birational motives of [K-S].
14.4.1 First description of Mrat (t2 (S), t2 (S 0 ))
We start with Definition 14.4.2. Let X = Xd and Y = Ye be smooth projective varieties over k: we denote by J(X, Y ) the subgroup of Ad (X × Y ) generated by the classes supported on subvarieties of the form X × N or M × Y , with M a closed subvariety of X of dimension < d and N a closed subvariety of Y of dimension < e. In other words: J(X, Y ) is generated by the classes of correspondences which are not dominant over X and Y by either the first or the second projection. Note that J(X, Y ) = Ad (X ×Y ) if d < e (project to Y ). In the case X = Y J(X, X) is a two-sided ideal in the ring of correspondences Ad (X × X) (see [Fu, p. 309]). Now let S and S 0 be smooth projective surfaces over k and let {πi = πi (S)} and {πi0 = πi (S 0 )}, for 0 ≤ i ≤ 4, be projectors giving C-K decompositions respectively for S and for S 0 as in Proposition 14.2.1. Then, as in Proposition 14.2.3, π2 (S) = π2alg (S) + π2tr (S), h2 (S) ' ρL ⊕ t2 (S), where t2 (S) = (S, π2tr (S), 0) and ρ is the Picard number of S. Similarly π2 (S 0 ) = π2alg (S 0 ) + π2tr (S 0 ), h2 (S 0 ) ' ρ0 L ⊕ t2 (S 0 ) where ρ0 is the Picard number of S 0 . Let us define a homomorphism Φ : A2 (S × S 0 ) → Mrat (t2 (S), t2 (S 0 )) as follows: Φ(Z) = π2tr (S 0 ) ◦ Z ◦ π2tr (S). Then we have the following result.
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Theorem 14.4.3. The map Φ induces an isomorphism 0 ¯ : A2 (S × S ) ' Mrat (t2 (S), t2 (S 0 )). Φ J(S, S 0 )
Proof From the definition of the motives t2 (S) and t2 (S 0 ) it follows that Mrat (t2 (S), t2 (S 0 )) = {π2tr (S 0 ) ◦ Z ◦ π2tr (S) | Z ∈ A2 (S × S 0 )}. We first show that J(S, S 0 ) ⊂ Ker Φ. Let Z ∈ J(S, S 0 ): we may assume that Z is irreducible and supported either on S × Y 0 with dim Y 0 ≤ 1 or on Y × S 0 with dim Y ≤ 1. Suppose Z is supported on S × Y 0 . The case dim Y 0 = 0 being easy, let us assume that Y 0 is a curve which, by possibly taking a desingularization (compare proof of Proposition 14.2.1), we may take to be smooth and irreducible. Let j : Y 0 → S 0 ; then Z = j∗ ◦ D, where j∗ is the graph Γj and D ∈ A1 (S × Y 0 ). Using the identity π2 (S) ◦ π2tr (S) = π2tr (S) ◦ π2tr (S) = π2tr (S) we get π2tr (S 0 ) ◦ Z ◦ π2tr (S) = π2tr (S 0 ) ◦ j∗ ◦ D ◦ π2 (S) ◦ π2tr (S). Let ∆Y 0 = π0 (Y 0 )+π1 (Y 0 )+π2 (Y 0 ) be a C-K decomposition. By Corollary 14.3.8 (a) π1 (Y 0 ) ◦ D ◦ π2 (S) = π0 (Y 0 ) ◦ D ◦ π2 (S) = 0 hence D ◦ π2 (S) = π2 (Y 0 ) ◦ D ◦ π2 (S). Let R0 be a rational point on Y 0 such that π2 (Y 0 ) = [R0 × Y 0 ]; then π2 (Y 0 ) ◦ D = [D(R0 )t × Y 0 ] and D ◦ π2tr (S) = D ◦ π2 (S) ◦ π2tr (S) = [D(R0 )t × Y 0 ] ◦ π2tr (S) = [π2tr (S)(D(R0 )t ) × Y 0 ]. From A2 (t2 (S)) = T (S) it follows that π2tr (S) acts as 0 on divisors, hence [π2tr (S)(D(R0 )t ) × Y 0 ] = 0 and π2tr (S 0 ) ◦ Z ◦ π2tr (S) = 0. This completes the proof in the case Z has support on S × Y 0 . Let us now consider the case when Z is supported on Y × S 0 , Y a curve on S. In order to show that π2tr (S 0 ) ◦ Z ◦ π2tr (S) = 0 we can just take the transpose. Then we get π2tr (S) ◦ Z t ◦ π2tr (S 0 ) and this brings us back to the previous case. Therefore Φ induces a map ¯ : A2 (S × S 0 )/J(S, S 0 ) → Mrat (t2 (S), t2 (S 0 )) Φ ¯ is injective. which is clearly surjective, and we are left to show that Φ 0 tr 0 tr Let Z ∈ A2 (S × S ) be such that π2 (S ) ◦ Z ◦ π2 (S) = 0: we claim that Z ∈ J(S, S 0 ).
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Let ξ be the generic point of S. To prove our claim we are going to evaluate (π2tr (S 0 ) ◦ Z ◦ π2tr (S))(ξ) over k(ξ). By using Chow’s moving lemma on S × S 0 we may choose a cycle in the class of Z in A2 (S × S 0 ) (which we will still denote by Z) such that π2tr (S 0 ) ◦ Z ◦ π2tr (S) is defined as a cycle and π2tr (S 0 ) ◦ Z ◦ π2tr (S)(ξ) can be evaluated using the formula (α ◦ β)(ξ) = α(β(ξ)), for α, β ∈ A2 (S × S 0 ). From the definition of the projector π2tr (S) in Proposition 14.2.3, we have π2tr (S) = ∆S − π0 (S) − π1 (S) − π2alg (S) − π3 (S) − π4 (S) where π2alg (S), π3 (S) and π4 (S) act as 0 on 0-cycles, while π0 (S)(ξ) = P if π0 (S) = [S × P ] and π1 (S)(ξ) = Dξ , where Dξ is a divisor (defined over k(ξ)) on the curve C = C(S) used to construct π1 (S). Therefore π2tr (S))(ξ) = ξ − P − Dξ and (π2tr (S 0 ) ◦ Z ◦ π2tr (S))(ξ) = π2tr (S 0 )(Z(ξ) − Z(P ) − Z(Dξ )) = 0 By the same argument as before, applied to the projectors {πi (S 0 )}, we get 0 = (π2tr (S 0 )(Z(ξ) − Z(P ) − Z(Dξ )) = (Z(ξ) − Z(P ) − Z(Dξ )) − mP 0 − π1 (S 0 )(Z(ξ) − Z(P ) − Z(Dξ )) where P 0 is a rational point defining π0 (S 0 ) and m is the degree of the 0-cycle Z(ξ) − Z(P ) − Z(Dξ ). The cycle π1 (S 0 )(Z(P ) + Z(Dξ )) = Dξ0 is a divisor (defined over k(ξ)) on the curve C 0 = C(S 0 ) appearing in the construction of π1 (S 0 ). Therefore we get from π2tr (S 0 ) ◦ Z ◦ π2tr (S) = 0: Z(ξ) = Z(P ) + Z(Dξ ) + mP 0 + π1 (S 0 )(Z(ξ)) − Dξ0 . The cycle on the right hand side is supported on a curve Y 0 ⊂ S 0 , with Y the union of Z(C) and C 0 . Therefore, by taking the Zariski closure in S × S 0 of both sides of the above formula we get : 0
Z = Z1 + Z2 where Z1 , Z2 ∈ A2 (S × S 0 ), Z1 is supported on S × Y 0 with dim Y 0 ≤ 1 and Z2 is a cycle supported on Y × S with Y ⊂ S, dim Y ≤ 1. Therefore Z ∈ J(S, S 0 ).
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Remark 14.4.4. Theorem 14.4.3 is an analogue in the case of surfaces of a well-known result for curves, namely the isomorphism: A1 (C × C 0 ) ' Mrat (h1 (C), h1 (C 0 )) J(C, C 0 ) which immediately follows from the definitions of J(C, C 0 ) and of the motives h1 (C) and h1 (C 0 ). In this case J(C, C 0 ) is the subgroup generated by the classes which are represented by “horizontal” and “vertical” divisors on C × C 0 . The equivalence relation defined by J(C, C 0 ) is denoted in [Weil1, Chap. 6] as “three line equivalence”. In the case of curves, since h1 (C) and h1 (C 0 ) have a “realization” as the Jacobians J(C) and J(C 0 ), the following result in [Weil1, Ch. 6, Thm 22] holds: A1 (C × C 0 ) ' Ab(J(C), J(C 0 )) J(C, C 0 ) where J(C), J(C 0 ) are the Jacobians. P Corollary 14.4.5. Keep the same notation and let Πr = i+j=r πi (S) × πj (S 0 ) be the Chow-K¨ unneth projectors on S × S 0 deduced from those of S and S 0 . Let F • be the filtration on Aj (S × S 0 ) defined by the Πr . Then J(S, S 0 ) ∩ A2 (S × S 0 )hom ' F 1 A2 (S × S 0 ) = Ker Π4 . Therefore S × S 0 satisfies Conjecture 14.3.3 if and only if A2 (S × S 0 )hom ⊂ J(S, S 0 ). Proof For simplicity let us drop (S) and (S 0 ) from the notation for projectors and use πi etc. for those of S and πi0 etc. for those of S 0 . Let Γ ∈ A2 (S × S 0 )hom : from Theorem 14.3.10 πj0 ◦Γ◦πj = 0 for j 6= 2. Therefore Γ ∈ Ker Π4 if and only if π20 ◦ Γ ◦ π2 = 0. By Lemma 14.4.1, it suffices to consider separately the algebraic and transcendental parts. Let Γ ∈ J(S, S 0 ) ∩ A2 (S × S 0 )hom : then (π2tr )0 ◦ Γ ◦ π2tr = 0, because Γ ∈ J(S, S 0 ). Since Γ ∈ A2 (S×S 0 )hom we also have: (π2alg )0 ◦Γ◦π2alg = 0. This follows from the isomorphism h2 (S) = ρL ⊕ t2 (S) where ρL ' (S, π2alg , 0), and the same for S 0 . In fact we have Mrat (ρL, ρ0 L) ' Mhom (ρL, ρ0 L), so that, if Γ ∈ A2 (S ×S 0 )hom , then (π2alg )0 ◦Γ◦π2alg yields the 0 map in Mhom (ρL, ρ0 L), hence it is 0. Therefore π20 ◦ Γ ◦ π2 = 0 which proves that Γ ∈ Ker Π4 . Conversely let Γ ∈ F 1 A2 (S × S 0 ) = Ker Π4 : then Γ ∈ A2 (S × S 0 )hom by [Mu2, 1.4.4]. By Theorem 14.4.3, we also have Γ ∈ J(S, S 0 ) because π20 ◦ Γ ◦ π2 = (π2alg )0 ◦ Γ ◦ π2alg + (π2tr )0 ◦ Γ ◦ π2tr = 0 and (π2alg )0 ◦ Γ ◦ π2alg = 0, since Γ ∈ A2 (S × S 0 )hom .
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14.4.2 Second description of Mrat (t2 (S), t2 (S 0 )) Let us still keep the same notation. 0 Definition 14.4.6. We denote by H≤1 be the subgroup of A2 (Sk(S) ) gen2 0 erated by the subgroups A (SL ), when L runs through all the subfields of k(S) containing k and which are of transcendence degree ≤ 1 over k.
Theorem 14.4.8 below will give a description of Mrat (t2 (S), t2 (S 0 )) in terms 0 of T (Sk(S) ) and H≤1 . We need a preparatory lemma: Lemma 14.4.7. Let S, S 0 and H≤1 be as above. Let ξ be the generic point of S and let Z ∈ A2 (S × S 0 ). Then Z ∈ J(S, S 0 ) if and only if Z(ξ) ∈ H≤1 . Proof Let us denote – by abuse – with the same letter Z both a cycle class and a suitable cycle in this class. Let Z ∈ J(S, S 0 ) ⊂ A2 (S×S 0 ) = A2 (S×S 0 ). If Z has support on Y × S with Y closed in S and of dimension ≤ 1, then Z(ξ) = 0. Therefore we may assume that Z has support on S × Y 0 and by linearity we may take Z to be represented by a k-irreducible subvariety of S×Y 0 . Furthermore, by taking its desingularization if necessary, we may also assume that Y 0 is a smooth curve. Let j : Y 0 → S 0 be the corresponding 0 ) → A2 (S 0 morphism and j∗ : A1 (Yk(ξ) k(ξ) ) the induced homomorphim on Chow groups. Then Z(ξ) = j∗ D(ξ) where D is a k-irreducible divisor on S × Y 0 . Then D(ξ) has a smallest field of rationality L, in the sense of [Weil2, Cor 4 p. 269] with k ⊂ L ⊂ k(ξ). D(ξ) consists of a finite number of points P1 , . . . , Pm on Y 0 each one conjugate to the others over L, and with the same multiplicity. Moreover L is contained in the algebraic closure of k(P1 ), where P1 ∈ Y 0 . Therefore tr degk L ≤ 1. We have D(ξ) ∈ A1 (YL0 ) and Z(ξ) = ˜j∗ D(ξ) where ˜j∗ : A1 (YL0 ) → A2 (SL0 ). Therefore Z(ξ) ∈ H≤1 . Conversely suppose that Z(ξ) ∈ H≤1 ; because of the definition of H≤1 we may assume that Z(ξ) is a cycle defined over a field L with k ⊂ L ⊂ k(ξ) and t = tr degk L ≤ 1. If t = 0 then Z(ξ) is defined over an algebraic extension extension of k, hence Z ∈ J(S, S 0 ). Assume t = 1 and let C be a smooth projective curve with function field L. Since L ⊂ k(ξ) there is a dominant rational map f from S to C. Let U ⊂ S be an open subset such that f is a morphism on U . Then η = f (ξ) is the generic point of C. Moreover, 0 0 Z(ξ) ∈ A2 (Sk(η) ) ⊂ A2 (Sk(ξ) ). Let Z 0 be the closure of Z(ξ) in C × S 0 so that Z 0 (η) = Z(ξ). Let Y 0 ⊂ S 0 be the projection of Z 0 : then dim Y 0 ≤ 1. Consider the morphism (f|U × idS 0 )∗ : A2 (C × S 0 ) → A2 (U × S 0 ) and let Z1 be the cycle in A2 (S ×S 0 ) obtained by taking the Zariski closure of
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(f|U × idS 0 )∗ (Z 0 ). Then Z1 has support on S × Y 0 and Z(ξ) = Z 0 (η) = Z1 (ξ). Therefore Z = Z1 + Z2 where Z2 has support on Y × S 0 , Y a curve on S, hence Z ∈ J(S × S 0 ). Theorem 14.4.8. Let S and S 0 be smooth projective surfaces over k and 0 0 let T (Sk(S) ) and H≤1 be as above. Let us define H = T (Sk(S) ) ∩ H≤1 . Then there is an isomorphism 0
Mrat (t2 (S), t2 (S )) '
0 T (Sk(S) )
H
.
Proof Let us define a homomorphism β
0 ) A2 (S × S 0 ) → T (Sk(S)
by β(Z) = ((π2tr )0 ◦ Z ◦ π2tr )(ξ), with ξ the generic point of S. By Lemma 14.4.7, β induces a map 0 T (Sk(S) ) A2 (S × S 0 ) β¯ : → 0 J(S, S ) H
and, by Theorem 14.4.3 A2 (S × S 0 ) ' Mrat (t2 (S), t2 (S 0 )). J(S, S 0 ) Therefore we are left to show that β¯ is an isomorphism. T (S 0
)
k(S) 0 Let [σ] ∈ , σ a representative in T (Sk(S) ) and Z the Zariski closure H 0 tr 0 of σ in S ×S : then Z(ξ) = σ. Let Z1 = (π2 ) ◦Z ◦π2tr and Z2 = Z −Z1 : then (π2tr )0 ◦ Z2 ◦ π2tr = 0. From Theorem 14.4.3 and Lemma 14.4.7 we get Z2 ∈ 0 J(S, S 0 ) and Z2 (ξ) ∈ H≤1 . On the other hand, both Z(ξ) = σ ∈ T (Sk(S) ) 0 0 and Z1 (ξ) ∈ T (Sk(S) ), hence Z2 (ξ) ∈ H = T (Sk(S) ) ∩ H≤1 . Therefore we get ¯ β(Z) = [Z1 (ξ)] = [Z(ξ) − Z2 (ξ)] = [σ − Z2 (ξ)] = [σ] and this shows that β¯
is surjective. Let Z ∈ A2 (S × S 0 ) be such that β(Z) ∈ H. Let Z1 = (π2tr )0 ◦ Z ◦ π2tr : then Z1 ∈ H≤1 , and by Lemma 14.4.7 Z1 ∈ J(S, S 0 ). By taking Z2 = Z − Z1 as before we have Z2 ∈ J(S, S 0 ), hence Z = Z1 + Z2 ∈ J(S, S 0 ). Therefore β¯ is injective. Corollary 14.4.9. a) t2 (S) = 0 ⇔ T (Sk(S) ) ⊂ H≤1 ⇔ T (Sk(S) ) = 0. b) Suppose that k is algebraically closed and has infinite transcendance degree over its prime subfield. Then t2 (S) = 0 ⇔ T (S) = 0. 2 (S) = 0, c) With the same assumption as in b), T (S) = 0 implies Htr and pg = 0 if chark = 0.
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Proof From Proposition 14.2.3 we get the second implication in the following: t2 (S) = 0 ⇒ t2 (Sk(S) ) = 0 ⇒ T (Sk(S) ) = 0 ⇒ T (Sk(S) ) ⊂ H≤1 the other two being obvious. The implication T (Sk(S) ) ⊂ H≤1 ⇒ t2 (S) = 0 follows from Theorem 14.4.8, hence a). To see b), in view of Proposition 14.2.3 we need only show that T (S) = 0 ⇒ t2 (S) = 0. Note that there exists a finitely generated subfield k0 ⊂ k and a smooth projective k0 -surface S0 such that S ' S0 ×k0 k. The assumption on k implies that the inclusion k0 ⊂ k extends to an inclusion k0 (S0 ) ⊂ k. A standard transfer argument shows that T ((S0 )k0 (S0 ) ) → T (S) is injective. So t2 (S0 ) = 0 by a) and therefore t2 (S) = 0. Finally, c) follows from b) and Corollary 14.2.4.
14.5 The birational motive of a surface In this section we first recall some definitions and results from [K-S] on the category of birational Chow motives (with rational coefficients) over k: this category is denoted there (see 6.1) by CHo (k, Q) or Motorat (k, Q) while we shall denote it here by Morat (k) or even Morat . Then we compute the group ¯ ¯ ¯ Morat (h(S), h(S)) of a surface S, where h(S) is the image of h(S) in Morat . Lemma 14.5.1. For every smooth and projective varieties X and Y , with dim X = d, let I be the subgroup of Meff rat (h(X), h(Y )) = Ad (X × Y ) defined as follows: I(X, Y ) = {f ∈ Ad (X × Y ) | f vanishes on U × Y, U open in X}. Then I is a two-sided tensor ideal in Meff rat . In particular for any smooth projective variety X there is an exact sequence of rings: φ
0 → I(X, X) → Ad (X × X) −→ A0 (Xk(X) ) → 0
(14.3)
where d = dim X and k(X) is the function field of X. If we denote by • the multiplication in A0 (Xk(X) ) defined via (14.3) then, if P and Q are two rational points of X, we have: [P ] • [Q] = [P ] in A0 (Xk(X) ). Proof The fact that I is a tensor ideal in Meff rat is proven in [K-S, 5.3]. We review the proof:
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If X, Y, Z are smooth projective varieties and U ⊂ X is open, then the usual formula defines a composition of correspondences: Adim X (U × Y ) × Adim Y (Y × Z) → Adim X (U × Z) and this composition is compatible with the restriction to any open subset V ⊂ U . Passing to the limit, since: A0 (Yk(X) ) = Adim Y (Yk(X) ) = lim Adim Y (U × Y ) = lim Adim X (U × Y ) U ⊂X
U ⊂X
we get a composition A0 (Yk(X) ) × Adim Y (Y × Z) → A0 (Zk(X) ). If α ∈ A0 (Yk(X) ) and β ∈ I(Y, Z), i.e. if β has support on a closed subset M × Z of Y × Z then β ◦ α = 0 ∈ A0 (Zk(X) ) as one sees by moving α away from M . Therefore we get a a pairing A0 (Yk(X) ) × A0 (Zk(Y ) ) → A0 (Zk(X) )
(14.4)
which, in the case X = Y = Z yields a multiplication • in A0 (Xk(X) ) defined by β¯ • α ¯ =β◦α ¯ denotes its class in A0 (Xk(X) ). where for a correspondence Γ in Ad (X ×X), Γ Let η be the class of the generic point of X in A0 (Xk(X) ), which is the image of the cycle [∆X ] of Ad (X × X) under the map φ in (14.3): then η is the identity for •. Let P and Q be closed points in X, and let [P ] and [Q] be the corresponding elements in A0 (Xk(X) ). By choosing representatives [X × P ] and [X × Q] in Ad (X × X) we get [X × P ] ◦ [X × Q] = [X × P ] in Ad (X × X). This shows that [P ] • [Q] = [P ] in A0 (Xk(X) ). Definition 14.5.2. We denote by Morat the category of birational Chow motives, i.e the pseudo-abelian envelope of the factor category Meff rat /I and, o . We also denote by h ¯ the (covariant) ¯ if M ∈ Meff , by M its image in M rat rat h
o composite functor V −→ Meff rat → Mrat . o Note that under the functor Meff rat → Mrat the Lefschetz motive L goes to 0. By Lemma 14.5.1, one has the following isomorphism in Morat :
¯ ¯ )) ' A0 (Yk(X) ) Morat (h(X), h(Y for X, Y ∈ V. We also have:
(14.5)
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Proposition 14.5.3 ([K-S, 5.3 and 5.4]). A morphism f in Meff rat belongs to the ideal I if and only if it factors through an object of the form M (1). Remark 14.5.4. The proof in [K-S, 5.4] is not correct because Chow’s moving lemma is applied on a singular variety. However, N. Fakhruddin pointed out that it is sufficient to take the subvariety Z appearing in this proof minimal to repair it, and moreover Chow’s moving lemma is then avoided. This correction will appear in the final version. Definition 14.5.5. For all n ≥ 0, we let eff a) d≤n Meff rat denote the thick subcategory of Mrat generated by motives of varieties of dimension ≤ n (thick means full and stable under direct summands). o b) d≤n Morat denote the thick image of d≤n Meff rat in Mrat . eff c) K≤n denote the ideal of Mrat consisting of those morphisms that factor through an object of d≤n Meff rat . o d) K≤n denote the thick image of K≤n in Morat .
For simplicity, we write K≤n (X, Y ) and Ko≤n (X, Y ) for two varieties X, Y instead of K≤n (h(X), h(Y )) and Ko≤n (h(X), h(Y )). Lemma 14.5.6.
a) The functor D(n) : Mrat → Mrat M 7→ Hom(M, Ln ),
where Hom(M, Ln ) = M ∨ ⊗ Ln is the internal Hom in Mrat , sends d≤n Meff rat to itself and defines a self-duality of this category such that D(n) (h(X)) = h(X) for any n-dimensional X. Moreover, for X, Y purely of dimension n, b) The map D(n) : An (X × Y ) → An (Y × X) is the transposition of cycles and in particular D(n) (J(X, Y )) = J(Y, X) where J(X, Y ) is the subgroup of Definition 14.4.2. c) D(n) (I(X, Y )) = K≤n−1 (Y, X) and D(n) (K≤n−1 (X, Y )) = I(Y, X), where I is as in Lemma 14.5.1. d) For X, Y purely of dimension n we have J(X, Y ) = I(X, Y ) + K≤n−1 (X, Y ).
(14.6)
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Proof a) and b) are obvious. For c), the argument in [K-S, proof of 5.4] (see Remark 14.5.4) implies that I(X, Y ) consists of those morphisms that factor through some h(Z)(1), where dim Z = n − 1. A similar argument shows that K≤n−1 (X, Y ) consists of those morphisms that factor through some h(Z) with dim Z = n−1. The claim is now obvious. Finally, d) follows immediately from c) and the definition of J. Lemma 14.5.7. Let S, S 0 be smooth projective surfaces over a field k. For any C-K decompositions as in Proposition 14.2.1 M M hi (S 0 ), hi (S), h(S 0 ) = h(S) = 0≤i≤4
0≤i≤4
we have 0 0 0 ¯ 1 (S), h ¯ 2 (S 0 )) ⊕ Mo (h ¯ ¯ Morat (h rat 2 (S), h2 (S )) ' T (Sk(S) )/T (S )
and ¯ 2 (S), h ¯ 2 (S 0 )) = Mo (t2 (S), t2 (S 0 )). Morat (h rat ¯ i (S), h ¯ j (S 0 ), be the images in Mo of Proof Let π ¯i = π ¯i (S), π ¯j0 = π ¯j (S 0 ), h rat the projectors πi , πj0 and of the corresponding motives hi (S), hj (S 0 )) for the surfaces S and S 0 (as defined in Proposition 14.2.1). It follows from Proposition 14.5.3 and Proposition 14.2.1 (iii) that π ¯3 = π ¯30 = π ¯4 = π ¯40 = 0. From Proposition 14.2.3 we get isomorphisms: h2 (S) ' ρL ⊕ t2 (S) and ¯ 2 (S) ' t2 (S), in Mo and similarly h2 (S 0 ) ' ρ0 L ⊕ t2 (S 0 ). It follows that h rat ¯ 2 (S 0 ) ' t2 (S 0 ). for S 0 : h Therefore in Morat we have ¯ ¯ 1 (S) ⊕ h ¯ 2 (S) = 1 ⊕ h ¯ 1 (S) ⊕ t2 (S) h(S) =1⊕h and ¯ 0) = 1 ⊕ h ¯ 1 (S 0 ) ⊕ h ¯ 2 (S 0 ) = 1 ⊕ h ¯ 1 (S 0 ) ⊕ t2 (S 0 ). h(S According to (14.5) we have ¯ 0 ), h(S ¯ 0 )) = A0 (S 0 ) Morat (h(S k(S) and ¯ 0 )) = Mo (h(Spec ¯ ¯ Morat (1, h(S k), h(S)) ' A0 (S 0 ). rat From Proposition 14.2.1 it follows: 0 eff 0 Meff rat (h1 (S), 1) = A (S)π1 = 0; Mrat (h2 (S), 1) = A (S)π2 = 0.
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Therefore we get: 0 ¯ 1 (S) ⊕ h ¯ 2 (S), h ¯ 1 (S 0 ) ⊕ h2 (S 0 )). )/A0 (S 0 ) ' Morat (h A0 (Sk(S)
Theorem 14.3.10 (i) yields : Mrat (h2 (S), h1 (S 0 )) = 0 while from [Sch, prop.4.5] it follows: Mrat (h1 (S), h1 (S 0 )) ' Ab(AlbS , AlbS 0 ). Therefore we have: 0 )/A0 (S 0 ) ' Ab(AlbS , AlbS 0 ) A0 (Sk(S) 0 ¯ 1 (S), h ¯ 2 (S 0 )) ⊕ Mo (h ¯ ¯ ⊕ Morat (h rat 2 (S), h2 (S )). (14.7) 0 There is a canonical map α : A0 (Sk(S) ) → Ab(AlbS , AlbS 0 ) which is 0 on 0 A0 (S ) (see [K-S, (9.5))]) as well as an isomorphism:
Ab(AlbS , AlbS 0 ) '
AlbS 0 (k(S))Q . AlbS 0 (k)Q
Therefore we get the following exact sequence: 0 0 0 → T (Sk(S) )/T (S 0 ) → A0 (Sk(S) )/A0 (S 0 ) →
AlbS 0 (k(S))Q → 0. AlbS 0 (k)Q
Hence: 0 0 A0 (Sk(S )/A0 (S 0 ) ' (AlbS 0 (k(S))/ AlbS 0 (k))Q ⊕ T (Sk(S) )/T (S 0 ).
(14.8)
From (14.7) and (14.8) we get; 0 0 0 ¯ 1 (S), h ¯ 2 (S 0 )) ⊕ Mo (h ¯ ¯ Morat (h rat 2 (S), h2 (S )) ' T (Sk(S) )/T (S ).
Proposition 14.5.8. With the same notation as in Lemma 14.5.7, the projection map 0 o 0 Ψ : Meff rat (t2 (S), t2 (S )) → Mrat (t2 (S), t2 (S ))
is an isomorphism. Proof The map Ψ of the proposition is clearly surjective, and we have to show that it is injective. 0 Let f ∈ Meff rat (t2 (S), t2 (S )) be such that Ψ(f ) = 0. Then f , as a correspondence in A2 (S × S 0 ), belongs to the subgroup I(S, S 0 ): from the definition of I(S, S 0 ) and J(S, S 0 ) (see Definition 14.4.2) it follows that I(S, S 0 ) ⊂ J(S, S 0 ). Thus f ∈ J(S, S 0 ) and from Theorem 14.4.3 we get that (π2tr )0 ◦ f ◦ π2tr = 0.
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0 tr 0 tr Since f ∈ Meff rat (t2 (S), t2 (S )), we also have (π2 ) ◦ f ◦ π2 = f , hence f = 0.
Lemma 14.5.9. Let S be a smooth projective surface and C a smooth proL jective curve. Then for any C-K decompositions h(S) = 0≤i≤4 hi (S) and L h(C) = 0≤j≤2 hj (C) as in Proposition 14.2.1, we have ¯ 1 (S), h ¯ 1 (C)) A0 (Ck(S) )/A0 (C) ' Morat (h ¯ i (X) and h ¯ j (C) are the images in Mo . where h rat Proof We have in Morat : ¯ ¯ 1 (S) ⊕ h ¯ 2 (S); h(S) =1⊕h
¯ ¯ 1 (C) h(C) =1⊕h
and, by Proposition 14.5.3, ¯ ¯ A0 (Ck(S) ) ' Morat (h(S), h(C));
¯ A0 (C) ' Morat (1, h(C))
with A0 (C) ' Q ⊕ JC (k)Q , where JC is the Jacobian of C. Therefore ¯ 1 (S), h(C)) ¯ ¯ 2 (S), h(C)) ¯ A0 (Ck(S) )/A0 (C) ' Morat (h ⊕ Morat (h and ¯ i (S), h(C)) ¯ ¯ i (S), 1) ⊕ Mo (h ¯ ¯ Morat (h = Morat (h rat i (S), h1 (C)) ¯ i (S), h ¯ 1 (C)) = Morat (h ¯ i (S), 1) = 0 for i = 1, 2. because Morat (h From Corollary 14.3.9 (ii) we get Mrat (h2 (S), h1 (C)) = 0 hence ¯ 2 (S), h(C)) ¯ Morat (h = 0. Therefore ¯ 1 (S), h ¯ 1 (C)). A0 (Ck(S) )/A0 (C) ' Morat (h
The following Theorem 14.5.10 is a reintepretation of Theorem 14.4.8 in ¯ 2 (S) and h ¯ 2 (S 0 ). terms of the birational motives h Let d≤1 Morat be the thick subcategory of Morat generated by motives of curves: by a result in [K-S, 9.5], d≤1 Morat is equivalent to the category AbS(k) of abelian k-schemes (extensions of a lattice by an abelian variety) with rational coefficients.
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Theorem 14.5.10. Let S, S 0 be smooth projective surfaces over k. Given any two refined C-K decompositions as in Propositions 14.2.1 and 14.2.3, there are two isomorphisms 0
Mrat (t2 (S), t2 (S )) '
Morat (t2 (S), t2 (S 0 ))
'
0 T (Sk(S) ) 0 ) Ko≤1 (S, S 0 ) ∩ T (Sk(S)
and Ko≤1 (S, S 0 ) = H≤1 . (See Definition 14.5.5 (iv) for the definition of Ko≤1 and Definition 14.4.6 for the definition of H≤1 .) Proof Let {πi } and {πi0 }, for 0 ≤ i ≤ 4, be the projectors giving a refined C-K decomposition respectively for S and S 0 . From Proposition 14.5.8 it follows that Mrat (t2 (S), t2 (S 0 )) ' Morat (t2 (S), t2 (S 0 )). From Lemma 14.5.1 we get: 0 A0 (Sk(S) )'
A2 (S × S 0 ) I(S, S 0 )
and from Lemma 14.5.6 (d): J(S, S 0 ) = I(S, S 0 ) + K≤1 (S, S 0 ). From Theorems 14.4.3 and 14.4.8, the map 0 β : A2 (S × S 0 ) → T (Sk(S) )
defined by β(Z) = ((π2tr )0 ◦ Z ◦ π2tr )(ξ), where ξ is the generic point of S, induces isomorphisms: 0 T (Sk(S) ) A2 (S × S 0 ) Mrat (t2 (S), t2 (S )) ' ' . 0 J(S, S 0 ) H≤1 ∩ T (Sk(S) ) 0
Moreover, it follows from Lemma 14.4.7 that, if T ∈ A2 (S × S 0 ) then T ∈ J(S, S 0 ) if and only if T (ξ) ∈ H≤1 . Hence 0 β(Z) ∈ H≤1 ∩ T (Sk(S) ) ⇐⇒ (π2tr )0 ◦ Z ◦ π2tr ∈ J(S, S 0 )
⇐⇒ (π2tr )0 ◦ Z ◦ π2tr = Γ1 + Γ2 where Γ1 ∈ I(S.S 0 ) and Γ2 ∈ K≤1 (S, S 0 ). Since Γ1 (ξ) = 0 we get, for any Z ∈ A2 (S × S 0 ) 0 β(Z) ∈ H≤1 ∩ T (Sk(S) ) ⇐⇒ (π2tr )0 ◦ Z ◦ π2tr (ξ) = Γ2 (ξ)
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with Γ2 ∈ K≤1 (S, S 0 ). This proves that the image of K≤1 (S, S 0 ) under the 0 0 map β is Ko≤1 (S, S 0 ) ∩ T (Sk(S) ) and coincides with H≤1 ∩ T (Sk(S) ). Therefore we get: 0 T (Sk(S) )
0
Mrat (t2 (S), t2 (S )) '
0 ) Ko≤1 (S, S 0 ) ∩ T (Sk(S)
0 0 and Ko≤1 (S, S 0 ) ∩ T (Sk(S) ) = H≤1 ∩ T (Sk(S) ). So we are left to show that
Ko≤1 (S, S 0 ) = H≤1 . From the definitions of Ko≤1 (S, S 0 ) and H≤1 it follows that H≤1 ⊂ Ko≤1 (S, S 0 ). ¯ ¯ 0 )) ' A0 (S 0 ), Mo (1, h(S ¯ 0 )) ' A0 (S 0 ) and We have Morat (h(S), h(S rat k(S) 0 ¯ 1 (S), h ¯ 1 (S 0 )) = {¯ ¯◦π Morat (h π10 ◦ Γ ¯1 |Γ ∈ A0 (Sk(S) )}.
From the construction of the projector π1 (S 0 ) as in Proposition 14.2.3, it follows that there exists a curve C 0 ⊂ S 0 such that π1 (S 0 ), as a map in Mrat (h(S 0 ), h(S 0 )), factors through the motive h1 (C 0 ). Therefore, every ¯ 1 (S), h ¯ 1 (S 0 )) factors trough the birational motive of a curve map α in Morat (h 0 C , i.e. it is in the image Ko≤1 (S, S 0 ) of K≤1 (S, S 0 ). Moreover, the same argument, as in the proof of lemma 14.4.7 shows that α ∈ H≤1 . From Corollary 14.3.9 (ii) it follows that the only map in the group ¯ 2 (S), h ¯ 2 (S 0 )) that factors through h(C) ¯ Morat (h for some curve C is 0. Therefore we get: 0 o ¯ 0 ¯ ¯ ¯ ¯ 0 )) + Mo (h Ko≤1 (S, S 0 ) = Morat (1, h(S rat 1 (S), h1 (S )) + Mrat (h1 (S), h2 (S ))
¯ 2 (S), h ¯ 1 (S 0 )) = 0. Furthermore because Morat (h 0 ¯ 0 )) + Mo (h ¯ ¯ Morat (1, h(S rat 1 (S), h1 (S ) ⊂ H≤1 .
From Lemma 14.5.7 0 0 0 ¯ 1 (S), h ¯ 2 (S 0 )) ⊕ Mo (h ¯ ¯ Morat (h rat 2 (S), h2 (S )) ' T (Sk(S) )/T (S )
hence: ¯ 1 (S), h ¯ 2 (S 0 )) Morat (h
=
0 Ko≤1 (S, S 0 ) ∩ T (Sk(S) )
T (S 0 )
This proves that Ko≤1 (S, S 0 ) ⊂ H≤1 .
=
0 H≤1 ∩ T (Sk(S) )
T (S 0 )
.
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Remarks 14.5.11. 1) According to Proposition 14.3.6 (ii), if S and S 0 are surfaces such that S×S 0 satisfies Conjecture 14.3.3 then the group Mrat (t2 (S), t2 (S 0 )) has finite rank. From Theorem 14.4.3 and Theorem 14.5.10 it follows that 0 this group is isomorphic to a quotient of the group T (Sk(S) )/T (S 0 ). The following example, suggested to us by Schoen and Srinivas, shows that, if S is a surface, the group T (Sk(S) )/T (S) may have infinite rank. Let E ⊂ P2Q¯ denote the elliptic curve defined by X 3 + Y 3 + Z 3 = 0, ¯ L = Q(E) and S = E × E. Then from the results in [Schoen] it follows that the group A2 (SL )deg 0 /A2 (S)deg 0 has infinite rank. Now, applying the exact ¯ we get an exact sequence (14.8) with X = E, Y = S = E × E and k = Q, sequence 0 → T (SL )/T (S) → A0 (SL )/A0 (S) → Ab(E, E × E) → 0. Since Ab(E, E × E) has finite rank, T (SL )/T (S) has infinite rank; since L ⊂ k(S), so does T (Sk(S) )/T (S). 2) In [B2, 1.8] (see also [J2, 1.12]) Bloch conjectured that, if S is a smooth projective surface and Γ ∈ A2 (S × S)hom , then Γ acts trivially on T (SΩ ), where Ω is a universal domain containing k. This conjecture implies that, if 2 (S) = 0, then the Albanese kernel T (S) vanishes. We claim that, from Htr the results in §§14.4 and 14.5, it follows that the above conjecture also implies A2 (S × S)hom ⊂ J(S, S), hence that EndMrat (t2 (S)) ' A2 (S × S)/J(S, S) (Theorem 14.4.3) is finite-dimensional as a quotient of A2hom (S × S) (at least in characteristic 0 for a “classical” Weil cohomology in Bloch’s conjecture). To show the claim, observe that if α ∈ A0 (Sk(S) ), then α(β) = β ◦ α for every α ∈ A0 (Sk(S) ) (see (14.4)). Therefore, if Γ ∈ A2 (S × S)hom , then ¯ tr ) = 0 because π tr (ξ) ∈ T (Sk(S) ) and k(S) ⊂ Ω. This implies that Γ(π 2 2 ¯ ◦ π tr = 0 in EndMo (t¯2 (S)). From Theorem 14.4.3 and Proposition π ¯2tr ◦ Γ 2 rat 14.5.8 it follows that Γ ∈ J(S, S). 14.6 Finite-dimensional motives In this section we first recall from [Ki] and [G-P2] some definitions and results on finite dimensional motives. Then we relate the finite dimensionality of the motive of a surface S with Bloch’s Conjecture on the vanishing of the Albanese kernel and with the results in §§14.4 and 14.5. Let C be a pseudoabelian, Q-linear, rigid tensor category and let X be an object in C. Let Σn be the symmetric group of order n: any σ ∈ Σn defines a map σ : (x1 , . . . , xn ) → (xσ(1) , . . . , xσ(n) ) on the n-fold tensor product X ⊗n of X by itself. There is a one-to-one correspondence between all irreducible representations of the group Σn (over Q) and all partitions of the integer n.
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Let Vλ be the irreducible representation corresponding to a partition λ of n and let χλ be the character of the representation Vλ . Let dim(Vλ ) X dλ = χλ (σ) · Γσ n! σ∈Σn
where Γσ is the correspondence associated to σ. Then {dλ } is a set of P pairwise orthogonal idempotents in HomC (X ⊗n , X ⊗n )) such that dλ = ∆X ⊗n . The category C being pseudoabelian, they give a decomposition of X ⊗n . The n-th symmetric product S n X of X is then defined to be the image Im(dλ ) when λ corresponds to the partition (n), and the n-th exterior power ∧n X is Im(dλ ) when λ corresponds to the partition (1, . . . , 1). If C = Mrat and M = h(X) ∈ Mrat for a smooth projective variety X, then P ∧n M is the image of M (X n ) under the projector (1/n!)( σ∈Σn sgn(σ)Γσ ), P while S n M is its image under the projector (1/n!)( σ∈Σn Γσ ). Definition 14.6.1 (see [Ki] and [G-P1]). The object X in C is said to be evenly (oddly) finite-dimensional if ∧n X = 0 (S n X = 0) for some n. An object X is finite-dimensional if it can be decomposed into a direct sum X+ ⊕ X− where X+ is evenly finite-dimensional and X− is oddly finitedimensional. Kimura’s nilpotence theorem [Ki, 7.2] says that if M is finite-dimensional, any numerically trivial endomorphism of M is nilpotent. We shall need the following more precise version in the proof of Theorem 14.6.9: Theorem 14.6.2. Let M ∈ Mrat be a finite-dimensional motive. Then the ideal of numerically trivial correspondences in EndMrat (M, M ) is nilpotent. Recall [A-K, 9.1.4] that the proof is simply this: Kimura’s argument shows that the nilpotence level is uniformly bounded. On the other hand, a theorem of Nagata and Higman says that if I in a non unital and not necessarily commutative ring such that there exists n > 0 for which f n = 0 for all f ∈ I, then I is nilpotent. Examples 14.6.3. 1) If two motives are finite-dimensional so is their direct sum and their tensor product. 2) A direct summand of an evenly (oddly) finite-dimensional motive is evenly (oddly) finite-dimensional. If a motive M is evenly and oddly finite-dimensional then M = 0 [Ki, 6.2]. A direct summand of a finitedimensional motive is finite-dimensional [Ki, 6.9]. 3) The dual motive M ∗ is finite-dimensional if and only if M is finitedimensional.
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4) The motive of a smooth projective curve is finite-dimensional: hence the motive of an abelian variety X is finite-dimensional. Also if X is the quotient of a product C1 × · · · × Cn of curves under the action of a finite group G acting freely on C1 × · · · × Cn then h(X) is finitedimensional. More generally: Proposition 14.6.4 (Kimura’s lemma, [Ki, 6.6 and 6.8]). If f : X → Y is a surjective morphism of smooth projective varieties, then h(Y ) is a direct summand of h(X). Hence, by Example 14.6.3 2), if h(X) is finitedimensional then h(Y ) is also finite-dimensional. Here is a simple proof, in the spirit of Kimura’s: let g = 1Y × f : Y × X → Y × Y and T = g −1 (∆), where ∆ is the diagonal of Y × Y . Pick a closed point p of the generic fibre of g|T : T → ∆: the closure of p in T defines a closed subvariety Z in Y × X which is finite surjective over ∆. Then Z defines a correspondence [Z] from Y to X, and one checks immediately that the composition [Z]
f∗
h(Y ) −→ h(X) −→ h(Y ) is multiplication by the generic degree of Z over ∆. We also have Kimura’s conjecture: Conjecture 14.6.5 ([Ki]). Any motive in Mrat is finite-dimensional. Lemma 14.6.6. For any smooth projective surface S, the motives h0 (S), h1 (S), halg 2 (S), h3 (S) and h4 (S) appearing in Propositions 14.2.1 and 14.2.3 are finite-dimensional. Hence all direct summands of h(S) appearing in these propositions are finite-dimensional, except perhaps t2 (S). Proof The lemma is clear for h0 (S), halg 2 (S) and h4 (S) since these motives are tensor products of Artin motives and Tate motives. Since h3 (S) ' h1 (S)(1), it remains to deal with h1 (S). But the construction of the projector defining h1 (S) in [Mu1, Sch] shows that it is a direct summand of the motive of a curve; the claim therefore follows from Examples 14.6.3 2) and 4). Lemma 14.6.7. Let U be a group acting transitively on a set E. Suppose that the following condition is verified: (*) If e ∈ E, u ∈ U and n ≥ 1 are such that un e = e, then ue = e.
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On the other hand, let G be a group “acting on this action”: there is an action of G on U and an action of G on E such that g
(ue) = g ug e
for any (g, u, e) ∈ G × U × E. Suppose moreover that G is finite, U has a finite G-invariant composition series {1} = Zr ⊂ · · · ⊂ Z1 = U such that, for all i, a) Zi / U ; b) Zi /Zi+1 is central in U/Zi+1 and uniquely divisible. Then G has a fixed point e on E. If f is another fixed point, there exists u ∈ U , invariant by G, such that f = ue. Proof We argue by induction on r, the case r = 1 being trivial. Suppose r > 1 and let Z = Zr−1 . Since G preserves Z, it acts on U/Z and E/Z, preserving the induced action. Moreover, the fact that Z is central in U and divisible implies that Condition (*) is preserved. By induction, there is e ∈ E such that g
e = zg e ∀g ∈ G
with zg ∈ Z. Let Ue be the stabilizer of e in U . Since the zg are central, Ue is stable under the action of G; in particular, G acts on Z/Z ∩ Ue . An easy computation shows that, for all g, h ∈ G: −1 g zgh zh zg ∈ Z ∩ Ue .
Now Z/Z ∩ Ue is divisible as a quotient of Z, and moreover Condition (*) implies that it is torsion-free. Therefore, H 1 (G, Z/Z ∩ Ue ) = 0 and there is some z ∈ Z such that zg ≡ g z −1 z (mod Z ∩ Ue ) for all g ∈ G. Then ze is G-invariant. For uniqueness, we argue in the same way. By induction, there exists u0 ∈ U such that f = u0 e and g u0 = zg u0 for all g ∈ G, with zg ∈ Z. Applying g ∈ G to the equation f = u0 e shows that zg ∈ Uf . Thus, g 7→ zg defines a 1-cocycle with values in Z ∩Uf . Since Z and Z/Z ∩Uf are uniquely divisible, so is Z ∩ Uf , hence this 1-cocycle is a 1-coboundary and we are done. Lemma 14.6.8. Let A be a Q-algebra, π a subset of A and ν a nilpotent P element of A. Suppose that there exists a polynomial P = ai ti , with a1 6= 0, such that P (ν) commutes with all the elements of π. Then ν commutes with all the elements of π.
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Proof Let us denote by C the centralizer of π and let r be such that ν r = 0. We prove that ν i ∈ C for all i by descending induction on i. The case i ≥ r is clear. Note that we may (and do) assume that P (0) = 0. Then P (ν) is nilpotent. Let i < r: then C 3 P (ν)i = ai1 ν i + . . . where the next terms are higher powers of ν. By induction, ai1 ν i ∈ C, hence ν i ∈ C. The following is a slight improvement of [G-P2, Th. 3]: Theorem 14.6.9. Let X be a smooth projective variety over k of dimension d, such that the the K¨ unneth components of the diagonal are algebraic. Assume that the motive h(X) ∈ Mrat is finite-dimensional. Then a) h(X) has a Chow-K¨ unneth decomposition M h(X) = hi (X) 0≤i≤2d
with hi (X) = (X, πi , 0). If {˜ πi } is another set of such orthogonal idempotents, then there exists a nilpotent correspondence n on X such that π ˜i = (1 + n)πi (1 + n)−1
(14.9)
for all i. In particular, ˜ i (X) hi (X) ' h ˜ i (X) = (X, π in Mrat , where h ˜i , 0). b) Moreover, the πi may be chosen so that πit = π2d−i . If {˜ πi } is another such choice, there exists a nilpotent correspondence n on X such that (14.9) holds and moreover, (1 + n)t = (1 + n)−1 . Proof a) The existence and “uniqueness” of the πi follow immediately from Kimura’s nilpotence theorem (Theorem 14.6.2) and from [J2, 5.4]. For b), let N = Ad (X × X)hom : this is a nilpotent ideal of EndMrat (h(X)) by Theorem 14.6.2. We apply Lemma 14.6.7 with U =1+N Zi = 1 + Ni E = {{πi } | πi 7→ πihom } G ' Z/2.
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We let U act on E by conjugation: this action is transitive by a). We let G act on this action as follows: if g is the nontrivial element of G, then g
u = (u−1 )t ;
g
{πi } = {πi t }.
Note that the action on E exists because the πihom are stable under transposition (Poincar´e duality). We now check the hypotheses of Lemma 14.6.7: clearly the Zi are normal in U , G-invariant and verify the centrality assumption. Moreover, Zi /Zi+1 ' Ni /Ni+1 is uniquely divisible. It remains to verify Condition (*): but this follows from Lemma 14.6.8 applied to P (ν) = (1 + ν)n . The proof is complete. Theorem 14.6.10 ([G-P2, Th. 7]). Let S be a smooth projective surface over an algebraically closed field k of characteristic 0 with pg (S) = 0, and suppose that k has infinite transcendence degree over Q: then the motive h(S) is finite-dimensional if and only if the Albanese kernel T (S) vanishes. Proof “If” follows from Corollary 14.4.9 b) (see proof of (1) ⇔ (2) in Theorem 14.6.12 below). For “only if”, note that the hypothesis pg = 0 implies 2 (S) = 0 and therefore (π tr (S))hom = 0. By Kimura’s nilpotence theoHtr 2 rem (Theorem 14.6.2), the finite-dimensionality hypothesis now implies that π2tr (S) = 0, and we conclude by Proposition 14.2.3. The following corollary may be viewed as a “birational” version of a result by S. Bloch in [B2, Lect. 1, Prop. 2]. Corollary 14.6.11. Let S be a smooth projective surface over an algebraically closed field k of characteristic 0 and infinite transcendence degree over Q. Then the following conditions are equivalent: i) pg (S) = 0 and the motive h(S) is finite-dimensional; ii) the Albanese Kernel T (S) vanishes; iii) t2 (S) = 0; iv) t¯2 (S) = 0 in Morat ; ¯ v) the motive h(S) in Morat is a direct summand of the birational motive of a curve. Proof By Theorem 14.6.10, (i) ⇒ (ii). The equivalence of (ii) and (iii) has been seen in Corollary 14.4.9 b) and the equivalence of (iii) and (iv) follows from Proposition 14.5.8. If t2 (S) = 0, then pg = 0 by Corollary 14.4.9 c) and h(S) is finite-dimensional by Lemma 14.6.6. Thus we have (i) ⇔ (ii) ⇔ (iii) ⇔ (iv). ¯ alg (S) = h ¯ 3 (S) = h ¯ 4 (S) = 0 by Proposition 14.2.1 Note that in general, h 2
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(iv). Therefore (iv) ⇒ (v) (see proof of Lemma 14.6.6). Conversely, if ¯ h(S) is a direct summand of the birational motive of a (not necessarily connected) curve D, then so is t¯2 (S). But Corollary 14.3.9 (ii) implies that ¯ Mrat (t2 (S), h(D))) = 0, hence Morat (t¯2 (S), h(D)) = 0, which implies that ¯ ¯ o EndMrat (t2 (S)) = 0 and therefore that t2 (S) = 0. So (v) ⇒ (iv) and the proof is complete. Theorem 14.6.12. Let S be a smooth projective surface and let h(S) = L L unneth decomposition 0≤i≤4 hi (S) = 0≤i≤4 (S, πi , 0) be a refined Chow-K¨ as in Prop. 14.2.1 and 14.2.3. Let us consider the following conditions: (i) the motive h(S) is finite-dimensional; (ii) the motive t2 (S) is evenly finite-dimensional; (iii) every endomorphism f ∈ EndMrat (h(S)) which is homologically trival is nilpotent; (iv) for every correspondence Γ ∈ A2 (S × S)hom , αi,i = πi ◦ Γ ◦ πi = 0, for 0 ≤ i ≤ 4; (v) for all i, the map EndMrat (hi (S)) → EndMhom (hhom (S)) is an isoi morphism (hence EndMrat (hi (S)) has finite rank in characteristic 0); (vi) the map EndMrat (t2 (S)) → EndMhom (thom (S)) is an isomorphism; 2 (vii) let J(S) be the 2-sided ideal of A2 (S × S) defined in Definition 14.4.2: then A2 (S × S)hom ⊂ J(S). Then (i) ⇔ (ii) ⇒ (iii) ⇐ (iv) ⇔ (v) ⇔ (vi) ⇔ (vii). Proof (i) ⇔ (ii) by Lemma 14.6.6.(t2 (S) is evenly finite dimensional because it is a direct summand of h2 (S).) (i) ⇒ (iii) follows from [Ki, 7.2] (see Theorem 14.6.2). (iv) ⇒ (iii): (iv) implies that hi (S), for 0 ≤ i ≤ 4, satisfy (i) and (ii) in Theorem 14.3.10. Therefore, by [G-P2, Cor. 3], every endomorphism f ∈ EndMrat (h(S)) which is homologically trivial is nilpotent. (iv) ⇒ (v). We have EndMrat (h1 (S)) ' EndAb (AlbS ) ' EndMhom (hhom (S)). 1 By duality the same result holds for EndMrat (h3 (S)). From (4) it follows that also the map EndMrat (h2 (S)) → EndMhom (hhom (S)) 2 is an isomorphism. (v) ⇒ (vi) is obvious. (vi) ⇒ (vii). If Γ ∈ A2 (S × S)hom then π2tr ◦ Γ ◦ π2tr yields the 0 map in
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EndMhom (thom (S)), therefore it is 0. Since Γ ∈ A2 (S × S)hom we also have 2 π2alg ◦ Γ ◦ π2alg = 0, hence Γ ∈ J(S) (see Lemma 14.4.1). (vii) ⇒ (iv). Let Γ ∈ A2 (S × S)hom : then Γ ∈ J(S) which proves that π2tr ◦ Γ ◦ π2tr = 0. Remark 14.6.13 (Abelian varieties and Kummer surfaces). Let A be an abelian variety of dimension d. Then h(A) has a Chow-K¨ unneth decomposition (see L [Sch]) h(A) = 0≤i≤2d hi (A) where hi (A) = (A, πiA , 0) and n∗ ◦ πiA = ni πiA , for every n ∈ Z. Here n∗ = (id × n)∗ is the correspondence induced by multiplication by n on A. The motive h(A) is finite-dimensional, hence the above decomposition is unique (up to isomorphism). Now suppose that d = 2, and let S be the Kummer surface associated to the involution a → −a on A (with singularities resolved). The rational map f : A → S induces an isomorphism between the Albanese kernels : L T (A) ' T (S) (see [B-K-L, A.11]). Let h(S) = 0≤i≤4 hi (S), with hi (S) = (S, πiS , 0). Reasoning as in [A-J, Th 3.2], we get that the formula πiS = (1/2)(f × f )∗ πiA defines a C-K decomposition on S. From the exact sequence in (14.3) and from Proposition 14.5.3 it follows that the map f induces homomorphisms f ∗ : A0 (Sk(S) ) → A0 (Ak(A) ) and f∗ : A0 (Ak(A) ) → A0 (Sk(S) ). Then f ∗ (f∗ (α)) = α + [−1] · α for all α ∈ A0 (Ak(A) ). From the equality n∗ ◦ π2A = n2 π2A it follows that ¯2A is the image of π2A under the map in (14.3). From π ¯2A ∈ A0 (Sk(S) ), where π the isomorphism ¯ A0 (Sk(S) ) ' EndMorat (h(S)) ¯ 2 (S) ' h ¯ 2 (A) in Mo . The Kummer surface S has q = dim H 1 (S, OS ) = we get h rat 0, hence h1 (S) = h3 (S) = 0. Therefore we get: ¯ ¯ 2 (A) h(S) '1⊕h and: ¯ 2 (A) ' 1 ⊕ h ¯ 1 (A) ⊕ h ¯ 2 (A). h In particular, f induces an isomorphism f∗ : t2 (A)t2 (S).
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14.7 Higher-dimensional refinements The next results extend, under the assumption of certain conjectures, some of the properties proven in Propositions 14.2.1 and 14.2.3 for the refined Chow-K¨ unneth decomposition of the motive of a surface to varieties of higher dimension. In particular these results apply to abelian varieties. To avoid questions of rationality we shall assume that k is separably closed; the reader will have no difficulties to extend these results to the general case along the lines of the proof of Proposition 14.2.3. In the following we will denote by A¯i (X) ⊂ H 2i (X) the image of the cycle class map cli : Ai (X) → H 2i (X) for a smooth projective variety X (see §14.1.3). Definition 14.7.1. We say that the Hard Lefschetz theorem holds for the Weil cohomology H if, for any smooth projective variety X of dimension d, any smooth hyperplane section W ⊂ X and any i ≤ d, the Lefschetz operator Ld−i : H i (X) → H 2d−i (X) given by cup product by cl(W )d−i , is an isomorphism. It is known that every classical Weil cohomology satisfies the Hard Lefschetz theorem. Let us choose a classical Weil cohomology theory H. Following [Kl], let B(X) and Hdg(X) denote respectively the Lefschetz standard conjecture and the Hodge standard conjecture for a smooth projective variety X. The conjecture B(X) is equivalent to the following for any L as in Definition 14.7.1 (see [Kl, 4.1]): θ(X) For each i ≤ d, there exists an algebraic correspondence θi inducing the isomorphism H 2d−i (X) → H i (X) inverse to Ld−i .
Recall also the conjectures: A(X, L) The restriction Ld−2i : A¯i (X) → A¯d−i (X) is an isomorphism for all i. C(X) The K¨ unneth projectors are algebraic. D(X) Numerical equivalence equals homological equivalence.
Under D(X), A∗hom (X) is a finite-dimensional Q-vector space. By [Kl, 4.1
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and 5.1], we have the following implications (for any L): A(X × X, L ⊗ 1 + 1 ⊗ L) ⇒ B(X) ⇒ A(X, L) B(X) ⇒ C(X)
(14.10)
A(X, L) + Hdg(X) ⇒ D(X) ⇒ A(X, L). Finally, B(X) is satisfied by curves, surfaces†, abelian varieties and it is stable under products and hyperplane sections [Kl, 4.1 and 4.3]. Also Hdg(X) is true in characteristic 0 and holds in arbitrary characteristic if X is a surface [Kl, §5]. We shall also need the following easy lemma: Lemma 14.7.2. Let H be a classical Weil cohomology theory. Let M = (Xd , p, m) ∈ Meff hom . Then a) m ≥ −d. b) If p∗ H i (X) 6= 0 then we have the sharper inequality m ≥ −[i/2]. Proof a) Let α : M → h(Y ) and β : h(Y ) → M be two morphisms such that β ◦ α = 1M . In particular, 0 6= α ∈ Corrm (X, Y ) = Ad+m (X × Y ), hence d + m ≥ 0. b) We have H i+2m (M ) = p∗ H i (X) 6= 0. On the other hand, the correspondence α of a) realises H i+2m (M ) as a direct summand of H i+2m (Y ). The inequality follows. Theorem 14.7.3. Let X be a smooth projective variety of dimension d such that Conjecture B(X) holds and that the ideal Ker(EndMrat (h(X)) → EndMhom (hhom (X))) is nilpotent (by Theorem 14.6.2, this is true if the motive h(X) is finite-dimensional). Let X ,→ PN be a fixed projective embedding. Then L i) There exists a self-dual C-K decomposition h(X) = hi (X) (πit = π2d−i ). ii) Let i : W ,→ X be a smooth hyperplane section of X and L = i∗ i∗ : h(X) → h(X)(1) be the corresponding “Lefschetz operator”. Then, for each i ∈ [0, d], the composition Ld−i
`i : h2d−i (X) → h(X) −−−→ h(X)(d − i) → hi (X)(d − i)
(14.11)
is an isomorphism. If moreover Conjecture D(X × X) holds, then: † In [Kl, 4.3], Kleiman requires that dim H 1 (X) = 2 dim Pic0X , but this assumption is verified by all classical Weil cohomologies.
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iii) For each i ∈ [0, 2d] there exists a further decomposition hi (X) '
[i/2] M
hi,j (X)(j)
(14.12)
j=0 hom such that, for each j, hhom i,j (X) is effective but hi,j (X)(−1) is not effective. Moreover, the isomorphism from (ii) induces isomorphisms
h2d−i,d−i+j (X)hi,j (X). iv) Let (πi,j ) be the orthogonal set of projectors defining this decompo0 ) is another such set of projectors, then there exists a sition. If (πi,j correspondence n, homologically equivalent to 0, such that 0 πi,j = (1 + n)πi,j (1 + n)−1 for all (i, j).
In particular, the hi,j (X) are unique up to isomorphism. Proof We first prove (i), (ii), (iii) and (iv) modulo homological equivalence. (i) is immediate since B(X) ⇒ C(X) (see (14.10)). The homological version of (ii) follows immediately from the form θ(X) of Conjecture B(X). We now come to the homological versions of (iii) and (iv). By D(X × X) and Jannsen’s theorem [J1], the algebra EndMhom (hhom (X)) is semi-simple. Given i ∈ [0, 2d], write M = hhom (X) as the direct sum of its isotypical i components Mα : for each α, we have Mα ' Sαnα where Sα is a simple motive and nα > 0. By Lemma 14.7.2 b), the largest integer jα such that Sα (−jα ) is effective exists and verifies jα ≤ i/2. We set M hhom Mα (−j). i,j (X) = jα =j
This proves the first claim of (iii). Moreover, this construction shows that the homological version of (14.12) is unique; in particular, the correhom are central in End hom (X)). This proves sponding projectors πi,j Mhom (hi [the homological version of] (iv). To see the second claim in (iii) (still in its homological version), let Sα be a simple summand of hhom 2d−i (X); then clearly `i (Sα )(−jα ) is effective but `i (Sα )(−jα − 1) is not effective. This proves that `i (hhom 2d−i,d−i+j (S)(d − i + hom j)) = hi,j (S)(j), hence an isomorphism hom hhom 2d−i,d−i+j (S)hi,j (S).
Lifting these results from homological equivalence to rational equivalence follows from the nilpotency hypothesis, as in the proof of Theorem 14.6.9
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a). The fact that `i still induces an isomorphism h2d−i,d−i+j (S)hi,j (S) is also a standard consequence of nilpotence (cf. [An, 5.1.3.3]): we first note that numerically trivial endomorphisms of h2d−i,d−i+j (S) and hi,j (S) are nilpotent as both motives are direct summands of h(X), up to Tate twists. Let θ be a cycle giving the inverse isomorphism to `i in Mhom . Then, in Mrat , m = `i ◦ θ − 1 and n = θ ◦ `i − 1 are numerically equivalent to 0, hence nilpotent. But then, 1 + m and 1 + n are isomorphisms. It follows that `i is left and right invertible, hence is an isomorphism. Moreover, we have: Theorem 14.7.4. Let X verify the hypotheses of Theorem 14.7.3. Then, L for a C-K decomposition h(X) = 0≤i≤2d hi (X) as in this theorem, for every i < d, the projector πi factors through h(Yi ), where Yi = X · H1 · H2 · · · · · Hd−i is a smooth hyperplane section of X of dimension i (with Hi hyperplanes). Hence hi (X) is a direct summand of h(Yi ) for all i < d. Similarly, h2d−i is a direct summand of h(Yi ). Proof By B(X), for each i ≤ d there exists an algebraic correspondence θi inducing the isomorphism H 2d−i (X) → H i (X) inverse to the isomorphism Ld−i : H i (X) → H 2d−i (X). Let j : Yi → X be the closed embedding and let Γi = j ∗ ◦ θi ∈ Ad−i (X × Yi ) and qi = j∗ ◦ Γi ∈ Ad (X × X).Then Γi and hence also qi factor trough Yi : furthermore qihom operates as the identity on H 2d−i (X) because j∗ · j∗ = Ld−i . Let fi = πi ◦ qi ◦ πi ∈ Mrat (hi (X), hi (X)). Then fihom is a projector on H ∗ (X) and in fact is the (i, 2d − i)-K¨ unneth projector. Therefore the map ai = πi − fi is homologically trivial, hence nilpotent by hypothesis, i.e. ani = 0 for some n > 0. Let bi = (1 + ai + a2i + · · · + an−1 ) = (1 − ai )−1 . We have: ai ◦ πi = πi ◦ ai = ai . Therefore i (1 − ai ) ◦ πi = πi − ai = fi and we get πi = (1 − ai )−1 ◦ fi = (1 + ai + a2i + · · · + an−1 ) ◦ fi = bi ◦ fi ; i πi = fi ◦ (1 − ai )−1 = fi ◦ bi . Since qi and therefore also fi factor trough h(Yi ) it follows that πi factors trough h(Yi ). Let gi = Γi ◦ πi : h(X) → h(Yi )and gi0 = bi ◦ πi ◦ j∗ : h(Yi ) → hi (X): then we have gi0 ◦ gi = πi , hence gi has a left inverse. Therefore hi (X) is a direct summand of h(Yi ) for all i < d. The case of π2d−i follows from the above since the C-K decomposition of Theorem 14.7.3 is self-dual.
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Remark 14.7.5. Theorem 14.7.3 notably applies to abelian varieties in characteristic 0. Also note that Theorem 14.7.4 answers – in a slightly weaker form – a question raised in [Mu2, p. 187]. We shall complement Theorem 14.7.3 with a somewhat more explicit result. For this we need lemma 14.7.6 which is just a reformulation of a result in [Ki, Prop. 2.11]: Lemma 14.7.6. Let X be a smooth projective variety of dimension d and let a ∈ Ai (X), b ∈ Ad−i (X), with i ≤ d, be such that hb, ai = deg(a · b) = 1. Let α = p∗1 a · p∗2 b ∈ Ad (X × X), where pi : X × X → X are the projections. Then α is a projector and the motive M = (X, α, 0) is isomorphic to Li . Proof We have α ◦ α = ha, biα = α, hence α is a projector. On the other hand Mrat (M, Li ) = Ad−i (X) ◦ α = Ai (X) ◦ α and Mrat (Li , M ) = α ◦ Ai (X) = α ◦ Ad−i (X). Therefore α ◦ b ∈ Mrat (Li , M ) and a ◦ α ∈ Mrat (M, Li ). Considering a as an element of Ad−i (X × Spec k) and b as an element of Ai (Spec k × X), we have a ◦ α = ha, bia = a and α ◦ b = ha, bib = b. Moreover a ◦ b = 1Spec k and b ◦ a = p∗1 a · p∗2 b = α = 1M . Hence a and b yield an isomorphism between M and Li . Theorem 14.7.7. Keep the notation and hypotheses of Theorem 14.7.3. For all i ∈ [0, d], the motive h2i,i (X) contains h(A¯i,X ) as a direct summand, where h(A¯i,X )is the Artin motive associated to the finite-dimensional vector space A¯i,X . Proof We prove this for i ≤ d/2; the result then follows for i ≥ d/2 by Poincar´e duality. We proceed as in the proof of Proposition 14.2.3. By B(X), the homomorphism Ld−2i : H 2i (X) → H 2d−2i (X) restricts to an isomorphism Ld−2i : A¯i (X)A¯d−i (X) where we use the same notation for the restriction of Ld−2i .
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From D(X × X) it follows that the restriction of the Poincar´e duality pairing on H 2i (X) × H 2d−2i (X) is still nondegenerate on A¯i (X) × A¯d−i (X). Therefore there exist elements ai,α ∈ Ai (X) and bi,α ∈ Ad−i (X) (α = 0, . . . , ρi ) such that 1) e˜i,α = cli (ai,α ) and e˜i,α = cli (bi,α ) form dual bases for Poincar´e duality. Now we claim that we can choose the ai,α and bi,α so that they also satisfy t (a ) = π 2) πkt (ai,α ) = πk (bi,α ) = 0 for k 6= 2i; π2i,j i,α 2i,j (bi,α ) = 0 for j < i. P P 0 To prove the claim, let π 0 = k6=2i πk and π 00 = j
pi,α = ai,α × bi,α ∈ Ad (X × X). We have hai,α , biα i = 1 and, by Lemma 14.7.6, the pi,α are projectors and each motive Mi,α = (X, pi,α , 0) is isomorphic to Li . Moreover the pi,α are P alg pairwise orthogonal. Let π2i = 1≤α≤ρi pi,α : by Condition 2) we have alg alg πk ◦ π2i = π2i ◦ πk = 0 for k 6= 2i; alg alg π2i,j ◦ π2i = π2i ◦ π2i,j = 0 for j < i.
Therefore alg alg alg π2i,i ◦ π2i = π2i ◦ π2i,i = π2i alg and we can split π2i,i as a sum π2i + p of two orthogonal projectors. The theorem is proven.
14.8 Return to birational motives Throughout this section, we assume that k is perfect. We recover and strengthen some of the previous results in two steps: (i) In §14.8.2 we show that, for a surface S provided with a refined CK decomposition as in Propositions 14.2.1 and 14.2.3, the image of t2 (S) under the full embedding of [Voev2, p. 197 and 3.2] eff eff eff Φ : Meff rat → DMgm → DM− := DMgm (k, Q)
(14.13)
is a birational motive. See Theorem 14.8.4. This gives back some of the previous computations.
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(ii) In §14.8.3, we interpret the computation of the endomorphism ring of t2 (S) as the existence of adjoints between certain categories of motives: see Theorem 14.8.8. Finally, in §14.8.4 we show that “nothing more happens” for surfaces when we pass from the category of pure motives to Voevodsky’s triangulated category of motives: see Corollary 14.8.13. 14.8.1 Categorical trivialities Let A be a pseudo-abelian additive category and B be a thick subcategory of A (thick means full and closed under direct summands). To B one may associate the following ideal I of A (cf. [A-K, 1.3.1]): I(A, A0 ) = {f : A → A0 | f factors through an object of B}. Let C = (A/I)\ be the pseudo-abelian envelope of the corresponding factor category, and P : A → C the corresponding projection functor. Recall that, for two objects A, A0 ∈ A, C(P A, P A0 ) = A(A, A0 )/I(A, A0 ). Let us now define ⊥
I = {A ∈ A | I(A, A) = 0}
(14.14)
= {A ∈ A | A(B, A) = 0} = {A ∈ A | ∀A0 ∈ A, P : A(A0 , A)C(P A0 , P A)}. Note that ⊥ I is stable under direct sums and direct summands: we view it as a thick subcategory of A. As usual, we say that “the” right adjoint of P is defined at an object C ∈ C if the functor A 3 A 7→ C(P A, C)
(14.15)
is representable. Let C0 be the full subcategory of C consisting of such objects: it is a thick subcategory of C. The following is an abstraction of the arguments in [K-S, proof of 9.5]: Proposition 14.8.1. a) If P # is “the” partial right adjoint of P (defined on C0 ), then P # (C0 ) ⊆ ⊥ I. b) For any C ∈ C0 , the counit map of the adjunction ε : P P #C → C is an isomorphism.
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c) C0 coincides with the essential image of P 0 = P|⊥ I . d) For any B ∈ ⊥ I, the unit morphism η : B → P #P B is an isomorphism. In particular, P # (C0 ) = ⊥ I and the functors P 0 : ⊥ I → C0 P # : C0 → ⊥ I form a pair of quasi-inverse equivalences of categories. Proof a) is obvious. b) By definition of adjunction, for any A ∈ A the composition ε
P
∗ A(A, P # C) −→ C(P A, P P # C) −→ C(P A, C)
is an isomorphism. Since P # C ∈ ⊥ I by a), the left map is an isomorphism and hence so is the right one. It follows that, for any C 0 ∈ C (which may be written as a direct summand of P A for some A), the map ε
∗ C(C 0 , P P # C) −→ C(C 0 , C)
is an isomorphism. By Yoneda’s lemma, this implies that ε is an isomorphism. c) Let for a moment C00 denote the essential image of P 0 . If C = P B ∈ C00 , with B ∈ ⊥ I, then clearly the functor (14.15) is represented by B, so C00 ⊆ C0 . Conversely, if C ∈ C0 , then C ∈ C00 by a) and b). d) Note that, by c), P # P B is defined. Let A ∈ A. As in any adjunction the composition η∗
A(A, B) −→ A(A, P # P B)C(P A, P B) is equal to P . Since it is an isomorphism, so is η∗ and hence η is an isomorphism by Yoneda. The other conclusions follow immediately. Corollary 14.8.2. P has an everywhere defined right adjoint if and only if A = B ⊕ ⊥ I. For future reference, we state the dual results (same proofs): Proposition 14.8.3. Let I⊥ = {A ∈ A | I(A, A) = 0}, viewed as a thick subcategory of A. Then the thick subcategory C0 of C of those objects where a left adjoint # P of P is defined coincides with the essential image of P 0 = P|I⊥ ; P 0 is an equivalence of categories, # P (C0 ) = I⊥ and # P is a quasiinverse of P 0 . Finally, # P is everywhere defined if and only if A = B ⊕ I⊥ .
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14.8.2 t2 (S) as a birational motive Recall from [Voev2] that the category DM−eff (k) admits a partially defined internal Hom eff Hom : DMgm (k) × DM−eff (k) → DM−eff (k)
which extends by Q-linearity to an internal Hom eff Hom : DMgm (k, Q) × DM−eff (k)Q → DM−eff (k)Q .
This gives a meaning to the following Theorem 14.8.4.
a) For any smooth projective variety X, one has Hom(Q(1), Φ(hi (X))) = 0 for i = 0, 1
in DM−eff = DM−eff (k, Q), where Φ is the full embedding of (14.13). b) Let S be a surface provided with a refined C-K decomposition as in Propositions 14.2.1 and 14.2.3. Then Hom(Q(1), Φ(t2 (S))) = 0. Therefore, Φ(h0 (X)), Φ(h1 (X)) and Φ(t2 (S)) belong to the image of the inclusion functor i : DM−o → DM−eff , where DM−o := DM−o (k)Q is the category of [K-S, 6.1]. Proof We do the proof for t2 (S): the other cases are similar and easier (reduce to X a curve). By definition, Hom(Q(1), Φ(t2 (S))) is a complex of sheaves on the category of smooth k-schemes provided with the Nisnevich topology. The fact that it is 0 may be checked locally; moreover, using [Voev1, Prop. 4.20], it suffices to check that for any function field extension K/k we have H∗ (K, Hom(Q(1), Φ(t2 (S)))) = 0. (H∗ denotes Nisnevich hypercohomology.) Since t2 (S) is a direct summand of h(S), Hom(Q(1), Φ(t2 (S))) is a direct summand of Hom(Q(1), M (S)). By [H-K, Lemma B.1], we have an isomorphism Hom(Q(1), M (S)) ' Hom(M (S), Q(1)[4]). This isomorphism is induced by the duality isomorphism M (S) ' M (S)∗ (2)[4].
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The latter is the image under Φ of the duality isomorphism θ : h(S) ' ∨ h(S)∨ (2) in Meff rat ; since θ carries t2 (S) to t2 (S) (2), Φ(θ) carries Φ(t2 (S)) to Φ(t2 (S))∗ (2)[4] and thus Hom(Q(1), Φ(t2 (S))) is isomorphic to the direct summand Hom(t2 (S), Q(1)[4]) of the complex Hom(M (S), Q(1)[4]). Let U be a smooth k-scheme. We have q+4 HqN is (U, Hom(M (S), Q(1)[4])) ' HN is (U × S, Q(1)) q+4 q+3 = HZar (U × S, Q(1)) = HZar (U × S, Gm )Q .
Passing to the function field K of U we get that, for q ∈ Z, the group q+3 Hq (K, Hom(Q(1), t2 (S))) is a direct summand of HZar (SK , Gm )Q . It is therefore 0 except perhaps for q = −3, −2. But for q = −3, H 0 (SK , Gm ) = K ∗ is “caught” by the direct summand 1 of h(S). For q = −2, H 1 (SK , Gm )Q = Pic(SK )Q decomposes into Pic0 (SK )Q ⊕ NS(SK )Q . The first summand is obtained from h1 (S) and the second from h(NSS )(1) as a direct summand of h2 (S). Hence the vanishing. The last claim now follows from [K-S, 6.2]. Corollary 14.8.5. Keep the notation of Theorem 14.8.4. Then (i) h0 (X), h1 (X), t2 (S) ∈ ⊥ I, where I is the ideal of Lemma 14.5.1 and ⊥ I is defined in (14.14). (Here, A = Meff , B = A ⊗ L.) rat ⊥ , where K (ii) hi (X) ∈ K⊥ (i = 0, 1) and t (S) ∈ K 2 ≤n is the ideal ≤i−1 ≤1 ⊥ of Definition 14.5.5 (iii) and K≤n is defined in Proposition 14.8.3. Moreover, for any smooth projective variety Y of dimension d, one has num Meff (Xk(Y ) ) rat (h(Y ), h0 (X)) ' A0
Meff rat (h(Y ), h1 (X)) ' AlbX (k(Y )) Meff rat (h(Y ), t2 (S)) ' T (Sk(Y ) ). Proof (1) is just a special case of Theorem 14.8.4 by the full faithfulness of (14.13), via Proposition 14.5.3. (2) follows from (1) by duality. For the isomorphisms, let us treat the case of t2 (S). We first observe that Meff rat (1, t2 (S)) = T (S)
(14.16)
(see Proposition 14.2.3). Then the isomorphism follows from (1), Proposition 14.5.3 and (14.16) applied over the function field of Y . The cases of h0 (X) and h1 (X) are similar. o Corollary 14.8.6. a) Let P : Meff rat → Mrat denote the projection functor. Then the right adjoint P # of P is defined on d≤2 Morat .
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b) Let S : Morat → (Morat /Ko≤1 )\ be the projection functor. The the left adjoint # S of S is defined on the thick image of d≤2 Morat by S. eff Proof a) Consider the thick subcategory do≤2 Meff rat of Mrat generated by those motives of the form h0 (X), h1 (X) and t2 (S) as in Theorem 14.8.4. o Clearly P (do≤2 Meff rat ) = d≤2 Mrat , and Corollary 14.8.5 (1) gives the inclusion ⊥ do≤2 Meff rat ⊂ I. The conclusion now follows from Proposition 14.8.1. b) The proof is the same, using this time Corollary 14.8.5 (2) (or rather its projection into Morat ) and Proposition 14.8.3.
Remark 14.8.7. It is natural to ask what is the largest full subcategory of Morat on which P # is defined. We don’t know the answer to this question ¯ but at least, P # is not defined on h(X) for any 3-fold X such that the group 2 Aalg (X) of codimension 2 cycles modulo algebraic equivalence is not finitely generated (cf. Griffiths’ examples). This will be proven in the final version of [K-S], the core of the argument being due to Joseph Ayoub. On the other hand we expect that the functor Sn of (14.18) below always has a left adjoint: this will be the object of a further work.
14.8.3 Birational motives and motives at the generic point Let \
=
d≤n Meff rat K≤n−1 ∩ d≤n Meff rat
!\
dn Morat =
d≤n Morat Ko≤n−1 ∩ d≤n Morat
dn Meff rat
(14.17)
where \ means taking the pseudo-abelian envelope (cf. Definition 14.5.5 for the definitions of the objects appearing in (14.17).) We thus have a diagram of categories and functors P
n d≤n Meff −→ d≤n Morat rat − Sn y Rn y
(14.18)
Qn
dn Meff −→ dn Morat . rat − ¯ for M ∈ d≤n Meff (With a previous notation, Pn (M ) = M rat .) (n) Note that the duality D of Lemma 14.5.6 acts on (14.18) by exchanging the categories d≤n Morat and dn Meff rat which are therefore anti-equivalent, and o also induces a duality on dn Mrat .
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¯ If dim X ≤ n, then we write hgen (X) for Sn (h(X)) = Sn Pn (h(X)): this is the motive of X at the generic point (relative to dimension n). We have hgen (X) = 0 if dim X < n. Lemma 14.5.6 c) shows that, for two n-dimensional smooth projective k-varieties X, Y , dn Morat (hgen (X), hgen (Y )) = An (X × Y )/J(X, Y )
(14.19)
where J is the ideal of Definition 14.4.2. The name “motive at the generic point” is in reference to Beilinson’s paper [Bei], where he calls the right hand side of (14.19) correspondences at the generic point. His prediction (conjecture) (*) on p. 35†, deduced from some “standard” conjectures on mixed motives, implies that, for X = Y , this Qalgebra is semi-simple finite-dimensional and that An (X ×X)hom ⊂ J(X, X). (Compare with Theorem 14.6.12 (7) in the case of a surface.) Theorem 14.8.8. a) Suppose that n ≤ 2. In (14.18), Pn and Qn have a right adjoint while Rn and Sn have a left adjoint. All these adjoints are right inverse to the corresponding functors. In particular, the functor o Sn Pn = Qn Rn : d≤n Meff rat → dn Mrat
has a canonical section Σn = Pn# ◦ # Sn = # Rn ◦ Q# n. b) Suppose n = 1: if C is a smooth projective curve, then for any C-K decomposition of C we have ¯ P1# h(C) ' h0 (C) ⊕ h1 (C) Q# 1 hgen (C) ' R2 (h1 (C)) #
R1 R1 (h(C)) ' h1 (C) ⊕ h2 (C) # ¯ 1 (C) S1 hgen (C) ' h Σ1 (hgen (C)) ' h1 (C).
c) Suppose n = 2: if S is a smooth projective surface, then for any refined C-K decomposition of S as in Propositions 14.2.1 and 14.2.3, † Also due to Rovinski and Bloch as he points out.
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we have ¯ P2# h(S) ' h0 (S) ⊕ h1 (S) ⊕ t2 (S) Q# 2 hgen (S) ' R2 (t2 (S)) #
R2 R2 (h(S)) ' t2 (S) ⊕ h3 (S) ⊕ h4 (S) # S2 hgen (S) ' t¯2 (S) Σ2 (hgen (S)) ' t2 (S).
Note that, as a composite of a left and a right adjoint, Σn has no special adjunction property. Proof a) For Pn and Sn this follows immediately from Corollary 14.8.6; the cases of Qn and Rn follow by using the duality D(n) . b) and c) Let us prove the first formula in c): the other cases are similar. L ¯ ¯ 0 (S)⊕ h ¯ 1 (S)⊕ t¯2 (S) (see proof 'h Writing h(S) = 4i=0 hi (S), we have h(S) of Lemma 14.5.7). By Corollary 14.8.5 (1), h0 (S) ⊕ h1 (S) ⊕ t2 (S) ∈ ⊥ I; the conclusion then follows from Proposition 14.8.1 d). Corollary 14.8.9. The functors Pn , Qn , Rn , Sn are essentially surjective for n ≤ 2. (This fact is not obvious a priori since we added projectors when defining the quotient categories: it amounts to saying that this operation was not necessary.) From Theorem 14.8.8 a), we have for n ≤ 2 a commutative diagram of natural transformations in d≤n Meff rat : id x
Pn# Pn x
→ −
# # Rn# Rn − → Rn# Q# n Qn Rn = Pn Sn Sn Pn
given by the units and counits of the respective adjunctions. Applying this diagram to h(C) (resp. h(S)) for C a curve (resp. S a surface) and using Theorem 14.8.8 b) (resp. c)), we get the following corollary, which gives a partial positive answer to Conjecture 14.3.4: Corollary 14.8.10. c) Given a curve C, in the diagram h(C) x
→ h0 (C) ⊕ h1 (C) − x
h1 (C) ⊕ h2 (C) − →
h1 (C)
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all maps and objects are independent of the choice of a C-K decomposition, and are natural in C for the action of correspondences. b) Given a surface S, in the diagram h(S) x
→ h0 (S) ⊕ h1 (S) ⊕ t2 (S) − x
t2 (S) ⊕ h3 (S) ⊕ h4 (S) − →
t2 (S)
all maps and objects are independent of the choice of a refined C-K decomposition as in Propositions 14.2.1 and 14.2.3, and are natural in S for the action of correspondences. ¯ In the case of a surface S, we may think of hbir (S) := P2# h(S) as the largest birational quotient of h(S) and think of Σ2 (hgen (S)) as the largest submotive at the generic point of hbir (S). Similarly, R2# R2 (h(S)) may be thought of as the largest subobject of h(S) “purely of dimension 2”. Note that both maps h(S) → t2 (S) and t2 (S) → h(S) given by a projector π2tr from a refined C-K decomposition do depend on the choice of this decomposition. Nevertheless, it is unambiguous to write t2 (S) for Σ2 (hgen (S)), viewed as a functor in S. Corollary 14.8.11. Let d≤2 Sm be the category of smooth (open) varieties of dimension ≤ 2 over k. Assume that k is of characteristic 0. There are functors hbir , t2 : d≤2 Sm → Meff rat extending the above functors. These functors are (stably) birationally invariant. There are similar contravariant functors starting from the category d≤2 place of function fields of transcendence degree ≤ 2 over k, with morphisms the k-places. Proof By [K-S, 5.6] there are canonical functors d≤2 T −1 placeop → d≤2 Sr−1 Sm → d≤2 Morat the latter extending the natural functor from smooth projective varieties. Here T −1 place denotes the category of finitely generated extensions of k with morphisms the k-places, localized by inverting morphisms of the form K ,→ K(t), Sr−1 Sm denotes the category of smooth k-varieties localized by inverting the dominant morphisms which induce a purely transcendental extension of function fields, and d≤2 denotes the full subcategories respectively of function fields of transcendence degree ≤ 2 and of varieties of dimension ≤ 2.
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14.8.4 The triangulated birational motive of a surface Suppose k perfect. Recall from [K-S] that the projection functor P : Meff rat → o Mrat inserts into a naturally commutative diagram of categories and functors Φ
Meff → DM−eff rat − ν≤0 Py y ¯ Φ
Morat − → DM−o where DM−o is a birational analogue of DM−eff , and that ν≤0 has an everywheredefined right adjoint/right inverse i (in fact DM−o is a priori defined as the full subcategory of DM−eff consisting of those objects C such that Hom(Q(1), C) = 0, and it is then proven that the inclusion functor i has a left adjoint ν≤0 which inserts itself in the above commutative diagram). If X is a smooth projective variety, we set ¯ ¯ (X) = Φ ¯ h(X) M = ν≤0 M (X) ∈ DM−o . ¯ ∈ Mo . Suppose that P # is defined at some Chow birational motive M rat ¯ Starting from the natural isomorphism ν≤0 Φ = ΦP , the two adjunctions give a “base change” morphism ¯ → iΦ ¯M ¯. ¯ = iΦP ¯ P #M ¯ → iν≤0 ΦP # M ΦP # M
(14.20)
¯ is defined. Then ¯ ∈ Morat be such that P # M Proposition 14.8.12. Let M the following conditions are equivalent: i) (14.20) is an isomorphism. ¯ is in the essential image of i (i.e. Hom(Q(1), ΦP # M ¯ ) = 0). ii) ΦP # M ¯ ' iN for some Proof (i) ⇒ (ii) is obvious. Conversely, suppose that ΦP # M eff N ∈ DM− . Since i is right inverse to ν≤0 , we find first ¯ ' ΦP ¯ P #M ¯. N ' ν≤0 iN ' ν≤0 ΦP # M ¯M ¯ comes from a unique Since i is fully faihtful, the morphism iN → iΦ # ¯ ¯ ¯ ¯ morphism ΦP P M ' N → ΦM , which clearly is the morphism induced ¯ →M ¯ . This counit is an isomorphism by Proposition by the counit P P # M 14.8.1 b), hence (14.20) is an isomorphism. ¯ is defined and Condition By Theorem 14.8.4 and Corollary 14.8.6 a), P # M ¯ (ii) of Proposition 14.8.12 is verified for all M ∈ d≤2 Morat . Hence (i) holds ¯ ¯ ¯ = h(C) for them. Taking M (h(S)) for a curve C (a surface S), we get:
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¯ ¯ (C) ' Φ(P # h(C)). Corollary 14.8.13. a) For any curve C, iM Given a C-K decomposition, this motive is isomorphic to M0 (C) ⊕ M1 (C). ¯ ¯ (S) ' Φ(P # h(S)). b) For any surface S, iM Given a C-K decomposition as in Propositions 14.2.1 and 14.2.3, this motive is isomorphic to M0 (S) ⊕ M1 (S) ⊕ Φ(t2 (S)). ¯ (S) is a direct summand of M (S) for any surface S, In particular, iM which answers positively a question of Ayoub.
References [A-J] R. Akhtar and R. Joshua K¨ unneth decomposition for quotient varieties, preprint, 2003, 18 pages. [An] Y. Andr´e, Une introduction aux motifs, Panoramas et synth`eses, SMF, 2004. [A-K] Y. Andr´e and B. Kahn, Nilpotence, radicaux et structures mono¨ıdales (with an appendix by P. O’Sullivan), Rend. Sem. Mat. Univ. Padova 108 (2003), 106–291. [Bei] A. Beilinson Remarks on n-motives and correspondences at the generic point, in Motives, Polylogarithms and Hodge Theory, F. Bogomolov and L. Katzarkov, eds., International Press, 2002, 33–44. [B1] S. Bloch K2 of Artinian Q-algebras, with application to algebraic cycles. Comm. Algebra 3 (1975), 405–428. [B2] S. Bloch Lectures on Algebraic Cycles, Duke University Mathematics Series IV, Duke University Press, Durham U.S.A., 1980. [B-K-L] S. Bloch, A. Kas and D. Lieberman Zero cycles on surfaces with pg = 0, Compositio Math. 33 (1976), 135–145. ´ ements de math´ematique: Alg`ebre, Ch. II, Hermann, 1970. [Bo] N. Bourbaki El´ ´ 43 (1974), 273–307. [Del-I] P. Deligne La conjecture de Weil, I, Publ. Math. IHES [Fu] W. Fulton Intersection Theory, Ergeb. Math. Grenzgeb., Springer-Verlag, Berlin, 1984. [G-P1] V. Guletskii and C. Pedrini The Chow motive of the Godeaux Surface, in Algebraic Geometry: A volume in memory of Paolo Francia, 179–195 W. de De Gruyter, 2002. [G-P2] V. Guletskii and C. Pedrini Finite-dimensional Motives and the Conjectures of Beilinson and Murre, K-Theory 30 (2003), 243–263. [Ha] R. Hartshorne Algebraic geometry, Springer, 1977. [H-K] A. Huber, B. Kahn The slice filtration and mixed Tate motives, preprint, 2005, http://www.math.uiuc.edu/K-theory/0719. [J1] U. Jannsen Motives, numerical equivalence and semi-simplicity, Invent. Math. 107 (1992), 447–452. [J2] U. Jannsen Motivic Sheaves and Filtrations on Chow groups, Proceedings of Symposia in Pure mathematics 55 (I) (1994), 245–302. [J3] U. Jannsen Equivalence realtions on algebraic cycles, in The Arithmetic and Geometry of Agebraic Cycles, NATO Sci. Ser. C Math. Phys. Sc. 548 Kluwer Ac. Publ. Co., 2000, 225–260. [K-S] B. Kahn and R. Sujatha Birational Motives, I, preliminary version, preprint, 2002.
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[Ki] S.I. Kimura Chow groups can be finite-dimensional, in some sense, Math. Ann. 331 (2005), 173–201. [Kl] S. Kleiman The standard conjectures, Proceedings of Symposia in Pure mathematics 55 (I) (1994), 3–20. [Mum1] D. Mumford Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 1968, 195–204. [Mum2] D. Mumford Abelian Varieties, Oxford University Press, Ely house, London, 1974. [Mu1] J. Murre On the motive of an algebraic surface, J. Reine angew. Math. 409 (1990), 190–204. [Mu2] J. Murre On a conjectural filtration on the Chow groups of an algebraic variety, Part I and II, Indagationes Mathem., n.S. 4(2) (1993), 177–201. [Schoen] C. Schoen Zero cycles modulo rational equivalence for some varieties over fields of transcendence degree 1, Proc. Symposia in Pure Math. 46 (1987), 463–473. [Sch] A.J. Scholl Classical motives, Proc. Symposia in Pure Math. 55 (I) (1994), 163–187. [Voev1] V. Voevodsky Cohomological theory of presheaves with transfers, in Cycles, transfers and motivic homology theories, Ann. Math. Studies 143, Princeton University Press, 2000. [Voev2] V. Voevodsky Triangulated categories of motives over a field, in E. Friedlander, A. Suslin, V. Voevodsky Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, 2000, 188–238. [Voev3] V. Voevodsky Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 2002, 351–355. [Voi] C. Voisin Sur les z´ero-cycles de certaines hypersurfaces munies d’un automorphisme, Ann. Scuola Norm. Sup. Pisa 19 (1992), 473–492. [Weil1] A. Weil Vari´et´es ab´eliennes et courbes alg´ebriques, Hermann, Paris, 1948. [Weil2] A. Weil Foundations of Algebraic Geometry, American Math. Soc. Providence, Rhode Island U.S.A., 1962. ´ ements de g´eom´etrie alg´ebrique, IV: ´etude [EGA4] A. Grothendieck, A. Dieudonn´e El´ locale ds sch´emas et des morphismes de sch´emas (quatri`eme partie), Publ. ´ 32 (1967). Math. IHES
15 A note on finite dimensional motives Shun-ichi Kimura
[email protected]
To Jacob for his 75th birthday Abstract Let C be a pseudo-abelian tensor Q-linear category. We consider the following problems. (i) Characterize 1-dimensional objects in C. (ii) When A ∈ C is Schur finite, study the set {λ|Sλ A = 0}. For (1), we prove that if the unit object 1 has no non-trivial direct summand, then invertible objects are 1-dimensional. If moreover C is rigid, then the converse holds. For (2), under some technical condition, we prove the existence of a minimal Young diagram that kills A. 15.1 Introduction Let C be a pseudo-abelian tensor category with Q-coefficients, namely each idempotent endo-morphism has Kernel and Cokernel, each Hom(A, B) is a Q-vector space with Q-bilinear compositions (A, B ∈ C), and has a tensor product functor ⊗ : C×C → C : (A, B) → A⊗B so that (1) for A1 , . . . , AN ∈ C, the tensor product A1 ⊗ · · · ⊗ AN is well-defined, independent of the order of taking the tensor product, and (2) there is a natural isomorphism A ⊗ B ' B ⊗ A such that for each σ ∈ SN , the morphism A1 ⊗ · · · ⊗ AN → Aσ(1) ⊗ · · · ⊗ Aσ(N ) is well-defined, independent of the order of composing the transpositions. In [5], we have introduced the notion of Kimura finiteness. For A ∈ 508
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C and a permutation σ ∈ SN , we define σA : A⊗N → A⊗N by sending 1 X a1 ⊗ · · · ⊗ aN to aσ(1) ⊗ · · · ⊗ aσ(N ) . Then sgn(σ)σA : A⊗N → A⊗N N! σ∈SN 1 X (resp. σA : A⊗N → A⊗N ) is an idempotent, and its image, which N! σ∈SN V exists by the assumption that C is pseudo-abelian, is denoted by N A (resp. SymN A). An object A is called evenly finite dimensional (resp. oddly finite V dimensional) when N A = 0 (resp. SymN A = 0) for some N > 0. An object A ∈ C is called Kimura finite if it is a direct sum of an evenly finite dimensional object and an oddly finite dimensional object. For example, V when C is the category of Z/2-graded vector spaces, d+1 V = 0 for an even degree d-dimensional vector space V , and Syme+1 W = 0 for an odd degree e-dimensional vector space W . Hence Kimura finite objects in C are exactly finite dimensional objects in C as vector spaces. The notion of Kimura finiteness is applied to Chow motives in [5], also see [4]. In [7], C. Mazza introduced a weaker finiteness condition based on Schur idempotents. When λ = (λ1 , . . . , λk ) with λ1 ≥ · · · ≥ λk is a partition of N (or equivalently, a Young X diagram with N boxes), then λ determines a Young symmetrizer cλ := cλ,σ σ ∈ Q[SN ] so that Q[Sn ] · cλ is isomorphic σ∈SN
to the irreducible representation of SN corresponding to λ. The Young symmetrizer cλ is an idempotent up to scalar, hence X Im cλ,σ σA : A⊗N → A⊗N σ∈SN
is well-defined, which image is denoted by Sλ A. When λ = (1, 1, . . . , 1), | {z } N -times V then Sλ A = N A, and when λ = (N ), then Sλ A = SymN A. We define A ∈ C to be Schur finite if Sλ A = 0 for some λ. Kimura finite objects are Schur finite, but not vice versa (see [7, Example 3.3] for such an example due to P. O’Sullivan). In this paper, we consider the following two problems. 1) Characterize 1-dimensional objects. 2) For a Schur finite object A, describe {λ|Sλ A = 0}. For (1), we prove that if the unit object 1 has no non-trivial direct summand, then invertible objects are 1-dimensional (Proposition 15.2.6). If moreover C is rigid, then the converse holds (Proposition 15.2.9). In particular, in
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the category of Chow motives, a motive is 1-dimensional if and only if it is invertible. For (2), assume that in C, A⊗N = 0 implies A = 0 (for example when C is rigid, see Lemma 15.3.13). Then we prove that for a Schur finite object A ∈ C, there exists a minimal Young diagram λ that kills A. Namely, for a Young diagram µ, we have Sµ A = 0 if and only if µ ⊃ λ (Corollary 15.3.10). We call this λ as the Schur dimension of A (Definition 15.3.11). Finally in the last section, we apply these results to study the Chow motives of smooth hypersurfaces. Acknowledgements The author is very much grateful to the invaluable discussion with Prof. Yves Andr´e. He is also grateful to the discussion with Prof. Uwe Jannsen and Prof. Carlo Mazza. The referee’s comments, which greatly clarified the arguments and generalized the results, were so much helpful that the author feels this paper is virtually coauthored by the referee. Special thanks goes to the referee.
15.2 1-dimensional finite objects Definition 15.2.1. Let C be a tensor category. An object 1 ∈ C is called a unit object if a natural isomorphism gX : X ' 1 ⊗ X is equipped, which is compatible with the associativity law and the commutativity law (see [8, 1.3, 2.2, 2.3, 2.4]). Remark 15.2.2. An identity object in a tensor category is unique up to a canonical isomorphism which preserves the isomorphism 1 → 1 ⊗ 1, if it exists (see [2, Prop. 1.3], and [8]). In this section, we work in the category of pseudo-abelian Q-linear tensor category C with a fixed unit object 1. Definition 15.2.3. A unit A ∈ C is invertible when for some B ∈ C, we have A ⊗ B ' 1. One can easily see that A ∈ C is invertible, if and only if the functor of tensoring A, namely C → C defined by X → X ⊗ A, is an equivalence of categories. Definition 15.2.4. Non-zero object A ∈ C is called evenly 1-dimensional if V2 A = 0, and oddly 1-dimensional if Sym2 A = 0. A is called 1-dimensional when A is either evenly or oddly 1-dimensional. Example 15.2.5. The unit object 1 is 1-dimensional. In fact, when σ : 1 ⊗ 1 → 1 ⊗ 1 is the commutativity law morphism, then the compatibility
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of gX and the commutativity law says that the following diagram commutes (see [8, 2.3]): y 1 EEE EEg1 yy y E y ∼ EEE yy ∼ y |y " σ /1⊗1 1⊗1 g1
The commutativity of the diagram implies that σ = id1⊗1 : 1 ⊗ 1 → 1 ⊗ 1, V and hence 2 1 = 0. Proposition 15.2.6. Suppose that End(1) has no non-trivial idempotent. Then every invertible objects of C is 1-dimensional. Proof Let A be invertible in C. Then A⊗A is again invertible. In particular, End(A ⊗ A) ' End(1). So A ⊗ A has no non-trivial direct factors. As V V A ⊗ A ' Sym2 A ⊕ 2 A, one of Sym2 A or 2 A is zero. Remark 15.2.7. In an earlier version of this paper, the author did not notice that 1 is always 1-dimensional. Moreover, he assumed only this condition for Proposition 15.2.6. The referee pointed out that the condition always holds, and gave the counterexample (Q, Q[+1]) in the category of a pair of graded Q-vector spaces, which is invertible but not 1-dimensional. Finally the referee gave the above elegant proof. Remark 15.2.8. The converse of Proposition 15.2.6 does not hold in general. In the category of R-modules, for a non-zero ideal I ⊂ R, the R-module R/I is 1-dimensional, but not invertible. Also in the construction of Chow motives, if one forgets to invert the Lefschetz motive L := h1 (P1 ), then L is 1-dimensional, but not invertible. Once we invert the Lefschetz motive, the category becomes rigid, and under the assumption of the rigidness, the converse of Proposition 15.2.6 holds, as Proposition 15.2.9 below. (In an earlier version, Proposition 15.2.9 was proved only for Chow motives, and the referee generalized it to rigid tensor categories.) Proposition 15.2.9. Let A be a 1-dimensional object of a rigid tensor category C. Then the evaluation morphism Aˇ ⊗ A → 1 identifies Aˇ ⊗ A with a non-zero direct summand of 1. In particular, if End(1) has no non-trivial idempotents, then A is invertible. Proof Let σ : Aˇ ⊗ A → A ⊗ Aˇ be the permutation of the factors, sending (x, y) to (y, x). This σ is isomorphism, so it is enough to show that the
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composition Aˇ ⊗ A
ev
/1
/A ⊗ A ˇ
δ
is equal to the permutation of the factors σ, up to signature. Let c be the composition of these morphisms. ˇ ⊗ A), ˇ c is idenSending c by the adjunction Hom(Aˇ ⊗ A, ) → Hom(A, tified with the composition Aˇ
Aˇ
δ⊗idAˇ
/A ⊗ A ˇ.
On the other hand, sending σ by the adjunction, one gets Aˇ
idAˇ ⊗δ
/A ˇ ⊗ A ⊗ Aˇ
σ⊗idAˇ
/A ⊗ A ˇ ⊗ Aˇ .
These two adjunction morphisms are equal, after composing idA ⊗ σ ˇ , where σ ˇ : Aˇ ⊗ Aˇ → Aˇ ⊗ Aˇ is the permutation of the factors. But because A is 1-dimensional, Aˇ is also 1-dimensional, and hence σ ˇ is either identity (when Aˇ is evenly 1-dimensional) or the multiplication by −1 (when Aˇ is oddly 1-dimensional). Hence we obtain c = ±σ. We are done.
15.3 Schur dimension of a Schur finite object P Definition 15.3.1. For a partition λ, we write |λ| for λi . If µ is another partition, we define λ∩µ to be the partition corresponding to the intersection of two Young diagrams, as in the diagram below.
λ
μ
λ∩μ
In other words, for λ = (λ1 , . . . , λk ) and µ = (µ1 , . . . , µ` ), let m = min(k, `), then λ ∩ µ := (min(λ1 , µ1 ), min(λ2 , µ2 ), . . . , min(λm , µm )). When λ = λ ∩ µ, we write λ ⊂ µ. Proposition 15.3.2. If A ∈ C is an objet of a tensor category C with Sλ A = 0, then for a partition µ with λ ⊂ µ, we have Sµ A = 0
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Proof Set N = |µ| − |λ|, then by Littlewood-Richardson rule below (or simply by Pieri’s formula), Sµ A is a direct summand of Sλ A ⊗ A⊗N , hence it is 0. Let us recall Littlewood-Richardson rule, the key tool in this section. Definition 15.3.3. (Littlewood-Richardson rule) Let λ, µ, ν be Young diagrams, and µ = (µ1 , µ2 , . . . , µk ). Then the Littlewood-Richardson number Nλµν is the number of ways to add |ν| − |λ| boxes to the Young diagram λ to make the Young diagram ν, to each of which boxes some integer is attached, subject to the following conditions: 1) The attached numbers are from 1 to k, and the number i is attached to µi boxes. 2) If two numbered boxes are in the same column, then their numbers are different. 3) In each row, the attached numbers are weakly increasing from the left to the right. 4) If the numbered boxes are listed from right to left, starting with the top row and working down, and one looks at the first t entries in this list (for any t between 1 and ν), each integer p between 1 and k − 1 occurs at least as many times as the next integer p + 1. Theorem 15.3.4. When A ∈ C is an object of a tensor category C, and ⊕N λ, µ are Young diagrams, then we have Sλ A ⊗ Sµ A = ⊕Sν λµν . See [6, I-9] for a proof, where Theorem 15.3.4 is proved in terms of Schur polynomials. Definition 15.3.5. For a Young diagram λ = (λ1 , . . . , λk ) and a pair of k X non-zero integers (a, b) ∈ Z2≥0 , we define n(a,b) (λ) := max(λi − b, 0). This i=a
is the number of boxes in the Young diagram λ, below the a−1-st row and be-
λ
b
a n (a, b) (λ) is the
number of boxes yond the b−1-st column.
in the shadowed area.
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Lemma 15.3.6. Let λ and µ be two Young diagrams with n(a,b) (µ) > 0. Let τ be a Young diagram such that the coefficient Nλµτ in the LittlewoodRichardson rule is non-zero. Then n(a,b) (τ ) ≥ n(a,b) (λ) + 1. Proof By the assumption n(a,b) (µ) > 0, we have µa ≥ b. Because the Littlewood-Richardson coefficient Nλµτ is positive, one can add µk boxes, numbered k, according to the Littlewood-Richardson rule (k = 1, 2, . . . , ), to the Young diagram λ to make the Young diagram τ . In particular, we add at least b boxes numbered a to λ by the rule (1). By the rules (3) and (4), these boxes are below the a − 1-st row. And they are in different columns by the rule (2), so at least one of these boxes should be beyond b − 1-st column. Therefore, τ has at least one more box than λ below the a − 1-st row and beyond b − 1-st row, which implies n(a,b) (τ ) > n(a,b) (λ). Corollary 15.3.7. Let λ be a Young diagram with n(a,b) (λ) > 0. DecomP pose Sλ⊗N = ν cν Sν . If cν > 0, then n(a,b) (ν) ≥ N . Proof We proceed by induction on N . When N = 1, there is nothing to P prove. Assume that this Corollary holds for N − 1. When Sλ⊗N = ν cν Sν P with cν > 0, there is some µ such that Sλ⊗N −1 = µ dµ Sµ with dµ > 0 and that Sν is a direct summand of Sµ ⊗ Sλ , namely Nν:λ,µ > 0. By inductive assumption, n(a,b) (µ) ≥ N − 1, hence by Lemma 15.3.6, it follows that n(a,b) (ν) ≥ N . Lemma 15.3.8. Let λ and µ be Young diagrams. For integer N , write P (Sλ∩µ )⊗N = cν Sν . If N is large enough and cν > 0, then ν contains either λ or µ. Proof Assume not. Then for each integer N , there exists a Young diagram νN such that νN is a direct summand of (Sλ∩µ )⊗N , and νN does not contain λ nor µ, namely for some iN and jN , we have λiN > (νN )iN and µjN > (νN )jN . Because there are finitely many possible iN ’s and jN ’s, some pair (iN , jN ) appears infinitely many times. Each νN 0 with N 0 > N contains some direct summand of (Sλ∩µ )⊗N , hence by replacing νN if necessary, we may assume that the same pair (i, j) = (iN , jN ) works for all N . Because ν1 = λ ∩ µ, we have λi > min(λi , µi ) (hence λi > µi ) and µj > min(λj , µj ) (hence µj > λj ). In particular, we have i 6= j. By symmetry, we may assume that i < j, and hence µi < λi . We have n(i,µi ) (λ ∩ µ) > 0, hence by Corollary 15.3.7, n(i,µi ) (νN ) ≥ N . On the other hand, because (νN )i < λi and (νN )j < µj ≤ µi , there are at most (λi − µi ) · (j − i) boxes in νN below
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the i − 1-th row and beyond the µi − 1-th column (see the picture below), hence n(i,µi ) (νN ) ≤ (λi − µi ) · (j − i), a contradiction.
λ
µ
ν
Ν
Without containing i=3
j=7
two blacked boxes, is confined to Ν the shadowed area. ν
Proposition 15.3.9. Let A ∈ C be a Schur finite object in a tensor category C. Suppose that two young diagrams λ and µ satisfy Sλ A = 0 and Sµ A = 0. ⊗N Then there exists some integer N > 0 such that Sλ∩µ A = 0. Proof By Lemma 15.3.8, there exists some N such that for each direct ⊗N summand Sν A of Sλ∩µ A, satisfying either ν ⊃ λ or ν ⊃ µ. By Proposition ⊗N 15.3.2, we have Sν A = 0 for each such ν, hence Sλ∩µ A = 0. Corollary 15.3.10. Assume that the tensor category C satisfies the condition that A⊗N = 0 implies A = 0 for A ∈ C and N > 0, Then for each Schur finite objet A ∈ C, there exists a smallest Young diagram λ which kills A, namely, a Young diagram µ satisfies Sµ A = 0 if and only if µ ⊃ λ. Proof Let λA be the intersection of all the Young diagrams µ with Sµ A = 0. Then by Proposition 15.3.9, we have SλA A = 0. By the construction, if Sµ A = 0, then µ ⊃ λ. Conversely, if µ ⊃ λA , then by Proposition 15.3.2, we have Sµ A = 0. The following definition is suggested by the referee. Definition 15.3.11. Let C be a tensor Q-linear category verifying the condition of Corollary 15.3.10. For a Schur finite object A ∈ C, we define the Schur dimension of A to be the smallest Young diagram λA which satisfies SλA A = 0, whose existence is guaranteed by Corollary 15.3.10. Remark 15.3.12. When V is a (d, e)-dimensional supervector space, then
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}| { z its Schur dimension is (e + 1, e + 1, . . . , e + 1), one more each, for row and column, than the classical dimension. For the condition that “A⊗N = 0 ⇒ A = 0”, we have the following criterion, which proof is due to the referee. Lemma 15.3.13. Let C be a rigid Q-linear tensor category . If A is an object of C such that A⊗N = 0 for some N > 0, then A = 0. ˇ and the Proof By adjunction, idA : A → A corresponds to δ : 1 → A ⊗ A, composition idA ◦ idA ◦ · · · ◦ idA corresponds to the morphism δ ⊗ · · · ⊗ δ : ˇ ⊗N , composed with the morphism idA ⊗ ev ⊗ · · · ⊗ ev ⊗ id ˇ , 1 → (A ⊗ A) A where ev : A ⊗ Aˇ → 1 is the evaluation morphism . But by the assumption, ˇ ⊗N is already zero, hence idA = idA ◦ idA ◦ · · · ◦ idA is zero, which (A ⊗ A) implies A = 0. Remark 15.3.14. Just assuming that C is pseudo-abelian closed tensor category is not good enough for Lemma 15.3.13. For example, in the category of Z-modules, we have (Q/Z)⊗2 = 0. Remark 15.3.15. Over C, if one assumes Hodge conjecture, one can even ⊗n say that a Chow motive is 1-dimensional √ if and only if M ' L for some n (see [5]). Over Q, let X = Spec Q[ −1], and ι : X → X be the complex conjugate, and define M = (X, 12 ([∆X ] − [Γι ]), 0). Then M is 1-dimensional, but not isomorphic to L⊗n for any n, because H ∗ (M ) = 0. By inverting the Lefschetz motive, the category of Chow motives becomes rigid, and it satisfies the assumption of Lemma 15.3.13 (and hence Corollary 15.3.10).
15.4 Chow Motive of a hypersurface Let X ⊂ section.
Pn+1
be a hypersurface of degree d, with h ∈ CH 1 X the hyperplane
Definition 15.4.1. Define the primitive part of the motive h(X) to be n
h(X)prim := (X, [∆X ] −
1X i h × hn−i , 0) d i=0
Remark 15.4.2. The cohomology of h(X)prim is concentrated on the middimensional part H n .
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Definition 15.4.3. Let us denote dim H ∗ (h(X)prim ) by N . We define the determinant of h(X)prim by ( VN h(X)prim (if n is even) det h(X)prim := . N Sym h(X)prim (if n is odd) Remark 15.4.4. If we assume the finite dimensionality of the motive h(X), then det h(X)prim is a 1-dimensional motive, because its cohomology group is 1-dimensional, because Kimura dimension of a Kimura finite motive is same as the dimension of its cohomology group ([5, Cor. 7.4]). In particular, det h(X)prim is invertible by Proposition 15.2.9. If moreover we work over C and assume the Hodge conjecture, then H ∗ (det h(X)prim ) ⊂ H ∗ (X N ) is spanned by an algebraic cycle α. When we choose its lift to the Chow group α ˜ ∈ CH∗ (X N ), we can write det h(X)prim = (X N , c(˜ α×α ˜ ), 0) for some c ∈ Q (see [5, Prop. 10. 3]). Let us consider the following question: If conversely we know that det h(X)prim can be written as (X N , c(α × α), 0) for some c ∈ Q and α ∈ CH∗ (X N ). Can we conclude that h(X) is finite dimensional? Proposition 15.4.5. When n is even, and we have det h(X)prim = (X N , cα× α, 0), then the motive h(X) is finite dimensional. When n is odd and, if moreover, det h(X)prim = (X N , c(α × α), 0), then h(X) is Schur finite. More precisely, we have S(N +1,1) h(X)prim = 0. n
1X i Proof Because h(X) = (X, h × hn−i , 0) ⊕ h(X)prim , the finite did i=0 mensionality (resp. Schur finiteness) of h(X) is equivalent to the finite dimensionality (resp. Schur finiteness) of h(X)prim . When n is even, the V assumption says that N h(X)prim = (X N , c(α × α), 0), which is evenly V V V 1-dimensional, hence we have 0 = 2 ( N h(X)prim ) 2N h(X)prim , therefore h(X)prim is evenly finite dimensional (actually, N dimensional). When n is odd, the assumption implies that 0=
2 ^
(SymN h(X)prim ) =
M
S(2N −i,i) h(X)prim
0
which implies the Schur finiteness of h(X)prim . By Corollary 15.3.10, we can conclude that S(N +1,1) h(X)prim = 0. Conjecture 15.4.6. When M is a Schur finite Chow motive, then Schur dimension of M is a rectangle.
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Assuming the finite dimensionality of M , using the nilpotency theorem, Schur dimension of M is the same as the Z/2-graded vector space H ∗ (M ), whose Schur dimension is always rectangle (Remark 15.3.12). Proposition 15.4.7. If we assume Conjecture 15.4.6, then assuming det h(X)prim = (X N , c(α × α), 0), we obtain the finite dimensionality of h(X). Proof By Proposition 15.4.5 and its proof, we may assume that n is odd, and prove the finite dimensionality of h(X)prim , which satisfies S(N +1,1) h(X)prim = 0. By Conjecture 15.4.6, the smallest Young diagram to kill h(X)prim must be a rectangle, which is contained in either (N + 1) or (1, 1), anyway h(X)prim is oddly of evenly finite dimensional, hence finite dimensional. In fact, then we have nilpotency theorem, so h(X)prim is oddly finite dimensional. References [1] Deligne, P.: Cat´egories tensorielles, Mosc. Math. J. 2 (2002) 227–248 [2] Deligne, P. and J. S. Milne Tannakian Categories, in Hodge Cycles, Motives and Shimura Varieties 101–228, Springer Lecture Notes 900 [3] Fulton, W. and J. Harris. Representation Theory, Springer GTM 129 [4] Guletskii, V. andC. Pedrini: The Chow motive of the Godeaux surface, in Algebraic geometry, 179–195, de Gruyter Berlin, 2002 [5] S. Kimura: Chow groups are finite dimensional, in some sense, Math. Ann. (2005) [6] Macdonald, I. G. : Symmetric Functions and Hall Polynomials, Oxford University Press (1995) [7] Mazza, C.: Schur functors and motives, K-theory 33 (2004) pp 89–106 [8] Saavedra, R.: Cat´egories Tannakiennes, Springer Lecture NoteS 265
16 Real Regulators on Milnor Complexes, II James D. Lewis † , 632 Central Academic Building University of Alberta Edmonton, Alberta T6G 2G1, CANADA
[email protected]
To Jacob Murre on the occasion of his 75th birthday. With admiration!
Abstract Let X be a projective algebraic manifold, and let KM k,X be the k-th Milnor K-theory sheaf on X. In some earlier work, we constructed a real regulator r from the Zariski cohomology of KM k,X (in all degrees) to a certain quotient of real Deligne cohomology. For certain classes of X, and from the work of others, this real regulator is known to satisfy a Noether-Lefschetz property, viz., r is ‘trivial’. In this paper we construct a twisted variant of Milnor K-theory, and corresponding twisted regulator r and arrive at a corresponding complex whose regulator images are nontrivial, even in the cases where r has trivial image.
16.1 Introduction Let X/C be a projective algebraic manifold of dimension n, OX the sheaf of regular functions on X, with sheaf of units O× X . We put × × KM (Milnor sheaf) k,X := OX ⊗ · · · ⊗ OX J, | {z } k-times
where J is the subsheaf of the tensor product generated by sections of the form: τ1 ⊗ · · · ⊗ τk τi + τj = 1, for some i 6= j . × [For example, KM 1,X = OX .] For 0 ≤ m ≤ 2, it is known [MS2] that M
k−m CHk (X, m) ' HZar (X, Kk,X ), where CHk (X, m) is Bloch’s higher Chow † Partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada
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M M group [Blo1], and where Kk,X = Image KM k,X → Kk (C(X)) . There is a real regulator [Blo2] 2k−m rk,m : CHk (X, m) → HD (X, R(k)),
(16.1)
which is known to satisfy a Noether-Lefschetz theorem for X a sufficiently general complete intersection of high enough multidegree [MS1]. In the case where k = m = 2, the vanishing of the real regulator in (16.1) for a sufficiently general curve X of genus g(X) > 1 is due to Collino [Co]. (One can also consult [Ke1, Ke2] for similar vanishing results for the Milnor regulator.) Apart from one’s natural inclination to further extend the results k−m of [Lew1] on the real Milnor regulator on HZar (X, KM k,X ) to other geometrically interesting Chow groups, the purpose of this paper is to construct k−m M a twisted variant H k−m (K M k−•,X ) of HZar (X, Kk,X ), for all k and m, and which has nontrivial regulator images, even in those cases where the standard regulator in (16.1) fails to be “nontrivial”. This however, comes at a price of giving up some functorial properties. There is a natural map k−m k−m (K M (X, KM HZar k−•,X ) which turns out to be neither injective k,X ) → H (modulo torsion) nor surjective (see § 16.8). Towards this goal of constructk−m M ing a twisted variant H k−m (K M k−•,X ) of HZar (X, Kk,X ), we introduce the notion of a flat holomorphic line bundle over a complex projective variety Z, being essentially a pair (L, k kL ), where k kL is a flat metric, and a meromorphic section σ ∈ Rat∗ (L), where Rat∗ (L) denotes the non-zero rational sections of L over Z, and where we view holomorphic line bundles in terms of their algebraic counterparts. The role of flatness of lines bundles is crutial to our construction of a regulator current that descends to the level of cohomology (see § 16.3). We formulate a variant of Milnor K-theory based on contructing “symbols” out of rational sections of flat line bundles, and construct a twisted “Gersten-Milnor” complex M T (k) T (k−1) KM KM KM → ··· k−•,X : k,X → k−1,Z codimX Z=1 T (j+2)
T (j+1)
M
··· →
codimX Z=k−j−1 T (3)
M
··· →
T (2)
KM 2,Z →
codimX Z=k−2
T (j)
M
KM j+1,Z →
KM j,Z → · · ·
codimX Z=k−j
M codimX Z=k−1
T (1)
KM 1,Z →
M
KM 0,Z → 0,
codimX Z=k
and accordingly define H
k−m
(K M k−•,X )
:=
ker T (m) :
L
L M M codimX Z=k−m K m,Z → codimX Z=k−m+1 K m−1,Z . L M T (m+1) codimX Z=k−2 K m+1,Z
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The main result of this paper, which describes the twisted regulator map (with this group as its source), can be found in 16.6 (Theorem 16.6.6). In § 16.7, we provide some examples where the twisted regulator is nontrivial, in the situation where the real regulator in (16.1) vanishes (modulo D(k, m)). We are indebted to the referee for suggesting the need for improvements to this paper, and to the organizers Jan Nagel and Chris Peters for their excellent job in organizing this conference. This paper is the end result of some fruitful collaboration we had with Brent Gordon some years ago [GL]. Since then, Brent Gordon has moved into the area of artificial intelligence. We are grateful for his contributions. Notation. X/C is a projective√algebraic manifold, of dimension n. For a subring A ⊂ R, put A(k) = (2π −1)k A. πk−1 : C = R(k) ⊕ R(k − 1) → R(k − 1), k (resp. E p,q ), the (global) complex-valued is the projection. Denote by EX X C ∞ -forms of degree k (resp. Hodge type (p, q)) on X. The Picard variety of X is denoted by Pic0 (X). The subspace of algebraic cocycles in H 2k−2 (X, Q) k−m H 2k−m−1 (X, R) denotes the (k − is denoted by H 2k−2 alg (X, Q). Finally, N m)-th coniveau filtration subspace of H 2k−m−1 (X, R).
16.2 Real Deligne cohomology For a subring A ⊂ R, we introduce the Deligne complex AD (k) :
A(k) → OX → Ω1X → · · · → Ωk−1 X . | {z } call this Ω•
Definition 16.2.1. Deligne cohomology is given by the hypercohomology: i HD (X, A(k)) = Hi (AD (k)).
Note that there is a short exact sequence: 0 → Ω•
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For m ≥ 1 and from Hodge theory, we have the isomorphisms: 2k−m HD (X, R(k))
H 2k−m−1 (X, C) F k H 2k−m−1 (X, C) + H 2k−m−1 (X, R(k)) H 2k−m−1 X, R(k − 1) πk−1 F k H 2k−m−1 (X, C)
' πk−1
−→ '
For example: 2k−1 HD (X, R(k)) '
πk−1
−→
H 2k−2 (X, C) F k H 2k−2 (X, C) + H 2k−2 (X, R(k))
H k−1,k−1 (X, R) ⊗ R(k − 1)
'
=: H
k−1,k−1
∨ n−k+1,n−k+1 (X, R(k − 1)) ' H (X, R(n − k + 1)) .
16.3 A regulator Using the isomorphism: 2k−1 HD (X, R(k)) ' H k−1,k−1 (X, R(k − 1)),
in this section we give a description of the real regulator, r : z k (X, 1) → H k−1,k−1 (X, R(k − 1)) ' H n−k+1,n−k+1 (X, R(n − k + 1))∨ , where z k (X, 1) is a certain twisted group of K1 classes. Towards this goal, we introduce the notion of a flat holomorphic line bundle, being essentially a pair (L, k kL ), where k kL is a flat metric on L, and the group nX z k (X, 1) = (σi , k kLi ) ⊗ Ei codimX Ei = k − 1, Li /Ei flat, i
σi ∈ Rat∗ (Li ),
X
o div(σi ) = 0 ,
i
where div(σ) is the divisor associated to the meromorphic section σ ∈ Rat∗ (L), where Rat∗ (L) denotes the nonzero rational sections of L over E, and where we view holomorphic line bundles in terms of their algebraic counterparts, in the Zariski topology. Recall that [GH, Sou]: H n−k+1,n−k+1 (X) '
n−k+1,n−k+1 EX,d–closed n−k,n−k ∂∂EX
.
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Now assume given a subvariety Z ⊂ X of codimension k − 1, an algebraic line bundle L over Z with a hermitian metric, and σ ∈ Rat∗ (Li ). For n−k+1,n−k+1 ω ∈ EX ⊗ R(n − k + 1), one has the L1 -integral: Z 1 √ ω log kσkL ∈ R, (16.2) (2π −1)n−k+1 Z where we use the fact that Zsing has real codimension ≥ 2 in Z. This integral naturally extends to elements of the form X (σ, k kL ) ⊗ Z. ξ= codimX Z=k−1 {L,k kL }/Z n−k+1,n−k+1 Thus ξ defines a current on EX ⊗ R(n − k + 1). That this current descends on the cohomology level, viz., it is ∂∂-closed, requires some assumptions on L and the metric. For this, we introduce the notion of a flat line bundle. Let Z be a projective variety and L a(n algebraic) line bundle on Z. Recall that if Z is smooth, a metric on L is equivalent to a collection of C ∞ functions {ρα : Uα → (0, ∞)} satisfying ρβ = ρα |gαβ |2 on Uα ∩ Uβ , where {Uα } is taken to be a Zariski open cover of Z where L|Uα ' Uα × C trivializes, and where the transition functions of L with respect to this cover are {gαβ }. The curvature form is the global closed real (1, 1)-form νL given locally on Uα by
νL =
1 √
2π −1
∂∂ log ρα ,
and it’s image c1 (L) = [νL ] in H 2 (X, Z) is (up to torsion) the first Chern class of L. Lemma 16.3.1. Let us assume given a line bundle L as above on a smooth projective Z, with first Chern class c1 (L) = 0. Then one can find a corresponding metric for which the curvature form νL = 0. Proof This can be deduced from [GH, Proposition, p. 148]. Since the proof is easy, and for convenience to the reader, we supply it here. From Hodge theory one has νL ∈ ∂∂EZ0 . Thus νL = 2π√1 −1 ∂∂ψ for some realvalued global C ∞ function ψ on Z. Now if we set ρ = exp(ψ), then ∂∂ log ρα = ∂∂ log ρ over each Uα . Now replace each ρα by ρ˜α := ρα /ρ. Then ρ˜α transforms accordingly over Uα ∩ Uβ and hence defines a metric on L; moreover {∂∂ log ρ˜α } = 0, which is what we needed to show.
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Now assume that Z is a projective variety, not necessarily smooth, with smooth part Z reg ⊂ Z. Also assume given a line bundle L on Z. A metric k kL is defined in the same way as given by a collection of functions {ρα : Uα → (0, ∞)}, ρβ = ρα |gαβ |2 , where the ρα pullback to C ∞ functions with respect to any desingularization Z˜ → Z. The curvature form νL is defined in the same way on Zreg . Definition 16.3.2. The metric k kL is flat if νL = 0. A flat line bundle L is a pair (L, k kL ) where k kL is flat. Note that if the metric is flat, then for any desingularization Z˜ → Z, L pulls back to a line bundle with trivial curvature form, hence trivial first Chern class. The presence of the Tate twist R(j) for the remaining part of this paper does not form any √ useful role. Thus we will ignore twists by supressing the factor 1/(2π −1)n−k+1 in (16.2). The reason for working with flat line bundles comes from our next result, viz., P Proposition 16.3.3. Consider ξ = i (σi , k kLi ) ⊗ Zi ∈ z k (X, 1) and ω ∈ n−k+1,n−k+1 EX,d–closed . Then the current defined by XZ ω 7−→ r(ξ)(ω) := ω log kσi kLi , i
Zi
is ∂∂-closed. n−k,n−k Proof Let ω ∈ ∂∂EX . Then we can write ω = d∂η for some η ∈ n−k,n−k . Also let EX X ξ= (σi , k kLi ) ⊗ Zi ∈ z k (X, 1) i
be given as above, and consider the corresponding integral XZ ω log kσi kLi . i
Zi
By Stokes’ theorem and a standard calculation (below): Z Z (d∂η) log kσi kLi = ∂η ∧ d log kσi kLi Zi Zi Z Z dη ∧ ∂ log kσi kLi , = ∂η ∧ ∂ log kσi kLi = Zi
Zi
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where the latter two equalities follow by Hodge type, and the former uses d (∂η) log kσkL = (d∂η) log kσkL − ∂η ∧ d log kσkL . More specifically, we make use of the following facts. By taking -tubes about the components D ⊂ div(σ), and using that dim D = n − k and n−k,n−k+1 ∂η ∈ EX , and that log kσk is locally L1 , in the limit as → 0 Z (∂η) log kσkL = 0, lim →0
Tube ((σ))
(using estimates involving lim→0+ log = 0). n−k,n−k+2 (locally L1 forms) and that Note that ∂η ∧ ∂ log kσkL ∈ EX,L 1 R dim Zi = n − k + 1. Thus we are left with an integral of the form Z dη ∧ ∂ log kσkL as indicated above. Next, d(η ∧ ∂ log kσkL ) = dη ∧ ∂ log kσkL , since ∂∂ log kσkL = 0, by the key ingredient of flatness of L. Thus Z Z (d∂η) log kσi kLi = d(η ∧ ∂ log kσi kL ) Zi
Zi
Z η ∧ ∂ log kσi kL .
= lim
→0
Tube ((σi ))
If we put Z = Zi for a given i, with z a local coordinate on Zreg , then we have the residue integral: Z Z dz 2 lim η ∧ ∂ log |z| = lim η∧ →0 |z|= →0 |z|= z Z √ dz = 2π −1 η|{z=0}∩Z , η|{z=0}∩Z = Residue{z=0}∩Z η ∧ , z {z=0}∩Z (i.e. taking “tubes” is dual to taking “residues”). Then by a residue calculation, linearity, and Stokes’ theorem, we arrive at the formula: √ Z −2π −1 X X r(ξ)(ω) = νD (σi ) η . 2 D D
i
(We note that there remains the possibility that Z = Zi is singular along D. To remedy this, one may pass to a normalization of Z with the same P calculations above.) Therefore i div(σi ) = 0, hence r(ξ)(ω) = 0 and we are done.
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16.3.4. Independence of the metric. Now consider Z ⊂ X an irreducible subvariety of codimension k − 1 in X, and L/Z a flat (algebraic) line bundle given by {gαβ , Uα }, and {ρα } a flat metric, i.e., ∂∂ log ρα = 0. Let {˜ ρα } be another flat metric. Then ρ˜β ρ˜α |gαβ |2 = , ρβ ρα |gαβ |2 hence ρ := (˜ ρα /ρα ) > 0 extends globally over Z; moreover, ∂∂ log ρ = 0. It follows that λ := 21 log ρ ∈ R, being harmonic and global, is necessarily constant. Therefore Z Z ω λ + log kσkL . ω log kσkL˜ = Z
Z
But λ
R
Z (· · · )
H k−1,k−1 (X, R). alg
defines a class in Thus if X ξ= (σj , k kLj ) ⊗ Zj , j
then in H k−1,k−1 (X, R) H k−1,k−1 (X, R) alg
,
the regulator r(ξ) is independent of the choice of flat metric k kLj on Lj /Zj . 16.4 Some Hodge theory The goal of this section is to describe the ∂∂-closed regulator current r(ξ) given in Proposition 16.3.3, from the point of view of de Rham cohomology. A good reference for this section is [GH, Chapter 0] and [Sou, Chapter ` and E r,s II]. Let D` (X), Dr,s (X) be the spaces of currents acting on EX X rspectively, and write D2n−` (X) = D` (X), Dn−r,n−s (X) = Dr,s (X). One has a corresponding decomposition M Dk (X) = Dp,q (X). p+q=k
Lemma 16.4.1 (∂∂-Lemma). If T ∈ Dp,q X is a coboundary, then T = ∂∂T0 for some T0 ∈ Dp−1,q−1 (X). Corollary 16.4.2. H
p,q
(X) '
p,q EX,d–closed p−1,q−1 ∂∂EX
Dp,q d–closed (X) ' . ∂∂Dp−1,q−1 (X)
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527
Lemma 16.4.3. The natural inclusion p,q Dp,q d–closed (X) → D∂∂–closed (X),
induces an isomorphism Dp,q (X) Dp,q ∂∂–closed d–closed (X) ' . ∂∂Dp−1,q−1 (X) ∂Dp−1,q (X) + ∂Dp,q−1 (X) Proof Let T ∈ Dp,q (X). Then ∂T ∈ Dp+1,q (X) and ∂T ∈ Dp,q+1 (X) ∂∂–closed are both d-closed. Therefore, from Hodge theory, ∂T = dS1 and ∂T = dS2 for some S1 ∈ F p+1 Dp+q (X), and S2 ∈ F q+1 Dp+q (X). Thus d(T −S1 −S2 ) = 0 and moreover by the Hodge (p, q) decomposition theorem, we can modify Sj within it’s Hodge type, such that the cohomology class [T − S1 − S2 ] is of type (p, q). (More explicitly, we can write [T − S1 − S2 ] = [A1 ] ⊕ [B] ⊕ [A2 ], where [A1 ] ∈ F p+1 H p+q (X, C),
[A2 ] ∈ F q+1 H p+q (X, C),
[B] ∈ H p,q (X),
are represented by d-closed currents (or forms) A1 , A2 , B of the corresponding Hodge types. Now replace Sj by Sj − Aj , and relabel it Sj .) Hence there exists T0 such that T − S1 − S2 + dT0 ∈ Dp,q d–closed (X). This implies that T + ∂T0p−1,q + ∂T0p,q−1 is d-closed. Next, suppose that T ∈ Dp,q d–closed (X) is given such that T ∈ Im∂ + Im∂. By the Hodge theorem, T has no harmonic part, and being d-closed implies that it is a coboundary. The lemma easily follows from this. To arrive at the same sort of de Rham description of r(ξ) that appears in the twisted case in [Ja, p. 349], we for the moment include the twist factor √ 1/(2π −1)n−k+1 appearing in (16.2). It is obvious that r(ξ) given in Proposition 16.3.3 determines an element of Dn−k+1,n−k+1,∂∂− closed (X, R(n − k + 1)). It follows easily from the proof of Lemma 16.4.3, that there exists ψ ∈ D2n−2k+2,0 (X) ⊕ · · · ⊕ Dn−k+2,n−k (X) such that r(ξ) + πn−k+1 (ψ) is d-closed. It’s action on n−k+1,n−k+1 n−k,n−k EX,d–closed /∂∂EX
is the same as r(ξ). By duality, viz., H k−1,k−1 (X, R(k − 1)) ' H n−k+1,n−k+1 (X, R(n − k + 1))∨ ,
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we end up with a class r(ξ) ∈ H k−1,k−1 (X, R(k − 1)). Note that likewise [r(ξ)] ∈
k−1,k−1 D∂∂–closed (X)
∂Dk−2,k−1 (X) + ∂Dk−1,k−2 (X)
' H k−1,k−1 (X).
Let k−1,k−1
Q
=
k−1,k−1 (X) D∂∂–closed k−1,k−1 DX,d–closed
.
There is a commutative diagram of short exact sequences: 0
0
0
↓
↓
↓ →
Im∂+ Im∂ Im∂∂
↓
↓
Dk−1,k−1 ∂∂–closed
→
Qk−1,k−1
→
Im∂ + Im∂
↓ k−1,k−1 DX,d–closed
→
0 →
Im∂∂
0 →
↓
↓
0 → H k−1,k−1 (X)
=
H k−1,k−1 (X) →
↓
↓
0
0
→ 0
→ 0
↓ 0
Thus Qk−1,k−1 '
Im∂ + Im∂ . Im∂∂
Note that r(ξ) determines a class {r(ξ)} ∈ Qk−1,k−1 , which is a measure of how far r(ξ) is from being d-closed. Proposition 16.4.4. Let X be a projective algebraic manifold of dimension n, and D an algebraic cycle of dimension k − 1 on X. Next, let ξ ∈ z k (X, 1) ⊗ RR be given and consider the corresponding r(ξ). Let us write P r(ξ) = i,α ri Zi,α log kσi,α k ∧ (?), with ri ∈ R, and assume that D meets each Zi,α properly (i.e. in a 0-dimensional set), and that |D| ∩ |(σi,α )Zi,α | =
Real Regulators on Milnor Complexes, II
529
∅. Then if we put [D] to be the Poincar´e dual of D, the cup product is given by the formula: X Z log kσi,α k. hr(ξ), [D]i = ri i,α
Zi,α ∩D
Proof By desingularization and linearity, we reduce to the case where j : D ,→ X is a smooth subvariety of X. Let [γ] the Poincar´e dual of any given cycle γ on X. Then j∗ ◦ j ∗ [γ] = [γ ∩ D]. This follows from j∗ ◦ j ∗ [γ] = hj∗ ◦ j ∗ [γ], [X]i = j∗ hj ∗ [γ], j ∗ [X]i = j∗ hj ∗ [γ], [D]i = h[γ], j∗ [D]iX = [γ ∩ D]. In this case r(ξ) ∈ H k−1,k−1 (X) has a well-defined pullback j ∗Rr(ξ) ∈ H k−1,k−1 (D), P where dimX D = k − 1, and where in this case, j ∗ r(ξ) = α Zα ∩D log kσα k. Note that j∗ is just the trace. The proposition follows from this. Remarks 16.4.5. i) It is easy to show that r(ξ) is d-closed ⇔ it is a R combination of algebraic cycles. This is generalized in Theorem 16.6.6 below. ii) The formula in Proposition 16.4.4 can be interpreted in terms of S height pairings. Let us further assume that |D|∩ i,α Zi,α, Sing = ∅. Then for a suitable choice of flat metrics, we have: X hr(ξ), [D]i = ri h(σi,α )Z˜i,α , D ∩ Z˜i,α i ht , i,α
˜ The verwhere h , i ht is the height pairing on a desingularization Z. sion of the definition of height pairing we employ is given in [MS3, Def. 1)]. (One need only show that for a suitable flat metric, H(log kσk) = 0, where H(−) is the harmonic projection. However if we write c = H(log kσk), then we know c ∈ R is constant. Put λ = e−c > 0, and multiply the metric ρ by λ · ρ.)
16.5 A Tame symbol Now let Z ⊂ X be of codimension k − 2 with given flat bundles (Lj , k kLj ), j = 1, 2 and σf ∈ Rat∗ (L1 ), σg ∈ Rat∗ (L2 ). As a first step in the direction
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J. D. Lewis
of constructing a twisted Milnor complex, we define a generalization of the Tame symbol as follows. T ({(σf , k kL1 ), (σg , k kL2 )} ⊗ Z) =
X
(−1)νD (σf )νD (σg )
codimZ D=1
ν (σg )
σf D
ν (σf )
σg D
,k k D
⊗νD (σg )
L1
−νD (σf )
⊗L2
⊗ D.
Proposition 16.5.1. T ({(σf , k kL1 ), (σg , k kL2 )} ⊗ Z) ⊂ z k (X, 1). Proof One first shows that T takes flat line bundles over Z to flat line bundles over each such D ⊂ Z of codimension one. If {fαβ }, resp. {gαβ } are the transition functions for L1 , resp. L2 , over Z with common open trivializing (j) cover {Uα }α∈I , and with corresponding {ρα : Uα → (0, ∞)}, j = 1, 2 (j) i.e. ∂∂ρα = 0, we consider the following calculation for codimension one irreducible D ⊂ Z. First, for σf = {fα } and σg = {gα } local representations of nonzero meromorphic sections of L1 and L2 we have fα = fαβ fβ and gα = gαβ gβ over Uα ∩ Uβ . Hence over D ∩ Uα ∩ Uβ , ν (g)
ν (g)
fαD
ν (f )
gαD
= hαβ
ν (σg ) −νD (σf ) on D ∩ gαβ (1) νD (σg ) (2) −νD (σf ) {(ρα ) (ρα ) } associated
where hαβ = fαβD
fβ D
ν (f )
gβD
,
Uα ∩ Uβ . Further, we have the metric
to the line bundle {hαβ } over D, and with respect to the open cover {Uα ∩ D}α∈I . But νD (σg ) (2) −νD (σf ) ∂∂ log((ρ(1) (ρα ) ) α ) (2) = νD (σg )∂∂ log(ρ(1) α ) − νD (σf )∂∂ log(ρα ) = 0,
hence this metric is flat as well. Next, we show that the divisor associated to T ({(σf , k kL1 ), (σg , k kL2 )}⊗Z) is zero. Choose a Zariski open set U ⊂ Z for which L1 , L2 both trivialize over U . Then over U we have (the restriction of) the divisor associated to the usual tame symbol, T ({f, g}⊗Z), which is zero, as required. Here we have identified f = σf and g = σg for f, g ∈ C(Z)× . Since Z is covered by such U , we are done. Proposition 16.5.2. r( Im(T )) = 0. We prove this by first establishing two Lemmas. Lemma 16.5.3. Let Z be a smooth subvariety of codimension k − 2 in X, and let f, g ∈ C(Z)× be given. Then r T {f, g} = 0.
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Proof [Lev]. It is instructive to sketch the proof. By a pushforward of the relevant currents involved, and by a proper modification, it suffices to assume that Z is smooth and that f, g : Z → P1 are morphisms. Put F = (f, g) : Z → P1 ×P1 , and let (t, s) = (z1 /z0 , w1 /w0 ) be affine coordinates for P1 × P1 . Then F ∗ T {t, s} = T {f, g}. One can explicitly compute: T {t, s} = (∞ × P1 , s) − (0 × P1 , s) + (P1 × 0, t) − (P1 × ∞, t). Next, put Z η= ∞×P1
Z log |s|∧? −
Z log |s|∧? +
0×P1
Z log |t|∧? −
P1 ×0
log |t|∧?. P1 ×∞
It is easy to see that F ∗ η = r(T {f, g}), and that η defines the zero cohomology class. This proves the lemma. Now let σ be a section of a flat line bundle over a given subvariety Z ⊂ X. Then kσk has at worst pole like growth along the pole set of σ; moreover ∂∂ log kσk = 0. Lemma 16.5.4. In terms of local analytic coordinates, kσk is locally a product of the form ρ = hh, where h is meromorphic. Proof Since ∂∂ log kσk = 0, it follows that ∂ log kσk is a meromorphic 1form. Therefore by the holomorphic Poincar´e lemma, and away from divisor set |(σ)|, locally (in the strong topology) we have ∂ log kσk = ∂H, for some holomorphic function H. Therefore locally d log kσk = ∂H + ∂ H = (∂ + ∂)H + (∂ + ∂)H = d(H + H). Thus log kσk = H + H + K for some K ∈ R, and hence ρ := hh where K h = eH+ 2 . Next, locally over the divisor set |(σ)|, we can replace ρ by ρ˜ = |f |2 ρ, for some meromorphic function f , such that log ρ˜ is defined and ∂∂ log ρ˜ = 0. Since ρ˜ is a product (local) of a holomorphic function times it’s conjugate, it follows that ρ is a product of a meromorphic function times it’s conjugate. 16.5.5. Proof of Proposition 16.5.2 Observe that if hh = kk on an open set in Cn , with h, k meromorphic, then −1 h h = , k k is both ∂ and ∂-closed. Thus h = ck for some c ∈ C× with |c| = 1. Thus
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J. D. Lewis
over an analytic cover {∆α } of Z, we can write kσk ∆α = hα hα , where hα is a meromorphic function on ∆α . Suppose that there is a finite cover Z˜ → Z ˜ such that by analytic continuation, h becomes a rational function on Z. ˜ ˜ ˜ ˜ Let’s write this as h. Then k˜ σ k = hh, where σ ˜ is the pullback of σ to Z. This for example would be the case if the c’s above are m-th roots of unity for some m ∈ N. Arriving at this situation would imply Proposition 16.5.2, by putting us in the setting of Lemma 16.5.3. However by a limit argument, we can reduce to this situation. Put S 1 = {z ∈ C |z| = 1}. Then {hα } 1 naturally defines an element in H 0 (Z, M× Z /S ). We can assume that Z is 1 1 smooth (and projective). The map H (Z, S ) → H 1 (Z, M× Z ) = 0 factors × × 1 1 through H (Z, OZ ) → H (Z, MZ ), which is well-known to be zero [GH]. From the short exact sequence × 1 0 → S 1 → M× Z → MZ /S → 0,
we deduce that 1 1 1 1 H 0 (Z, M× Z /S ) → H (Z, S ) = H (Z, R/Z), √
∼
is surjective, where e −1t : R/Z → S 1 . Since the kernel of the map H 2 (Z, Z) → H 2 (X, Q) is a finite group, there is no loss of generality in identifying H 1 (Z, R/Z) with H 1 (Z, R)/H 1 (Z, Z), and H 1 (Z, Q/Z) with H 1 (Z, Q)/H 1 (Z, Z). Next, since kσk ∆α = hα hα , it follows that div({hα }) = div(σ), and hence the class of {hα } in H 1 (Z, R/Z), which we identify with H 1 (Z, R)/H 1 (Z, Z) ' H 0,1 (Z)/H 1 (Z, Z) ' Pic0(Z), is the class correspond0 ing to the flat line bundle associated to σ. Let L0 Pic (Z)×Z be the Poincar´e bundle, and let σ ˜ be a rational section of L Pic (Z)×Z which doesn’t vanish on {0} × Z.† Then for t in some polydisk neighbourhood of 0 ∈ Pic0 (Z), one has a family of flat metrics k kt on Lt := L {t}×Z , and if we put σt = σ ˜ {t}×Z , then locally kσt kt = ht,α ht,α , and we arrive at a deformation {ht,α } of {h0,α }. Next, by our identifications above, H 1 (Z, Q/Z) is a dense subset of the torus H 1 (Z, R/Z), and a point in H 1 (Z, Q/Z) corresponds to a class {hα } with corresponding c’s being m-th roots of unity. Thus with regard to kσk, we can write {hα } as a limit of classes corresponding to points in H 1 (Z, Q/Z). Namely, t→0 ht,α := ht,α · h−1 7→ {hα }, 0,α · hα where {ht,α } corresponds to a class in H 1 (Z, Q/Z) for t in a countably † This is easy to arrange. With respect to a projective embedding of Pic0 (Z) × Z, the twisted bundle L(m) is very ample for m 1. One simply chooses a general section of Γ(L(m)) and of Γ(O(m)) and assigns σ to be the quotient.
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dense subset of some polydisk neighbourhood of 0 ∈ Pic0 (Z). This proves the proposition.
16.6 A Milnor complex We first recall the definition of Milnor K-theory [BT]. Let F be a field with multiplicative group F× , and set M T n (F× ), T (F× ) = n≥0 ∼
the tensor algebra of the Z-module F× . (Here T 0 (F× ) := Z.) Then F× → T 1 (F× ) by a 7→ [a]. If a 6= 0, 1, set ra = [a] ⊗ [1 − a] in T 2 (F× ). Then the two-sided ideal R generated by the ra ’s is graded, and we put M K∗M F = T (F× ) R = KnM F. n≥0
Then K∗M F may be presented as a ring with generators `(a), for a ∈ F× , subject to the relations `(ab) = `(a) + `(b), `(a)`(1 − a) = 0,
a 6= 0, 1.
Observe that KjM (F) = Kj (F) for j = 0, 1, 2, where the latter is Quillen K-theory. 16.6.1. A twisted Milnor complex. As before let L be a (flat) line bundle over Z ⊂ X. We view L in terms of a corresponding Cartier divisor, viz., work on the Cartier divisor level. Then we first observe that although the set of nonzero rational sections Rat∗ (L/Z) of L over Z is not a group, by ` fixing Z, the union L/Z Rat∗ (L/Z) can be endowed with the structure of ` a group. More specifically, an element of L/Z Rat∗ (L/Z) is given by a pair (σ, L), σ ∈ Rat∗ (L/Z), L/Z flat, with product structure (σ1 , L1 ) ? (σ2 , L2 ) = (σ1 σ2 , L1 ⊗ L2 ). Thus we assign ∗ KM 1,L/Z (C(Z)) := Rat (L/Z).
In terms of local trivializations, σ ∈ Rat∗ (L/Z) ⇔ σ = {σα } | σα ∈ K1 (C(Z)) and gαβ σβ = σα , (16.3) Q where {gαβ } defines L, and where {σα } lies in the direct product α K1 (C(Z)). By passing to a direct limit over refining open covers of Z, the latter term in (16.3), which will still be denoted by K M 1,L/Z (C(Z)), is given the structure
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J. D. Lewis
we require, and we will assume this to be the case in the discussion below. Next, we consider the group a a M K Rat∗ (L/Z). = (C(Z)) = KM 1,Z 1,L/Z L/Z flat
L/Z flat
Put KM •,Z
=
X ∞
⊗Z j KM 1,Z
R• ,
j=0
where R• is the two-sided ideal generated by , (i) σ ⊗ (−σ) σ ∈ K M 1,Z (ii) (1 − f ) ⊗ f f ∈ C(Z)× − {1} . The first relation leads to a desired anticommutative property of products of “symbols”, and the second relation incorporates the “usual” Steinberg relation in the case of function fields. Since R• = ⊕j≥2 Rj is graded, we can write ∞ M KM = KM •,Z j,Z , j=0 M M where for example K M 0,Z = Z, K 1,Z is given above, and K 2,Z = group of symbols {σ1 , σ2 } [generalized] Steinberg relations .
Thus for example, {σ, −σ} = 1, hence one can easily check that {σ, σ} = {σ, −1} = {−1, σ} = {σ, σ −1 }. In general, given a symbol {σ1 , σ2 }, and up to rewriting this as a product of other symbols, one can always “factor out” common divisors in the divisor sets |(σ1 )|, |(σ2 )|. 16.6.2. We want to build a twisted Milnor-Gersten complex out of this, with the j-th term given by M KM j,Z . codimX Z=k−j
Thus M codimX Z=k
KM 0,Z =
M codimX Z=k
K0 (C(Z)),
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and the first two (generalized) Tame symbols div := T (1) :
M
T
:= T
:
M codimX Z=k−2
KM 0,Z ,
codimX Z=k
codimX Z=k−1 (2)
M
KM 1,Z → KM 2,Z
M
→
KM 1,Z ,
codimX Z=k−1
have already been defined. In order to define higher Tame symbols M M T (j+1) : KM −→ KM j+1,Z j,Z , codimX Z=k−j−1
codimX Z=k−j
we digress again [BT]. This time we will assume given a field F with a discrete valuation ν : F× → Z, and the corresponding discrete valuation ring O := {a ∈ F | ν(a) ≥ 0}, where we assign ν(0) = ∞. Let π ∈ O generate the unique maximal ideal (π), i.e., ν(π) = 1, and recall that all other nonzero ideals are of the form (π m ), for m ≥ 0. Note that F× = O× · π Z (direct product). Next let k = k(ν) be the residue field, and K•M Milnor K-theory. Then there is a map dπ : F× → (K•M k(ν))(Π) : uπ i 7→ `(u) + iΠ, where Π = `(π), and satisfies Π2 = `(−1)Π. This map induces ∂π : K•M F → (K•M k(ν))(Π). Next, one defines maps ∂π0 , ∂ν : K•M F → K•M k(ν), by ∂π (x) = ∂π0 (x) + ∂ν (x)Π,
(16.4)
which can be shown to be independent of the choice of π such that ν(π) = 1. This hinges on an explicit description of ∂ν [BT, Proposition 4.5]. The (genM F → K M k(ν). We also have eralized) Tame symbol is the map ∂ν : Km m−1 that multiplication is graded and skew commutative, and that deg `(a) = 1 = deg Π for a ∈ F× . For example, if we let a = a0 π i , b = b0 π j , so that ν(a) = i, ν(b) = j, and let a0 , b0 be the corresponding values in k(ν), then ∂π (a) = dπ (a) = `(a0 ) + iΠ,
∂π (b) = dπ (b) = `(b0 ) + jΠ,
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J. D. Lewis
so that ∂ν (a) = i and ∂ν (b) = j. Next, as shown in [BT], ∂π ({a, b}) = (`(a0 ) + iΠ)(`(b0 ) + jΠ) i = `(a0 )`(b0 ) + `(aj0 ) − `(b0 ) + ij`(−1) Π j j a ij a0 ∂ν ({a, b}) = ` (−1) i = ` (−1)ij i = ` T {a, b} , b b0 where T is the usual Tame symbol. Similarly, for a general product, we have (1)
(N )
∂π ({a(1) , . . . , a(N ) }) = (`(a0 ) + k1 Π) · · · (`(a0 ) + kj Π) (1) = `(a0 ) + k1 Π ∂π0 ({a(2) , . . . , a(N ) }) + ∂ν ({a(2) , . . . , a(N ) })Π (1) (N ) (2) (N ) = `(a0 ) · · · `(a0 ) + (−1)N −1 k1 `(a0 ) · · · `(a0 ) (1) (2) (N ) + `(a0 ) + k1 `(−1) ∂ν ({a , . . . , a }) Π, whence `(T (N ) {a(1) , . . . , a(N ) }) := ∂ν ({a(1) , . . . , a(N ) }) (−1)N −1 k1 (2) (N ) (1) = ` a0 , . . . , a0 (−1)k1 a0 , T (N −1) ({a(2) , . . . , a(N ) }) , where again T (2) = T is the usual Tame symbol above. For our purposes we define the Tame symbol on K M N,Z , by defining it’s value at the generic point of an irreducible codimension one D ⊂ Z, viz., (N ) (1) (N ) (N ) (1) (N ) TD ({σ , . . . , σ }) := T ({σ , . . . , σ }) D (−1)N −1 νD (σ(1) ) (2) (N ) (16.5) = σ0 , . . . , σ0 × (1) (1) (−1)νD (σ ) σ 0 , T (N −1) ({σ (2) , . . . , σ (N ) }) ν (σ (j) ) (j)
(j)
where σ (j) = πDD σ0 , πD a local equation of D in Z, and σ 0 the value (j) × of σ0 in C(D) . Here it is important to understand that if Lj /Z is the (j) flat line bundle associated to σ (j) , then σ 0 is a section of Lj D . In other words, this calculation occurs over the generic point of D, where one fixes the choice of local equation of D (but see Proposition 16.6.3(i) below). Now note that T (2) ({σg , σh }) = T ({σg , σh }) involves a section of a flat line bundle. From this it follows that T (3) can be defined with symbols of sections of flat bundles. By induction, it follows that if T (j+1) involves symbols of sections
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of flat bundles, since this is the case for T (j) . If we work modulo 2-torsion, the formula for TD in (16.5) simplifies somewhat, viz., (N )
TD ({σ (1) , . . . , σ (N ) }) ≡ (−1)N −1 νD (σ(1) ) (2) (N ) (1) (N −1) (2) (N ) σ0 , T ({σ , . . . , σ }) ≡ σ0 , . . . , σ0 ≡
N Y
d (1) (j) (N ) σ0 , . . . , σ0 , . . . , σ0
(−1)N −j νD (σ(j) ) ,
j=1
where the latter equivalence modulo 2-torsion follows by induction. Since the real regulator is blind to torsion, it makes sense to redefine the Tame symbol by the formula: (N ) TD ({σ (1) , . . . , σ (N ) }) (−1)N −j νD (σ(j) ) (16.6) QN d (1) (j) (N ) , := j=1 σ 0 , . . . , σ 0 , . . . , σ 0 and accordingly redefine KM •,Z
=
M ∞ j=0
KM j,Z
2–torsion . subgroup
(16.7)
Thus K M j,Z will now be interpreted as the corresponding group modulo 2torsion. Note that T (1) is still the divisor map, T (1) and T in Proposition 16.5.1 both agree on K M 2,Z (as we are working modulo 2-torsion), and that T (1) ◦ T (2) = 0. Quite generally, we prove the following: Proposition 16.6.3. Assume given our modified definition of T (N ) in (16.6) above. i) The definition of T (N ) does not depend on the local equations defining the codimension one D’s in Z. ii) Up to 2-torsion, T (N ) (RN ) ⊂ RN −1 for all N . L M iii) T (N ) ◦ T (N +1) = 0 ∈ K N −1,Z for all N . Proof of Proposition 16.6.3. The proof of Proposition 16.6.3 is a straightforward series of calculations. First of all, (i) is true for the same reasons as in the standard case in (16.4) above. If we let ≡ have the meaning “modulo
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J. D. Lewis
RN −1 and 2-torsion”, then the proof of (ii) follows from the calculations: TD {σ, −σ, σ (3) , . . . , σ (N ) } ≡
N Y
(j) N −j d (j) (N ) ≡ 0. {σ 0 , −σ 0 , . . . , σ 0 , . . . , σ 0 }νD (σ (−1)
j=3
Likewise, TD {f, 1 − f, σ (3) , . . . , σ (N ) } ≡ 0. Here we use the fact that for D ⊂ |(f )|, f itself is a local equation. Hence if νD (f ) ≥ 0 (1, 1) . (f 0 , [1 − f ]0 ) = (1, [±]1) := (1, −1) if νD (f ) < 0 To prove (iii), first observe that if σ (1) , σ (2) are rational sections of a bundle L/Z, then div ◦ T {σ (1) , σ (2) } = 0. This we proved earlier. This translates to saying that X (1) (2) νD (σ (2) )νE (σ 0 ) − νD (σ (1) )νE (σ 0 ) {E} = 0.
(16.8)
E⊂D (⊂Z)
We now consider E ⊂ D ⊂ Z and compute: (−1)N −j νD (σ(j) ) N Y d (1) (j) (N ) (1) (N ) TE ◦ TD {σ , . . . , σ } = TE σ 0 , . . . , σ 0 , . . . , σ 0 j=1
=
N Y Y j=1
×
d d (1) (i) (j) (N ) σ 00 , . . . , σ 00 , . . . , σ 00 , . . . , σ 00
(−1)i+j νD (σ(j) )νE (σ(i) 0 )
i<j
Y
d d (1) (j) (i) (N ) σ 00 , . . . , σ 00 , . . . , σ 00 , . . . , σ 00
(−1)i+j+1 νD (σ(j) )νE (σ(i) 0 )
i>j
=
N Y Y j=1
d (1) (i) σ 00 , . . . , σ 00 ,
i<j
d (j) (N ) . . . , σ 00 , . . . , σ 00
(i) )ν (σ (j) ) (−1)i+j [νD (σ(j) )νE (σ(i) E 0 )−νD (σ 0
.
By summing over E ⊂ D ⊂ Z and using (16.8), this proves the proposition.
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It follows then that one has a twisted Milnor complex of the form M KM KM KM k−•,X : k,X → k−1,Z → codimX Z=1
M
··· →
codimX Z=k−j−1
··· →
M
KM 2,Z →
M codimX Z=k−1
codimX Z=k−2
M
KM j+1,Z →
KM 1,Z →
KM j,Z → · · ·
codimX Z=k−j
M
KM 0,Z → 0.
codimX Z=k
(16.9) Let H k−m (K M k−•,X ) := ker T (m) :
L
L M M codimX Z=k−m K m,Z → codimX Z=k−m+1 K m−1,Z . L M T (m+1) codimX Z=k−2 K m+1,Z
Remarks 16.6.4. . i) One should expect the same story, viz., working with the unmodified definition of T (N ) in (16.5) and not working modulo 2-torsion, for the existence of such a complex above. Again, since the details would appear to be more involved, and that our regulator is blind to torsion, we opted for the simpler definition of H k−m (K M k−•,X ). ii) Warning: It would be pointless to attempt to sheafify the complex in (16.9), as it would lead to the Gersten-Milnor resolution of an untwisted Milnor sheaf (see 16.8). iii) It is reasonably clear that there is a surjective map H k (K M k−•,X ) → k k k CH (X)/CH alg (X), i.e. when m = 0. [This is because H (K M k−•,X ) is the free abelian group generated by cycles of codimension k in X, modulo the divisors of sections σ of flat line bundles L over codimension k − 1 subvarieties Z in X. Since a flat line bundle L corresponds to a line bundle with trivial first Chern class on any desingularization ≈ Z˜ → Z, the divisors of such σ are algebraically equivalent to zero.] In particular, one has a surjective map H k (K M k−•,X ) → Ln−k H2(n−k) (X), where Lm H` (X) := π`−2m Zm (X), ` ≥ 2m, is Lawson homology. (Here: Zm (X) = Cm (X)×Cm (X)/ ∼, with (x, y) ∼ (x0 , y 0 ) ⇔ x+y 0 = y + x0 , and where Cm (X) is the disjoint union of the Chow varieties of effective m-cycles of degree d = 0, 1, 2, . . ..) With respect to an unmodified definition of H k−m (K M k−•,X ) as suggested in (i), we pose the following.
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Question 16.6.5. Does there exist a map k H k−m (K M k−•,X ) → Ln−k H2n−2k+m (X) =: πm (Zn−k (X)) = πm (Z (X))?
We now state the main result: Theorem 16.6.6 (Main Theorem). Assume m ≥ 1. The current defined by Qm Qm ∗ 1 σj ∈ 1 Rat (Lj /Z) 7−→ codimX Z = k − m m Z X `−1 \ ω 7→ (−1) log kσ` k(d log kσ1 k ∧ · · · ∧ d log kσ` k ∧ · · · ∧ d log kσm k) ∧ ω `=1
Z
descends to a cohomological map k−1,k−m H (X) ⊕ H k−m,k−1 (X) ∩ H 2k−m−1 (X, R) M k−m , rk,m : H (K k−•,X ) → D(k, m) where D(k, m) =
( 2k−2 Halg (X, Q) ⊗ R 0
if m = 1 if m > 1
That is, this map does not depend on the choice of flat metrics on the respective flat bundles. Moreover, assume given D, smooth of dimension k − 1, and a morphism f : D → X such that f (D) is in “general position”. If ω ∈ H m−1,0 (D) ⊕ H 0,m−1 (D) and ξ ∈ H k−m (K M k−•,X ) are given, then rk,m (ξ)(f∗ ω) is induced by Qm Qm ∗ 1 Rat (Lj /Z) 7−→ 1 σj ∈ codimX Z = k − m m Z X \ ω 7→ (−1)`−1 log kσ` k(d log kσ1 k ∧ · · · ∧ d log kσ` k ∧ `=1
Z∩D:=f −1 (Z)
· · · ∧ d log kσm k) ∧ ω . 2n−2k+m+1 Furthermore, as a current acting on EX , if rk,m (ξ) is d-closed, then S 2k−m−1 it restricts to the zero class in H (X − V ) where V = α Zα is the P Qm support of ξ = α ∈ H k−m (K M k−•,X ). In this case, r k,m (ξ) 1 σj,α , Zα
lies in the Hodge projected image N k−m H 2k−m−1 (X, R) → H k−1,k−m (X) ⊕ H k−m,k−1 (X).
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Proof While it would be redundant to give a complete proof of this theorem, it is important to explain the new ingredients required to make the proof in the nontwisted case [Lew1] adaptible to the twisted situation. Firstly, the theorem is already proven in the case m = 1, this being the import of § 16.3–16.5. Next, if kσ` k ∈ R× is constant for some `, then some standard estimates together with a Stokes’ theorem argument implies that the corresponding regulator value on closed forms ω is zero for m ≥ 2. This leads to independence of the flat metric, after quotienting out by D(k, m), for m ≥ 1. To show for example that (σ, −σ, σ3 , . . . , σm ), Z or say (f, 1 − f, σ3 , . . . , σm ), Z goes to zero under the real regulator, amounts to reducing to the case where the σj ’s are rational functions via some finite covering Z 0 → Z, similar to what we did earlier in 16.5.5, and then applying the same arguments given in [Lew1]. That the current given in Thm 16.6.6 is ∂∂-closed now follows from the same proof as given in [Lew1]. Finally, the latter statement of the theorem is rather easy to prove. Being d-closed implies that we have a cohomology class on X, which restricts to a cohomology class on X − V ; moreover the current clearly vanishes on those forms compactly supported on X − V . 16.7 Some examples In contrast to the various vanishing results in the literature for the regulator in the nontwisted case [MS1, Ke1, Ke2, Co], etc., we exhibit some nonvanishing regulator results in the twisted case. 16.7.1. Regulator on H 0 (K M 2−•,X ) for a curve X. Let X be a compact Riemann surface of genus g ≥ 1, and let f, g ∈ C(X)× P × be given. Write T {f, g} = M j=1 (cj , pj ), where pj ∈ X and cj ∈ C . Then QM j=1 cj = 1 by Weil reciprocity. Now fix p ∈ X, and let Lj be a choice of line bundle corresponding to the zero cycle pj − p. Since deg(pj − p) = 0, Lj is a flat line bundle. There exists rational sections {σj } of the flat bundles {Lj } over X such that div(σj ) = pj − p. Thus one can easily check by a Q Tame symbol calculation that ξ := {f, g} j {σj , cj } ∈ H 0 (K M 2−•,X ). Note that d log |f | log |g| = log |f |d log |g| + log |g|d log |f |, and by a Stokes’ theorem argument together with standard estimates, Z Z log |f |d log |g| ∧ ω = − log |g|d log |f | ∧ ω, (16.10) X
for any d-closed form ω ∈
X 1 . EX,R
Thus if either f or g were constant, then
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both integrals in (16.10) would vanish. Applying the same reasoning to the Q terms j {σj , cj }, it easily follows that Z log |f |d log |g| − log |g|d log |f | ∧ ω, (16.11) r2,2 (ξ)(ω) = X
Q namely the contribution of the terms j {σj , cj } to the regulator current is zero. One can always find f and g [Lew1] such that the computation in (16.11) is nonzero for general X. For a simple example, consider this. Let E be a general elliptic curve, D another general curve, and X ⊂ E×D a general hyperplane section. Since X dominates E, and that the real regulator for E is nontrivial [Blo3], such an f and g can be found for X via pullback of corresponding rational functions on E, and then by continuity, the same story will hold as X varies with general moduli. Thus in summary, one can find general curves of genus g 1 for which the regulator 1 r2,2 : H 0 (K M 2−•,X ) → H (X, R),
is nontrivial. This is in complete constrast to the situation of the regulator in (16.1), viz., r2,2 : CH2 (X, 2) ' H 0Zar (X, K2,X ) → H 1 (X, R), where it is known ([Co]) that r2,2 is trivial for sufficiently general X of genus g > 1. Indeed, one can naively carry out the same construction above to arrive at a class ξ ∈ H 0Zar (X, K2,X ) arising from rational functions f, g on X with P × T {f, g} = M j=1 (cj , pj ), where pj ∈ X and cj ∈ C and Z log |f |d log |g| − log |g|d log |f | ∧ ω 6= 0. X
The issue boils down to finding a rational functions hj on X for which div(hj ) = N (pj − p), for some integer N 6= 0, which is not in general possible. In fact the difficulty of finding such hj amounts to finding torsion points on X, and is related to the known affirmative answer to the MumfordManin conjecture. For a discussion of the relation of the Mumford-Manin conjecture to this regulator calculation, the reader can consult [Lew1], as well as the references cited there. 16.7.2. Regulator on H 1 (K M 1−•,X ) for a surface X. For a simple example of a nontrivial twisted regulator calculation, where the usual regulator vanishes, consider the case X = M ×N , where M and N are smooth curves. If M and N are sufficiently general with g(M )g(N ) ≥ 2, then the image of the regulator in (16.1) vanishes (modulo the group of algebraic cocycles) [C-L1]. Now suppose we are given a curve C ⊂ M × N ,
Real Regulators on Milnor Complexes, II
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T f ∈ C(C)× , and ω ∈ H 1 (M, C) ⊗ H 1 (N, C) H 1,1 (X, R(1)) for which Z ω log |f | = 6 0. (16.12) C
Such a situation in (16.12) is fairly easy to arrive at for general M and N , by a deformation from a special case situation. (For example, one can use a construction in [Lew2, §7], or M × N can be a general deformation of a product of 2 curves dominating a general Abelian surface, together with the main results of [C-L2].) Write div(f )C =
m X
[(pj , qj ) − (sj , tj )].
j=1
We can write (pj , qj ) − (sj , tj ) = [(pj , qj ) − (pj , tj )] + [(pj , tj ) − (sj , tj )]. Now put Dj = {pj } × N and Kj = M × {tj }, and observe that the degree zero divisor (pj , tj ) − (pj , qj ) on Dj corresponds to a flat line bundle on Dj , and likewise the degree zero divisor (sj , tj ) − (pj , tj ) on Kj corresponds to a flat line bundle on Kj . Consider Cartier divisors σj on Dj , ηj on Kj , i.e. rational sections of the respective flat line bundles, with div(σj ) = (pj , tj ) − (pj , qj ), div(ηj ) = (sj , tj ) − (pj , tj ). Then ξ := (f, C) +
m X
(σj , Dj ) +
j=1
m X
(ηj , Kj ) ∈ H 1 (K M 1−•,X ).
j=1
[Note: As in the previous example, observe that one cannot replace the σj ’s (resp. ηj ’s) by rational functions on the Dj ’s (resp. Kj ’s), even if one replaces (pj , tj ) − (pj , qj ) and (sj , tj ) − (pj , tj ) by nonzero integral multiples.] Moreover the pullback of ω to Dj and Kj is zero. Thus: Z r(ξ)(ω) = ω log |f | = 6 0. C
Finally, observe that for general M and N , H 1 (M, Q) ⊗ H 1 (N, Q) ∩ 2 (M × N, Q) = 0. Thus while the image of the regulator in (16.1) for Halg M × N vanishes (modulo the group of algebraic cocycles), the twisted regulator does not vanish.
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16.8 Comparison to usual Milnor K-cohomology M M Recall that Kk,X = Image KM k,X → Kk (C(X)) . From the works of (ElbazVincent/M¨ uller-Stach, 1998) and (Gabber, 1992) [E-M], there is a flasque Gersten resolution for Milnor K-theory, viz., M M M 0 → Kk,X → KkM (X) → Kk−1 (Z) → · · · codimX Z=1
→
M
M
K1M (Z) →
codimX Z=k−1
K0M (Z) → 0.
codimX Z=k
The maps in this complex are the [higher] Tame symbols. Taking Zariski cohomologies, we arrive at natural maps M
k−m HZar (X, Kk,X ) → H k−m (K M k−•,X ), M
k−m k−m (X, KM HZar k,X ) → HZar (X, Kk,X ),
and hence a map k−m k−m (K M (X, KM HZar k−•,X ). k,X ) → H
(16.13)
The regulator rk,m on H k−m (K M k−•,X ) pullsback to a corresponding regulator k−m M on HZar (X, Kk,X ) via the map in (16.13). This corresponding regulator on M H k−m Zar (X, Kk,X ) is induced by the regulator in [Lew1]. The regulator in [Lew1] coincides, up to a real isomorphism on cohomology, (and up to a normalizing constant), with the regulator in (16.1) for m = 1, 2. But from the examples in § 16.7, it is clear that map in (16.13) cannot be surjective, for otherwise one would have the vanishing of the twisted regulator in the cases where the regulator in (16.1) vanishes. Secondly, for m = 0, we have k (X, KM ) ' CHk (X), whereas from Remarks 16.6.4(iii), H k (K M HZar k,X k−•,X ) k k looks more like CH (X)/CH alg (X). Thus the map in (16.13) cannot be injective either. It would be nice to have a more precise description of the image and kernel of the map in (16.13), and a possible connection between H k−m (K M k−•,X ) and Lawson homology as raised in Question 16.6.5 however that line of enquiry will not be pursued here.
References [BT] Bass, H. and J. Tate: The Milnor ring of a global field, in Algebraic K-theory II Lecture Notes in Math. 342 Springer-Verlag (1972) 349–446 [Bei] Beilinson, A.: Higher regulators and values of L-functions J. Soviet Math. 30 (1985) 2036–2070.
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[Blo1] Bloch, S.: Algebraic cycles and higher K-theory Adv. Math. 61 (1986) 267–304 [Blo2] : Algebraic cycles and the Beilinson conjectures, Contemporary Mathematics Vol. 58, Part I (1986) 65–79 [Blo3] , Lectures on Algebraic Cycles Duke University Mathematics Series IV (1980) [C-L1] Chen, X. and J. D. Lewis: Noether-Lefschetz for K1 of a certain class of surfaces, Bol. Soc. Mexicana (3) 10 (2004) [C-L2] : The Hodge-D-conjecture for K3 and Abelian surfaces J. Alg. Geometry 14, (2005) 213–240. [Co] Collino, A.: Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic jacobians J. Algebraic Geometry 6 (1997) 393–415 ¨ller-Stach: Milnor K-theory, higher Chow [E-M] Elbaz-Vincent, P. and S. Mu groups and applications Invent. math. 148 (2002) 177–206 [EV] Esnault, H. and E. Viehweg: Deligne-Beilinson cohomology, in Beilinson’s Conjectures on Special Values of L-Functions (Rapoport, Schappacher, Schneider, eds.) Perspectives in Math. 4 Academic Press, San Diego (1988) 43–91 [GL] Gordon B. and J. Lewis: Collaboration. [GH] Griffiths, P. and J. Harris: Principles of Algebraic Geometry John Wiley & Sons, New York (1978) [Ja] Jannsen, U.: Deligne cohomology, Hodge-D-conjecture, and motives, in Beilinson’s Conjectures on Special Values of L-Functions (Rapoport, Schappacher, Schneider, eds.) Perspectives in Math. 4 Academic Press, San Diego (1988) 305–372. [Ka] Kato, K.: Milnor K-theory and the Chow group of zero cycles, in Applications of K-theory to Algebraic Geometry and Number Theory, Part I Contemp. Math. 55 (1986) 241–253 [Ke1] Kerr, M.: Geometric construction of regulator currents with applications to algebraic cycles, Princeton University Thesis, (2003) [Ke2] : A regulator formula for Milnor K-groups. K-Theory 29 (2003) 175– 210 [Lev] Levine, M.: Localization on singular varieties Invent. Math. 31 (1988) 423– 464 [Lew1] Lewis, J.: Real regulators on Milnor complexes K-Theory 25 (2002) 277– 298 [Lew2] , Regulators of Chow cycles on Calabi-Yau varieties, in Calabi-Yau Varieties and Mirror Symmetry (N. Yui, J. D. Lewis, eds.), Fields Institute Communications 38, (2003), 87–117 ¨ller-Stach, S.: Constructing indecomposable motivic cohomology [MS1] Mu classes on algebraic surfaces J. Algebraic Geometry 6 (1997) 513–543 [MS2] , Algebraic cycle complexes, in Arithmetic and Geometry of Algebraic Cycles, (Gordon, Lewis, M¨ uller-Stach, S. Saito, Yui, eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000, 285–305 [MS3] : A remark on height pairings, in Algebraic Cycles and Hodge Theory, Torino, 1993 (A. Albano, F. Bardelli, eds.), Lecture Notes in Mathematics 1594 Springer-Verlag (1994), 253–259 ´, C.: Lectures on Arakelov Geometry Cambridge Studies in Advanced [Sou] Soule Mathematics 33 Cambridge University Press, Cambridge, England (1992)
17 Chow-K¨unneth decomposition for universal families over Picard modular surfaces A. Miller, Math. Inst. Univ. Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg
[email protected]
S. M¨ uller-Stach, Math. Inst., Johannes Gutenberg Univ. Mainz, Staudingerweg 9, 55099 Mainz
[email protected]
S. Wortmann, Math. Inst. Univ. Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg
[email protected]
Y.-H.Yang, Max Planck Inst. f¨ ur Mathematik, Inselstrasse 22, 04103 Leipzig
[email protected],
[email protected]
K. Zuo, Math. Inst., Johannes Gutenberg Univ. Mainz, †
Staudingerweg 9, 55099 Mainz
[email protected]
Dedicated to Jaap Murre
Abstract We prove existence results for Chow–K¨ unneth projectors on compactified universal families of Abelian threefolds with complex multiplication over a particular Picard modular surface studied by Holzapfel. Our method builds up on the approach of Gordon, Hanamura and Murre in the case of Hilbert modular varieties. In addition we use relatively complete models in the sense of Mumford, Faltings and Chai and prove vanishing results for L2 – Higgs cohomology groups of certain arithmetic subgroups in SU (2, 1) which are not cocompact. † Supported by: DFG Schwerpunkt–Programm, DFG China Exchange program, NSF of China (grant no. 10471105), Max–Planck Gesellschaft. ‡ 1991 AMS Subject Classification 14C25 Keywords: Chow motive, Higgs bundle, Picard modular surface
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17.1 Introduction In this paper we discuss conditions for the existence of absolute ChowK¨ unneth decompositions for families over Picard modular surfaces and prove some partial existence results. In this way we show how the methods of Gordon, Hanamura and Murre [12] can be slightly extended to some cases but fail in some other interesting cases. Let us first introduce the circle of ideas which are behind Chow–K¨ unneth decompositions. For a general reference we would like to encourage the reader to look into [26] which gives a beautiful introduction to the subject and explains all notions we are using. Let Y be a smooth, projective k –variety of dimension d and H ∗ a Weil cohomology theory. In this paper we will mainly be concerned with the case k = C, where we choose singular cohomology with rational coefficients as Weil cohomology. Grothendieck’s Standard Conjecture C asserts that the K¨ unneth components of the diagonal ∆ ⊂ Y × Y in the cohomology H 2d (Y × Y, Q) are algebraic, i.e., cohomology classes of algebraic cycles. In the case k = C this follows from the Hodge conjecture. Since ∆ is an element in the ring of correspondences, it is natural to ask whether these algebraic classes come from algebraic cycles πj which form a complete set of orthogonal idempotents ∆ = π0 + π1 + . . . + π2d ∈ CH d (Y × Y )Q summing up to ∆. Such a decomposition is called a Chow–K¨ unneth decomposition and it is conjectured to exist for every smooth, projective variety. One may view πj as a Chow motive representing the projection onto the j–the cohomology group in a universal way. There is also a corresponding notion for k–varieties which are relatively smooth over a base scheme S. See section 17.3, where also Murre’s refinement of this conjecture with regard to the Bloch–Beilinson filtration is discussed. Chow–K¨ unneth decompositions for abelian varieties were first constructed by Shermenev in 1974. Fourier– Mukai transforms may be effectively used to write down the projectors, see [18, 26]. The cases of surfaces was treated by Murre [27], in particular he gave a general method to construct the projectors π1 and π2d−1 , the so– called Picard and Albanese Motives. Aside from other special classes of 3–folds [1] not much evidence is known except for some classes of modular varieties. A fairly general method was introduced and exploited recently by Gordon, Hanamura and Murre, see [12], building up on previous work by Scholl and their own. It can be applied in the case where one has a modular parameter space X together with a universal family f : A → X of abelian varieties with possibly some additional structure. Examples are given by
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elliptic and Hilbert modular varieties. The goal of this paper was to extend the range of examples to the case of Picard modular surfaces, which are uniformized by a ball, instead of a product of upper half planes. Let us now describe the general strategy of Gordon, Hanamura and Murre so that we can understand to what extent this approach differs and eventually fails for a general Picard modular surface with sufficiently high level structure. Let us assume that we have a family f : A → X of abelian varieties over X. Since all fibers are abelian, we obtain a relative Chow—K¨ unneth decomposition over X in the sense of Deninger/Murre [6], i.e., algebraic cycles Πj in A ×X A which sum up to ∆ ×X ∆. One may view Πj as a projector related to Rj f∗ C. Now let f : A → X be a compactification of the family. We will use the language of perverse sheaves from [3] in particular also the notion of a stratified map. In [11] Gordon, Hanamura and Murre have introduced the Motivic Decomposition Conjecture : Conjecture 17.1.1. Let A and X be quasi–projective varieties over C, A smooth, and f : A → X a projective map. Let X = X0 ⊃ X1 ⊃ . . . ⊃ Xdim(X) be a stratification of X so that f is a stratified map. Then there are local systems Vαj on Xα0 = Xα − Xα−1 , a complete set Πjα of orthogonal projectors and isomorphisms X M ∼ Ψjα : Rf ∗ QA −→ ICXα (Vαj )[−j − dim(Xα )] j,α
j,α
in the derived category. This conjecture asserts of course more than a relative Chow–K¨ unneth decomposition for the smooth part f of the morphism f . Due to the complicated structure of the strata in general its proof in general needs some more information about the geometry of the stratified morphism f . In the course of their proof of the Chow–K¨ unneth decomposition for Hilbert modular varieties, see [12], Gordon, Hanamura and Murre have proved the motivic decomposition conjecture in the case of toroidal compactifications for the corresponding universal families. However to complete their argument they need the vanishing theorem of Matsushima–Shimura [21]. This theorem together with the decomposition theorem [3] implies that each relative projector Πj on the generic stratum X0 only contributes to one cohomology group of A and therefore, using further reasoning on boundary strata Xα , relative projectors for the family f already induce absolute projectors. The plan of this paper is to extend this method to the situation of Picard modular surfaces. These were invented by Picard in his study of the family
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of curves (called Picard curves) with the affine equation y 3 = x(x − 1)(x − s)(x − t). The Jacobians of such curves of genus 3 have some additional CM –structure arising from the Z/3Z deck transformation group. Picard modular surfaces are certain two dimensional ball quotients X = B2 /Γ and form a particular beautiful set of Shimura surfaces in the moduli space of abelian varieties of dimension 3. A nice class of Picard Modular Surfaces are the above mentioned Jacobians of Picard curves. Many examples are known through the work of Holzapfel [15, 16]. Unfortunately the generalization of the vanishing theorem of Matsushima and Shimura does not hold for Picard modular surfaces and their compactifications. The reason is that B2 is a homogenous space for the Lie group G = SU (2, 1) and general vanishing theorems like Ragunathan’s theorem [4, pg. 225] do not hold. If V is an irreducible, non–trivial representation of any arithmetic subgroup Γ of G, then the intersection cohomology group H 1 (X, V) is frequently non–zero, whereas in order to make the method of Gordon, Hanamura and Murre work, we would need its vanishing. This happens frequently for small Γ, i.e., high level. However if Γ is sufficiently big, i.e., the level is small, we can sometimes expect some vanishing theorems to hold. This is the main reason why we concentrate our investigations on one particular example of a Picard modular surface X in section 17.4. The necessary vanishing theorems are proved by using Higgs bundles and their L2 –cohomology in section 17.6. Such techniques provide a new method to compute intersection cohomology in cases where the geometry is known. This methods uses a recent proof of the Simpson correspondence in the non–compact case by Jost, Yang and Zuo [17, Thm. A/B]. But even in the case of our chosen surface X we are not able to show the complete vanishing result which would be necessary to proceed with the argument of Gordon, Hanamura and Murre. We are however able to prove the existence of a partial set π0 , π1 , π2 , π3 , π7 , π8 , π9 , π10 of orthogonal idempotents under the assumption of the motivic decomposition conjecture 17.1.1 on the universal family A over X: Theorem 17.1.2. Assume the motivic decomposition conjecture 17.1.1 for unneth projectors πi f : A → X. Then A supports a partial set of Chow–K¨ for i 6= 4, 5, 6. Unfortunately we cannot prove the existence of the projectors π4 , π5 , π6 due to the non–vanishing of a certain L2 –cohomology group, in our case H 1 (X, S 2 V1 ), where V1 is (half of) the standard representation. This is special to SU (2, 1) and therefore the proposed method has no chance to
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go through for other examples involving ball quotients. If H 1 (X, S 2 V1 ) would vanish or consist out of algebraic Hodge (2, 2)–classes only, then we would obtain a complete Chow–K¨ unneth decomposition. This is an interesting open question and follows from the Hodge conjecture, since all classes in .H 1 (X, S 2 V1 ) have Hodge type (2, 2). We also sketch how to prove the motivic decomposition conjecture in this particular case, see section 17.7.2, however details will be published elsewhere. This idea generalizes the method from [12], since the fibers over boundary points are not anymore toric varieties, but toric bundles over elliptic curves. We plan to publish the full details in a forthcoming publication and prefer to assume the motivic decomposition conjecture 17.1.1 in this paper. The logical structure of this paper is as follows: In section 17.2 we present notations, definitions and known results concerning Picard Modular surfaces and the universal Abelian schemes above them. Section 17.3 first gives a short introduction to Chow Motives and the Murre Conjectures and then proceeds to our case in § 17.3.2. The remainder of the paper will then be devoted to the proof of Theorem 17.1.2: In section 17.4 we give a description of toroidal degenerations of families of Abelian threefolds with complex multiplication. In section 17.5 we describe the geometry of a class of Picard modular surfaces which have been studied by Holzapfel. In section 17.6 we prove vanishing results for intersection cohomology using the non–compact Simpson type correspondence between the L2 –Higgs cohomology of the underlying VHS and the L2 –de Rham cohomology resp. intersection cohomology of local systems. In section 17.7 everything is put together to prove the main theorem 17.1.2. The appendix (section 17.8) gives an explicit description of the L2 –Higgs complexes needed for the vanishing results of section 17.6.
17.2 The Picard modular surface In this section we are going to introduce the (non–compact) Picard modular surfaces X = XΓ and the universal abelian scheme A of fibre dimension 3 over X. For proofs and further references we refer to [9]. Let E be an imaginary quadratic field with ring of integers OE . The Picard modular group is defined as follows. Let V be a 3-dimensional E-vector space and L ⊂ V be an OE -lattice. Let J : V × V → E be a nondegenerate Hermitian form of signature (2, 1) which takes values in OE if it is restricted to L × L. Now let G0 = SU(J, V )/Q be the special unitary group of (V, φ). This is a semisimple algebraic group over Q and for any Q-algebra A its
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group of A-rational points is G0 (A) = {g ∈ SL(V ⊗Q A) | J(gu, gv) = J(u, v), for all u, v ∈ V ⊗Q A}. In particular one has G0 (R) ' SU(2, 1). The symmetric domain H associated to G0 (R) can be identified with the complex 2-ball as follows. Let us fix once and for all an embedding E ,→ C and identify E ⊗Q R with C. This gives V (R) the structure of a 3-dimensional C-vector space and one may choose a basis of V (R) such that the form J is represented by the diagonal matrix [1, 1, −1]. As H can be identified with the (open) subset of the Grassmannian Gr1 (V(R)) of complex lines on which J is negative definite, one has H ' {(Z1 , Z2 , Z3 ) ∈ C3 | |Z1 |2 + |Z2 |2 − |Z3 |2 < 0}/C∗ . This is contained in the subspace, where Z3 6= 0 and, switching to affine coordinates, can be identified with the complex 2-ball B = {(z1 , z2 ) ∈ C2 | |z1 |2 + |z2 |2 < 1}. Using this description one sees that G0 (R) acts transitively on B. The Picard modular group of E is defined to be G0 (Z) = SU(J, L), i.e. the elements g ∈ G0 (Q) with gL = L. It is an arithmetic subgroup of G(R) and acts properly discontinuously on B. The same holds for any commensurable subgroup Γ ⊂ G0 (Q), in particular if Γ ⊂ G0 (Z) is of finite index the quotient XΓ (C) = B/Γ is a non-compact complex surface, the Picard modular surface. Moreover, for torsionfree Γ it is smooth. We want to describe XΓ (C) as moduli space for polarized abelian 3-folds with additional structure. For this we will give a description of XΓ (C) as the identity component of the Shimura variety SK (G, H). Let G = GU(J, V )/Q be the reductive algebraic group of unitary similitudes of J, i.e. for any Q-algebra A G0 (A) = {g ∈ GL(V ⊗Q A) | there exists µ(g) ∈ A∗ such that J(gu, gv) = µ(g)J(u, v), for all u, v ∈ V ⊗Q A}. As usual A denotes the Q-adeles and Af denotes the finite adeles. Let K be a compact open subgroup of G(Af ), which is compatible with the integral structure defined by the lattice L. I.e., K should be in addition a subgroup ˆ := {g ∈ G(Af ) | g(L ⊗Z Z) ˆ = L ⊗Z Z}. ˆ Then one can of finite index in G(Z) define SK (G, H)(C) := G(Q)\H × G(Af )/K. `n(K) This can be decomposed as SK (G, H)(C) = j=1 XΓj (C).
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The variety SK (G, H)(C) has an interpretation as a moduli space for certain 3-dimensional abelian varieties. Recall that over C an abelian variety A is determined by the following datum: a real vector space W (R), a lattice W (Z) ⊂ W (R), and a complex strucuture j : C× → AutR (W (R))), for which there exists a nondegenerate R−bilinear skew-symmetric form ψ : W (R) × W (R) → R taking values in Z on W (Z) such that the form given by (w, w0 ) 7→ ψ(j(i)w, w0 ) is symmetric and positive definite. The form ψ is called a Riemann form and two forms ψ1 , ψ2 are called equivalent if there exist n1 , n2 ∈ N>0 such that n1 ψ1 = n2 ψ2 . An equivalence class of Riemann forms is called a homogeneous polarization of A. An endomorphism of a complex abelian variety is an element of EndR (W (R)) preserving W (Z) and commuting with j(z) for all z ∈ C× . A homogenously polarized abelian variety (W (R), W (Z), j, ψ) is said to have complex multiplication by an order O of E if and only if there is a homomorphism m : O → End(A) such that m(1) = 1, and which is compatible with ψ, i.e. ψ(m(αρ )w, w0 ) = ψ(w, m(α)w0 ) where ρ is the Galois automorphism of E induced by complex conjugation (via our fixed embedding E ,→ C.) We shall only consider the case O = OE in the following. One can define the signature of the complex multiplication m, resp. the abelian variety (W (R), W (Z), j, ψ, m) as the signature of the hermitian form (w, w0 ) 7→ ψ(w, iw0 ) + iψ(w, w0 ) on W (R) with respect to the complex structure imposed by m via O ⊗Z R ' C. We write m(s,t) if m has signature (s, t). ˆ as before one has the Finally for any compact open subgroup K ⊂ G(Z) notion of a level-K structure on A. For a positive integer n we denote by An (C) the group of points of order n in A(C). This group can be identified with W (Z) ⊗ Z/nZ and taking the projective limit over the system (An (C))n∈N>0 defines the Tate module of A : ˆ T (A) := lim An (C) ' W (Z) ⊗ Z. ←− ˆ are called K-equivalent Now two isomorphisms ϕ1 , ϕ2 : W (Z)⊗ Zˆ ' L⊗ Z if there is a k ∈ K such hat ϕ1 = kϕ2 and a K-level structure on A is just a K-equivalence class of these isomorphisms. ˆ there is a Proposition 17.2.1. For any compact open subgroup K ⊂ G(Z) one-to-one correspondence between (i) the set of points of SK (G, H)(C) and (ii) the set of isomorphism classes of (W (R), W (Z), j, ψ, m(2,1),ϕ ) as above. Proof [9, Prop.3.2]
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Remark 17.2.2. If we take ˆ ˆ ⊂ N · (L ⊗Z Z)}, KN := {g ∈ G(Af ) | (g − 1)(L ⊗Z Z) then a level–K structure is just the usual level-N structure, namely an isomorphism AN (C) → L ⊗ Z/N Z. Moreover KN ⊂ G(Q) = ΓN , where ΓN is the principal congruence subgroup of level N, i.e. the kernel of the canonical map G0 (Z) → G0 (Z/N Z). In this case the connected component of the identity of SKN (G, H) is exactly XΓN (C). We denote with AΓ the universal abelian scheme over XΓ (C). In section 17.4 the compactifications of these varieties will be explained in detail. For the time being we denote them with X Γ and AΓ . As the group Γ will be fixed throughout the paper we will drop the index Γ if no confusion is possible.
17.3 Chow motives and the conjectures of Murre Let us briefly recall some definitions and results from the theory of Chow motives. We refer to [26] for details.
17.3.1 For a smooth projective variety Y over a field k let CHj (Y ) denote the Chow group of algebraic cycles of codimension j on Y modulo rational equivalence, and let CHj (Y )Q := CHj (Y )⊗Q. For a cycle Z on Y we write [Z] for its class in CHj (Y ). We will be working with relative Chow motives as well, so let us fix a smooth connected, quasi-projective base scheme S → Spec k. If S = Spec k we will usually omit S in the notation. Let Y, Y 0 be smooth projective varieties over S, i.e., all fibers are smooth. For ease of notation (and as we will not consider more general cases) we may assume that Y is irreducible and of relative dimension g over S. The group of relative correspondences from Y to Y 0 of degree r is defined as Corrr (Y ×S Y 0 ) := CHr+g (Y ×S Y 0 )Q . Every S-morphism Y 0 → Y defines an element in Corr0 (Y ×S Y 0 ) via the class of the transpose of its graph. In particular one has the class [∆Y /S ] ∈ Corr0 (Y ×S Y ) of the relative diagonal. The self correspondences of degree 0 form a ring, see [26, pg. 127]. Using the relative correspondences one proceeds as usual to define the category MS of (pure) Chow motives over S. The objects of this pseudoabelian Q-linear tensor category are triples
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(Y, p, n) where Y is as above, p is a projector, i.e. an idempotent element in Corr0 (Y ×S Y ), and n ∈ Z. The morphisms are 0
HomMS ((Y, p, n), (Y 0 , p0 , n0 )) := p0 ◦ Corrn −n (Y ×S Y 0 ) ◦ p. When n = 0 we write (Y, p) instead of (Y, p, 0), and h(Y ) := (Y, [∆Y ]). Definition 17.3.1. For a smooth projective variety Y /k of dimension d a Chow-K¨ unneth-decomposition of Y consists of a collection of pairwise orthogonal projectors π0 , . . . , π2d in Corr0 (Y × Y ) satisfying (i) π0 + . . . + π2d = [∆Y ] and (ii) for some Weil cohomology theory H ∗ one has πi (H ∗ (Y )) = H i (Y ). If one has a Chow-K¨ unneth decomposition for Y one writes hi (Y ) = (Y, πi ). A similar notion of a relative Chow-K¨ unneth-decomposition over S can be defined in a straightforward manner, see also introduction. Towards the existence of such decomposition one has the following conjecture of Murre: Conjecture 17.3.2. Let Y be a smooth projective variety of dimension d over some field k. (i) There exists a Chow-K¨ unneth decomposition for Y . (ii) For all i < j and i > 2j the action of πi on CH j (Y )Q is trivial, i.e. πi · CH j (Y )Q = 0. (iii) The induced j step filtration on F ν CH j (Y )Q := Kerπ2j ∩ · · · ∩ Kerπ2j−ν+1 is independent of the choice of the Chow–K¨ unneth projectors, which are in general not canonical. (iv) The first step of this filtration should give exactly the subgroup of homological trivial cycles CH j (Y )Q in CH j (Y )Q . There are not many examples for which these conjectures have been proved, but they are known to be true for surfaces [26], in particular we know that we have a Chow-K¨ unneth decomposition for X. In the following theorem we are assuming the motivic decomposition conjecture which was explained in the introduction. The main result we are going to prove in section 17.7 is: Theorem 17.3.3. Under the assumption of the motivic decomposition conjecture 17.1.1 A has a partial Chow–K¨ unneth decomposition, including the projectors πi for i 6= 4, 5, 6 as in Part (1) of Murre’s conjecture.
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Over the open smooth part X ⊂ X one has the relative projectors constructed by Deninger and Murre in [6], see also [18]: Let S be a fixed base scheme as in section 17.3. We will now state some results on relative Chow motives in the case that A is an abelian scheme of fibre dimension g over S. Firstly we have a functorial decomposition of the relative diagonal ∆A/S . Theorem 17.3.4. There is a unique decomposition ∆A/S =
2g X
Πi
in
CHg (A ×S A)Q
s=0
such that (idA × [n])∗ Πi = ni Πi for all n ∈ Z. Moreover the Πi are mutually orthogonal idempotents, and [t Γ[n] ] ◦ Πi = ni Πi = Πi ◦ [t Γ[n] ], where [n] denotes the multiplication by n on A. Proof [6, Thm. 3.1] Putting hi (A/S) = (A/S, Πi ) one has a Poincar´e-duality for these motives. Theorem 17.3.5. (Poincar´e-duality) h2g−i (A/S)∨ ' hi (A/S)(g) Proof [18, 3.1.2]
17.3.2 We now turn back to our specific situation. From Theorem 17.3.4 we have the decomposition ∆A/X = Π0 + . . . + Π6 . We will have to extend these relative projectors to absolute projectors. In order to show the readers which of the methods of [11], where Hilbert modular varieties are considered, go through and which of them fail in our case, we recall the main theorem (Theorem 1.3) from [11]: Theorem 17.3.6. Let p : A → X as above satisfy the following conditions: 1) The irreducible components of X − X are smooth toric projective varieties. 2) The irreducible components of A − A are smooth projective toric varieties. 3) The variety A/X has a relative Chow-K¨ unneth decomposition. 4) X has a Chow-K¨ unneth decomposition over k.
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5) If x is a point of X the natural map. top
2r (Ax (C), Q)π1 CH r (A) → HB
(X,x)
is surjective for 0 ≤ r ≤ d = dim A − dim X. i (A (C), Q)π1top (X,x) = 0. 6) For i odd, HB x 7) Let ρ be an irreducible, non-constant representation of π1top (X, x) and V the corresponding local system on X. Assume that V is contained in the i–th exterior power Ri p∗ Q = Λi R1 p∗ Q of the monodromy representation for some 0 ≤ i ≤ 2d. Then the intersection cohomology H q (X, V) vanishes if q 6= dim X. Under these assumptions A has a Chow-K¨ unneth decomposition over k. As it stands we can only use conditions (3),(4) and (5) of this theorem, all the other conditions fail in our case. As for conditions (1) and (2) we will have to weaken them to torus fibrations over an elliptic curve. This will be done in section 4. Condition (3) holds because of the work of Deninger and Murre ([6]) on Chow-K¨ unneth decompositions of Abelian schemes. Condition (4) holds in our case because of the existence of Chow-K¨ unneth projectors for surfaces (see [26]). In order to prove condition (5) and to replace conditions (6) and (7) we will from section 5 on use a non-compact Simpson type correspondence between the L2 -Higgs cohomology of the underlying variation of Hodge structures and the L2 -de Rham cohomology (respectively intersection cohomology) of local systems. This will show the vanishing of some of the cohomology groups mentioned in (6) of Theorem 17.3.6 and enable us to weaken condition (7).
17.4 The universal abelian scheme and its compactification In this section we show that the two conditions (1) and (2) of Theorem 17.3.6 fail in our case. Instead of tori we get toric fibrations over an elliptic curve as fibers over boundary components. The main reference for this section is [23].
17.4.1 Toroidal compactifications of locally symmetric varieties In this paragraph an introduction to the theory of toroidal compactifications of locally symmetric varieties as developed by Ash, Mumford, Rapoport and Tai in [2] is given. The main goal is to fix notation. All details can be found in [2], see the page references in this paragraph.
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Let D = G(R)/K be a bounded symmetric domain (or a finite number of bounded symmetric domains, for the following discussion we will assume D to be just one bounded symmetric domain), where G(R) denotes the Rvalued points of a semisimple group G and K ⊂ G(R) is a maximal compact ˇ be its compact dual. Then there is an embedding subgroup. Let D ˇ D ,→ D.
(17.1)
ˇ Note that G0 (C) acts on D. We pick a parabolic P corresponding to a rational boundary component F , by Z 0 we denote the connected component of the centralizer Z(F ) of F , by P 0 the connected component of P and by Γ a (torsion free, see below for this restriction) congruence subgroup of G. We will explicitly be interested only in connected groups, so from now on we can assume that G0 = G. Set N ⊂ P0
the unipotent radical
U ⊂N
the center of the unipotent radical
UC
its complexification
V = N/U Γ0 = Γ ∩ U Γ1 = Γ ∩ P 0 T = Γ0 \UC . Note that U is a real vector space and by construction, T is an algebraic torus over C. Set D(F ) := UC · D, ˇ This is an open set in D ˇ where the dot · denotes the action of G0 (C) on D. and we have the inclusions ˇ D ⊂ D(F ) = UC · D ⊂ D
(17.2)
and furthermore a complex analytic isomorphism UC · D ' UC × EP
(17.3)
where EP is some complex vector bundle over the boundary component corresponding to P . We will not describe EP any further, the interested reader is referred to [2], chapter 3. The isomorphism in (17.3) is complex ˇ from the left to the translation on analytic and takes the UC -action on D UC on the right.
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We once for all choose a boundary component F and denote its stabilizer by P . ¿From (17.2) we get (see [2], chapter 3 for details, e.g. on the last isomorphism) Γ0 \D ⊂ Γ0 \(UC · D) ' Γ0 \(UC × EP ) ' T × E, where E = Γ0 \EP . The torus T is the one we use for a toroidal embedding. Furthermore D can be realized as a Siegel domain of the third kind: D ' {(z, e) ∈ UC · D ' UC × E | Im(z) ∈ C + h(e)}, where h:E→U is a real analytic map and C ⊂ U is an open cone in U . A finer description of C which is needed for the most general case can be found in [2]. We pick a cone decomposition {σα } of C such that (Γ1 /Γ0 ) · {σα } = {σα } with finitely many orbits and [ C⊂ σα ⊂ C.
(17.4)
α
This yields a torus embedding T ⊂ X{σα } .
(17.5)
We can thus partially compactify the open set Γ0 \(UC · D): Γ0 \(UC · D) ' Γ0 \UC × E ,→ X{σα } × E. The situation is now the following: Γ0 \(UC · D) ' T × E ,→ X{σα } × E ∪
(17.6)
Γ0 \D. We proceed to give a description of the vector bundle EP in order to describe the toroidal compactification geometrically.
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Again from [2] (pp.233) we know that D∼ as real manifolds and =F ×C ×N D(F ) ∼ = F × V × UC , where V = N/U is the abelian part of N . Now set D(F )0 := D(F ) mod UC . This yields the following fibration: D(F ) π1 fibres UC
D(F )0
π
π2 fibres V
F.
Taking the quotient by Γ0 yields a quotient bundle Γ0 \D(F ) π1 fibres T := Γ0 \UC
D(F )0 So, T is an algebraic torus group with maximal compact subtorus Tcp := Γ0 \U. Take the closure of Γ0 \D in X{σα } × E and denote by (Γ0 \D){σα } its interior. Factor D → Γ\D by D → Γ0 \D → Γ1 \D → Γ\D. It is the following situation we aim at obtaining: (Γ0 \D){σα } ←Γ0 \D → Γ1 \D → Γ\D ∪ ∪ ∪ (Γ0 \D(c)){σα } ←- Γ0 \D(c) → Γ1 \D(c) ,→ Γ\D.
(17.7)
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Here D(c) is a neighborhood of our boundary component. More precisely for any compact subset K of the boundary component and any c ∈ C define Im z ∈ C + h(e) + c D(c, K) = Γ1 · (z, e) ∈ UC × E . and e lies above K Then by reduction theory for c large enough, Γ-equivalence on D(c, K) reduces to Γ1 -equivalence. This means that we have an inclusion Γ1 \D(c)
/
Γ\D
'
(Γ1 /Γ0 )\(Γ0 \D(c)) where the quotient by Γ1 /Γ0 is defined in the obvious way. Furthermore Γ0 \D ,→ (Γ0 \D){σα } directly induces Γ0 \D(c) ,→ (Γ0 \D(c)){σα } . Having chosen {σα } such that (Γ1 /Γ0 ) · {σα } = {σα }, we get Γ1 \D(c) ,→ (Γ1 /Γ0 )\(Γ0 \D(c)){σα } which yields the partial compactification and establishes the diagram ((17.7)). The following theorem is derived from the above. Theorem 17.4.1. With the above notation and for a cone decomposition {σα } of C satisfying the condition (17.4), the diagram Γ1 \D(c) v S
SSS SSS SSS SSS S)
/
Γ\D
(17.8)
(Γ1 /Γ0 )\(Γ0 \D(c)){σα }
yields a (smooth if {σα } is chosen appropriately) partial compactification of Γ\D at F .
17.4.2 Toroidal compactification of Picard modular surfaces We will now apply the results of the last paragraph to the case of Picard modular surfaces and give a finer description of the fibres at the boundary.
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Theorem 17.4.2. For each boundary component of a Picard modular surface the following holds. With the standard notations from [2] (see also the last paragraph and [23] for the specific choices of Γ0 , Γ1 etc.) (Γ1 /Γ0 )\(Γ0 \D) is isomorphic to a punctured disc bundle over a CM elliptic curve A. A toroidal compactification (Γ1 /Γ0 )\(Γ0 \D){σα } is obtained by closing the disc with a copy of A (e.g. adding the zero section of the corresponding line bundle). We now turn to the modification of condition (2). The notation we use is as introduced in chapter 3 of [8]. Let P˜ be a relatively complete model of an ample degeneration datum associated to our moduli problem. As a general reference for degenerations see [25], see [8] for the notion of relatively complete model and [23] for the ample degeneration datum we need here. In [23] the following theorem is proved. Theorem 17.4.3.(i) The generic fibre of P˜ is given by a fibre-bundle over a CM elliptic curve E, whose fibres are countably many irreducible components of the form P, where P is a P1 -bundle over P1 . (ii) The special fibre of P˜ is given by a fibre-bundle over the CM elliptic curve E, whose fibres consist of countably many irreducible components of the form P1 × P1 . Remark 17.4.4. In this paper we work with some very specific Picard modular surfaces and thus the generality of Theorem 17.4.3 is not needed. It will be needed though to extend our results to larger families of Picard modular surfaces, see section ??.
17.5 Higgs bundles on Picard modular surfaces In this section we describe in detail the Picard modular surface of Holzapfel which is our main object. We follow Holzapfel [15, 16] very closely. In the remaining part of this section we explain the formalism of Higgs bundles which we will need later.
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17.5.1 Holzapfel’s surface We restrict our attention to the Picard modular surfaces with compactification X and boundary divisor D ⊆ X which were discussed by Picard [28], Hirzebruch [14] and Holzapfel [15, 16]. These surfaces are compactifications of ball quotients X = B/Γ where Γ is a subgroup of SU (2, 1; O) with O = Z ⊕ Zω, ω = exp(2πi/3), i.e., O is the ring of Eisenstein numbers. In the case Γ = SU (2, 1; O), studied already by Picard, the quotient B/Γ is P2 − 4 points, an open set of which is U = P2 − ∆ and ∆ is a configuration of 6 lines (not a normal crossing divisor). U is a natural parameter space for a family of Picard curves y 3 = x(x − 1)(x − s)(x − t) of genus 3 branched over 5 (ordered) points 0, 1, s, t, ∞ in P1 . The parameters s, t are coordinates in the affine set U . If one looks at the subgroup Γ0 = Γ ∩ SL(3, C), then X = B/Γ0 has a natural compactification X with a smooth boundary divisor D consisting of 4 disjoint elliptic curves E0 + E1 + E2 + E3 , see [15, 16]. This surface X is birational to a covering of P2 − ∆ and hence carries a family of curves over it. If we pass to yet another subgroup Γ00 ⊂ Γ of finite index, then we obtain a Picard modular surface ^ ×E X = E^ with boundary D a union of 6 elliptic curves which are the strict transforms of the following 6 curves T1 , Tω , Tω2 , E × {Q0 }, E × {Q1 }, E × {Q2 } on E × E in the notation of [15, page 257]. This is the surface we will study in this paper. The properties of the modular group Γ00 are described in [15, remark V.5]. In particular it acts freely on the ball. X is the blowup of E × E in the three points (Q0 , Q0 ) (the origin), (Q1 , Q1 ) and (Q2 , Q2 ) of triple intersection. Note that E has the equation y 2 z = x3 − z 3 . On E we have an action of ω via (x : y : z) 7→ (ωx : y : z). E maps to P1 using the projection p : E → P1 ,
(x : y : z) 7→ (y : z).
This action has 3 fixpoints Q0 = (0 : 1 : 0) (the origin), Q1 = (0 : i : 1) and Q2 = (0 : −i : 1) which are triple ramification points of p. Therefore one has 3Q0 = 3Q1 = 3Q2 in CH 1 (E).
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In order to proceed, we need to know something about the Picard group of X. Lemma 17.5.1. In NS(E × E) one has the relation T1 + Tω + Tω2 = 3(0 × E) + 3(E × 0). Proof Since E has complex multiplication by Z[ω], the N´eron–Severi group has rank 4 and divisors T1 , Tω , 0 × E and E × 0 form a basis of NS(E × E). Using the intersection matrix of this basis, the claim follows. The following statement is needed later: Lemma 17.5.2. The log–canonical divisor is divisible by three: KX + D = 3L for some line bundle L. Proof If we denote by σ : X → E × E the blowup in the three points (Q0 , Q0 ), (Q1 , Q1 ) and (Q2 , Q2 ), then we denote by Z = Z1 + Z2 + Z3 the union of all exceptional divisors. We get: σ ∗ T = D1 + Z, σ ∗ Tω = D2 + Z, σ ∗ Tω2 = D3 + Z, and σ ∗ E × Q0 = D4 + Z1 , σ ∗ E × Q1 = D5 + Z2 , σ ∗ E × Q2 = D6 + Z3 . Now look at the line bundle KX + D. Since KX = σ ∗ KE×E + Z + D = Z + D, we compute KX + D =
6 X i−1
Di +
3 X
Zj .
j=1
The first sum, D1 + D2 + D3 = −3Z + σ ∗ (T1 + Tω + Tω2 ) = −3Z + 3σ ∗ (0 × E + E × 0). is divisible by 3. Using 3Q0 = 3Q1 = 3Q2 , the rest can be computed in NS(X) as D4 + D5 + D6 + Z = σ ∗ (E × 0 + E × Q1 + E × Q2 ) = 3σ ∗ (E × 0). Therefore the class of KX + D in NS(X) is given by KX + D = −3Z + 3σ ∗ (0 × E) + 6σ ∗ (E × 0)
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and divisible by 3. Since Pic0 (X) is a divisible group, KX + D is divisible by 3 in Pic(X) and we get a line bundle L with KX + D = 3L whose class in NS(X) is given by L = σ ∗ (0 × E) − Z + 2σ ∗ (E × 0). If we write σ ∗ (0 × E) = D0 + Z1 , we obtain the equation L = D0 + D5 + D6 in NS(X). Note that D0 intersects both D5 and D6 in one point. All Di , i = 1, . . . , 6 have selfintersection −1 and are disjoint. It is not difficult to see that L is a nef and big line bundle since X has logarithmic Kodaira dimension 2 [15]. L is trivial on all components of D by the adjunction formula, since they are smooth elliptic curves. The rest of this section is about the rank 6 local system V = R1 p∗ Z on X. The following Lemma was known to Picard [28], he wrote down 3 × 3 monodromy matrices with values in the Eisenstein numbers: Lemma 17.5.3. V is a direct sum of two local systems V = V1 ⊕ V2 of rank 3. The decomposition is defined over the Eisenstein numbers. Proof The cohomology H 1 (C) of any Picard curve C has a natural Z/3Z Galois action. Since the projective line has H 1 (P1 , Z) = 0, the local system V ⊗ C decomposes into two 3–dimensional local systems V = V1 ⊕ V 2 which are conjugate to each other and defined over the Eisenstein numbers.
Both local systems V1 , V2 are irreducible and non–constant.
17.5.2 Higgs bundles on Holzapfel’s surface Now we will study the categorical correspondence between local systems and Higgs bundles. It turns out that it is sometimes easier to deal with one resp. the other.
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Definition 17.5.4. A Higgs bundle on a smooth variety Y is a holomorphic vector bundle E together with a holomorphic map ϑ : E → E ⊗ Ω1Y which satisfies ϑ ∧ ϑ = 0, i.e., an End(E) valued holomorphic 1–form on Y . Each Higgs bundle induces a complex of vector bundles: E → E ⊗ Ω1Y → E ⊗ Ω2Y → . . . → E ⊗ ΩdY . Higgs cohomology is the cohomology of this complex. The Simpson correspondence on a projective variety Y gives an equivalence of categories between polystable Higgs bundles with vanishing Chern classes and semisimple local systems V on Y [29, Sect. 8]. This correspondence is very difficult to describe in general and uses a deep existence theorem for harmonic metrics. For quasi–projective Y this may be generalized provided that the appropriate harmonic metrics exist, which is still not known until today. There is however the case of VHS (Variations of Hodge structures) where the harmonic metric is the Hodge metric and is canonically given. For example if we have a smooth, projective family f : A → X as in our example and V = Rm f∗ C is a direct image sheaf, then the corresponding Higgs bundle is M E= E p,q p+q=m
where E p,q is the p–the graded piece of the Hodge filtration F • on H = V ⊗ OX . The Higgs operator ϑ is then given by the graded part of the Gauss–Manin connection, i.e., the cup product with the Kodaira–Spencer class. In the non–compact case there is also a corresponding log–version for Higgs bundles, where Ω1Y is replaced by Ω1Y (log D) for some normal crossing divisor D ⊂ Y and E by the Deligne extension. Therefore we have to assume that the monodromies around the divisors at infinity are unipotent and not only quasi–unipotent as in [17, Sect. 2]. This is the case in Holzapfel’s example, in fact above we have already checked that the log–canonical divisor KX + D is divisible by three. We refer to [29] and [17] for the general theory. In our case let E = E 1,0 ⊕ E 0,1 be the Higgs bundle corresponding to V1 with Higgs field 1 ϑ : E → E ⊗ ΩX (log D).
This bundle is called the uniformizing bundle in [29, Sect. 9]. Let us return to Holzapfel’s example. We may assume that E 1,0 is 2– dimensional and E 0,1 is 1–dimensional, otherwise we permute V1 and V2 .
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1 (log D) is an isomorphism. Lemma 17.5.5. ϑ : E 1,0 → E 0,1 ⊗ ΩX
Proof For the generic fiber this is true for rank reasons. At the boundary D this is a local computation using the definition of the Deligne extension. This has been shown in greater generality in [17, Sect. 2-4] (cf. also [20, Sect. 4]), therefore we do not give any more details. Let us summarize what we have shown for Holzapfel’s surface X: Corollary 17.5.6. KX (D) is nef and big and there is a nef and big line bundle L with L⊗3 ∼ = KX (D). The uniformizing bundle E has components E 1,0 = Ω1X (log D) ⊗ L−1 ,
E 0,1 = L−1 .
The Higgs operator ϑ is the identity as a map E 1,0 → E 0,1 ⊗ Ω1X (log D) and it is trivial on E 0,1 .
17.6 Vanishing of intersection cohomology Let X be Holzapfel’s surface from the previous section. We now want to discuss the vanishing of intersection cohomology H 1 (X, W) for irreducible, non–constant local systems W ⊆ Ri p∗ Q. Let V1 be as in the previous section. Denote by (E, ϑ) the corresponding Higgs bundle with 1 E = ΩX (log D) ⊗ L−1 ⊕ L−1 and Higgs field ϑ : E → E ⊗ Ω1X (log D). Our goal is to compute the intersection cohomology of V1 . We use the isomorphism between L2 – and intersection cohomology for C–VHS, a theorem of Cattani, Kaplan and Schmid together with the isomorphism between L2 –cohomology and L2 –Higgs cohomology from [17, Thm. A/B]. Therefore for computations of intersection cohomology we may use L2 –Higgs cohomology. We refer to [17] for a general introduction to all cohomology theories involved. Theorem 17.6.1. The intersection cohomology H q (X, V1 ) vanishes for q 6= 2.
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Proof We need only show this for q = 1, since V1 has no invariant sections, hence H 0 (X, V1 ) = 0 and the other vanishings follow via duality H q (X, V1 ) ∼ = H 2 dim(X)−q (X, V2 ) from the analogous statement for V2 . The following theorem provides the necessary technical tool. Theorem 17.6.2 ([17, Thm. B]). The intersection cohomology H q (X, V1 ) can be computed as the q–th hypercohomology of the complex 0 → Ω0 (E)(2) → Ω1 (E)(2) → Ω2 (E)(2) → 0 on X, where E is as above. This is a subcomplex of ϑ
ϑ
E →E ⊗ Ω1X (log D)→E ⊗ Ω2X (log D). In the case where D is smooth, this is a proper subcomplex with the property Ω1 (E)(2) ⊆ Ω1X ⊗ E. Proof This is a special case of the results in [17]. The subcomplex is explicitly described in section 17.8 of our paper. Lemma 17.6.3. Let E be as above with L nef and big. Then the vanishing 1 (log D) ⊗ Ω1X ⊗ L−1 ) = 0 H 0 (ΩX
implies the statement of theorem 17.6.1. Proof We first compute the cohomology groups for the complex of vector bundles and discuss the L2 –conditions later. Any logarithmic Higgs bundle E = ⊕E p,q coming from a VHS has differential ϑ : E p,q → E p−1,q+1 ⊗ Ω1X (log D). In our case E = E 1,0 ⊕ E 0,1 and the restriction of ϑ to E 0,1 is zero. The differential ϑ : E 1,0 → E 0,1 ⊗ Ω1X (log D) is the identity. Therefore the complex ϑ
ϑ
(E • , ϑ) : E →E ⊗ Ω1X (log D)→E ⊗ Ω2X (log D)
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looks like:
Ω1X (log D) ⊗ L−1 ⊗ Ω2X (log D)
⊕
Ω1X (log D)⊗2 ⊗ L−1 ↓ L−1 ⊗ Ω2X (log D) .
1 ΩX (log D) ⊗ L−1 ↓∼ = −1 ⊕ L ⊗ Ω1X (log D)
⊕ L−1 ↓ 0
Therefore it is quasi–isomorphic to a complex 0
0
1 1 L−1 −→S 2 ΩX (log D) ⊗ L−1 −→ΩX (log D) ⊗ Ω2X (log D) ⊗ L−1
with trivial differentials. As L is nef and big, we have H 0 (L−1 ) = H 1 (L−1 ) = 0. Hence we get 1 H1 (X, (E • , ϑ)) ∼ (log D) ⊗ L−1 ) = H 0 (X, S 2 ΩX
and H2 (X, (E • , ϑ)) is equal to 2 H 0 (X, KX ⊗L)∨ ⊕H 0 (X, Ω1X (log D)⊗ΩX (log D)⊗L−1 )⊕H 1 (X, S 2 Ω1X (log D)⊗L−1 ).
If we now impose the L2 –conditions and use the complex Ω∗(2) (E) instead of (E • , ϑ), the resulting cohomology groups are subquotients of the groups described above. Since Ω1 (E)(2) ⊆ Ω1X ⊗ E we conclude that the vanishing 1 H 0 (X, Ω1X (log D) ⊗ ΩX ⊗ L−1 ) = 0
is sufficient to deduce the vanishing of intersection cohomology. Now we verify the vanishing statement. Lemma 17.6.4. In the example above we have H 0 (Ω1X (log D) ⊗ Ω1X ⊗ L−1 ) = 0. Proof Let σ : X → E × E be the blow up of the 3 points of intersection. Then one has an exact sequence 1 0 → σ ∗ Ω1E×E → ΩX → i∗ Ω1Z → 0,
where Z is the union of all (disjoint) exceptional divisors. Now Ω1E×E is the
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1 (log D) ⊗ Ω1 ⊗ L−1 has as a subsheaf trivial sheaf of rank 2. Therefore ΩX X 1 −1 2 copies of ΩX (log D) ⊗ L . The group
H 0 (X, Ω1X (log D) ⊗ L−1 ) is zero by the Bogomolov–Sommese vanishing theorem (see [7, Cor. 6.9]), since L is nef and big. In order to prove the assertion it is hence sufficient to show that H 0 (Z, Ω1X (log D) ⊗ Ω1Z ⊗ L−1 ) = 0. But Z is a disjoint union of P1 ’s. In our example we have KX (D) ⊗ OZ ∼ = OZ (3) since (L.Z) = 1 and therefore Ω1Z ⊗ L−1 ∼ = OZ (−3). Now we use in addition the conormal sequence 0 → NZ∗ → Ω1X (log D)|Z → Ω1Z (log(D ∩ Z)) → 0. Note that NZ∗ = OZ (1). Twisting this with Ω1Z ⊗ L−1 ∼ = OZ (−3) gives an exact sequence 1 (log D) ⊗ Ω1Z ⊗ L−1 → OZ (−1) → 0. 0 → OZ (−2) → ΩX
On global sections this proves the assertion. So far we have only shown the vanishing of H q (X, V1 ) and hence of H q (X, V) for q 6= 2. In order to apply the method of Gordon, Hanamura and Murre, we also have to deal with the case Λi V. Theorem 17.6.5. Let ρ be an irreducible, non–constant representation of π1 (X), which is a direct factor in Λk (V1 ⊕ V2 ) for k ≤ 2. Then the intersection cohomology group H 1 (X, Vρ ) is zero. Proof Let us first compute all such representations: if k = 1 we have only V1 and its dual. If k = 2, we have the decomposition Λ2 (V1 ⊕ V2 ) = Λ2 V1 ⊕ Λ2 V2 ⊕ End(V1 ). Since V1 is 3–dimensional, Λ2 V1 ∼ = V2 and therefore the only irreducible, non–constant representation that is new here is End0 (V1 ), the trace–free endomorphisms of V1 . Since we have already shown the vanishing H 1 (X, V1,2 ), it remains to treat H 1 (X, End0 (V1 )). The vanishing of H 1 (X, End0 (V1 )) is a general and well–known statement: The representation End0 (V1 ) has regular highest weight and therefore contributes only to the middle dimension H 2 . A reference for this is [19, Main Thm.], cf. [4, ch. VII] and [30].
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The vanishing of H 1 (X, End0 (V1 )) has the following amazing consequence, which does not seem easy to prove directly using purely algebraic methods. In the compact case this has been shown by Miyaoka, cf. [24]. Lemma 17.6.6. In our situation we have 1 HL0 2 (X, S 3 ΩX (log D)(−D) ⊗ L−3 ) = 0.
Proof Write down the Higgs complex for End0 (E). In degree one, a term which contains 1 (log D)(−D) ⊗ L−3 S 3 ΩX
occurs. Since H 1 vanishes, this cohomology group must vanish too. Finally we want to discuss the case k = 3. Unfortunately here the vanishing techniques do not work in general. But we are able to at least give a bound for the dimension of the remaining cohomology group. Namely for k = 3, one has Λ3 (V1 ⊕ V2 ) = Λ3 V1 ⊕ Λ3 V2 ⊕ (Λ2 V1 ⊗ V2 ) ⊕ (Λ2 V2 ⊗ V1 ). Here the only new irreducible and non–constant representation is S 2 V1 ⊆ V1 ⊗ V1 and its dual. We would like to compute H 1 (X, S 2 V1 ) using a variant of the symmetric product of the L2 –complexes Ω∗ (S)(2) as described in the appendix. The Higgs complex without L2 –conditions looks as follows:
1 S 2 ΩX (log D) ⊗ L−2 ⊕ Ω1X (log D) ⊗ L−2 ⊕ L−2 ↓ 1 S 2 Ω1X (log D) ⊗ L−2 ⊗ Ω1X (log D) ⊕ Ω1X (log D) ⊗ L−2 ⊗ Ω1X (log D) ⊕ L−2 ⊗ ΩX (log D) ↓ 2 (log D) S 2 Ω1X (log D) ⊗ L−2 ⊗ Ω2X (log D) ⊕ Ω1X (log D) ⊗ L−2 ⊗ Ω2X (log D) ⊕ L−2 ⊗ ΩX
Again many pieces of differentials in this complex are isomorphisms or zero. For example the differential 1 S 2 Ω1X (log D) ⊗ L−2 ⊗ Ω1X (log D) → ΩX (log D) ⊗ L−2 ⊗ Ω2X (log D)
is a projection map onto a direct summand, since for every vector space W we have the identity S 2 W ⊗ W = S 3 W ⊕ W ⊗ Λ2 W .
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Therefore the Higgs complex for S 2 (E) is quasi–isomorphic to 0
0
1 1 L−2 →S 3 ΩX (log D) ⊗ L−2 →S 2 ΩX (log D) ⊗ L−2 ⊗ Ω2X (log D).
We conclude that the first cohomology is given by H 0 (X, S 3 Ω1X (log D) ⊗ L−2 ). If we additionally impose the L2 –conditions (see appendix), then we see that the first Higgs cohomology of S 2 (E, ϑ) vanishes, provided that we have H 0 (X, S 2 Ω1X (log D) ⊗ Ω1X ⊗ L−2 ) = 0. Using 1 0 → σ ∗ Ω1E×E → ΩX → i∗ Ω1Z → 0
we obtain an exact sequence 1 (log D) ⊗ Ω1X ⊗ L−2 ) → 0 → H 0 (X, S 2 Ω1X (log D) ⊗ L−2 ) → H 0 (X, S 2 ΩX 1 → H 0 (Z, S 2 ΩX (log D) ⊗ Ω1Z ⊗ L−2 ).
A generalization of [24, example 3] leads to the vanishing H 0 (X, S 2 Ω1X (log D) ⊗ L−2 ) = 0. 1 (log D)| = O (1) ⊕ O (2), we get Since ΩX Z Z Z
H 0 (Z, S 2 Ω1X (log D) ⊗ Ω1Z ⊗ L−2 ) = C3 , because Ω1X (log D) ⊗ Ω1Z ⊗ L−2 = OZ ⊕ 2OZ (−1) ⊕ OZ (−2). However we are not able to decide whether these 3 sections lift to X. When we restrict to forms with fewer poles, then the vanishing will hold for a kind of cuspidal cohomology.
17.7 Proof of the Main Theorem In paragraph 17.7.1 we prove our main theorem, in paragraph 17.7.2 we give some indication on the proof of the motivic decomposition conjecture in our case, however the details will be published in a forthcoming paper. We thus will drop the assumption on the motivic decomposition conjecture in Theorem 17.7.2.
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17.7.1 From Relative to Absolute We now state and prove our main theorem. Let p : A −→ X be the compactified family over Holzapfel’s surface. Assume the motivic decomposition conjecture 17.1.1 ([5], [10, Conj. 1.4]) for A/X. In the proof we will need an auxiliary statement which was implicitely proven in section 6: Lemma 17.7.1. Let x ∈ X be a base point. Then π1top (X, x) acts on the Betti cohomology group H 2j (Ax (C), Q). Then, for 0 ≤ j ≤ d = 3, the cycle top class map CH j (A) → H 2j (Ax (C), Q)π1 (X,x) is surjective. Proof By Lemma 5.3 the sheaf R1 p∗ C is a sum of two irreducible representation of π1top (X, x). By the proof of Theorem 6.5., R2 p∗ C decomposes into a one–dimensional constant representation and three irreducible ones. The constant part corresponds to the identity in End(V1 ) and therefore to the polarization class on the fibers, which is a Hodge class. Therefore the invariant classes in H 2 (Ax (C), Q), and by duality also in H 4 (Ax (C), Q), consist of Hodge classes and are hence in the image of the cycle class map by the Hodge conjecture for divisors (and curves). Now we can prove our main theorem: Theorem 17.7.2. Assuming the motivic decomposition conjecture 17.1.1, the total space of the family p : A −→ X supports a partial set of Chow– K¨ unneth projectors πi for i 6= 4, 5, 6. Proof The motivic decomposition conjecture 17.1.1 states that we have a relative Chow–K¨ unneth decomposition with projectors Πiα on strata Xα which is compatible with the topological decomposition theorem [3] M X ∼ = Ψjα : Rp∗ QA → ICXα (Vαj )[−j − dim(Xα )]. j,α
j,α
Now we want to pass from relative Chow–K¨ unneth decompositions to absolute ones. We use the notation of [11] and for the reader’s convenience we recall everything. Let P i /X and Pαi /X be the mutually orthogonal projectors adding up to the identity ∆(A/X) ∈ CHdim(A) (A ×X A) such that (P i /X)∗ Rp∗ QA = ICX (Ri p∗ QA )[−i], (Pαi /X)∗ Rp∗ QA = ICXα (Vαi )[−i−dim(Xα )], where the sheaves Vαi are local systems supported over the cusps. The projectors Pαi /X on the boundary strata decompose further into Chow– K¨ unneth components, since the boundary strata consist of smooth elliptic
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curves and the stratification has the product type fibers described in theorem 17.4.1. Let us now summarize what we know about the local systems Ri p∗ C on the open stratum X0 from section 6: R1 p∗ C is irreducible and has no cohomology except in degree 2 by Theorem 17.6.5. R2 p∗ C contains a trivial subsystem and the remaining complement has no cohomology except in degree 2 again by Theorem 17.6.5. R3 p∗ C also contains a trivial subsystem and its complement has cohomology possibly in degrees 1, 2, 3, see section 6. By duality similar properties hold for Ri p∗ C with i = 4, 5, 6. Using these properties together with Lemma 17.7.1 we can follow closely the proof of Thm. 1.3 in [11]: First construct projectors (P 2r /X)alg which are constituents of (P 2r /X) for 0 ≤ r ≤ 3. This follows directly from Lemma 17.7.1 as in Step II of [11, section 1.7.]. Step III from [11, section 1.7.] is valid by the vanishing observations above. As in Step IV of loc. cit. this implies that we have a decomposition into motives in CHM(k) 2r 2r M 2r−1 = (A, P 2r−1 , 0) (1 ≤ r ≤ d), Mtrans = (A, Ptrans , 0) (0 ≤ r ≤ d), 2r 2r , 0) (0 ≤ r ≤ d), = (A, Palg Malg
plus additional boundary motives Mαj for each stratum Xα . As in Step 2r further. The projectors constructed in this V of [11] we can split Malg way define a set of Chow–K¨ unneth projectors πi for i 6= 4, 5, 6, since the relative projectors which contribute to more than one cohomology only affect cohomological degrees 4, 5 and 6. Remark 17.7.3. If H 1 (X, S 2 V1 ) vanishes or consists of algebraic (2, 2) Hodge classes only, then we even obtain a complete Chow–K¨ unneth decomposition in the same way, since algebraic Hodge (p, p)–classes define Lefschetz motives Z(−p) which can be split off by projectors in a canonical way. In fact all classes in H 1 (X, S 2 V1 ) have Hodge type (2, 2), as one can see from the computation of the Higgs cohomology. Therefore the Hodge conjecture on A would imply a complete Chow–K¨ unneth decomposition. However the Hodge conjecture is not very far from proving the total decomposition directly.
17.7.2 Motivic decomposition conjecture The goal of this paragraph is to sketch the proof of the motivic decomposition conjecture 17.1.1 in the case we treat in this paper. The complete details for the following argument will be published in a future publication. First note that since A is an abelian variety we can use the work of Deninger and Murre ([6]) on Chow-K¨ unneth decompositions of Abelian schemes to
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obtain relative Chow-K¨ unneth projectors for A/X. To actually get relative Chow-K¨ unneth projectors for A/X, we observe the following. Recall our results in section 17.4. We showed that the special fibres over the smooth elliptic cusp curves Di are of the form Ys = E × P1 × P1 . We do not need the cycle class map CH∗ (Ys × Ys ) → H∗ (Ys × Ys ) to be an isomorphism as in [10, Thm. I]. Since the boundary strata on X are smooth elliptic curves it is sufficient to know the Hodge conjecture for the special fibres. But the special fibres are composed of elliptic curves and rational varieties by our results in section 17.4. Therefore the methods in [10] can be refined to work also in this case and we can drop the assumption in theorem 17.7.2. Remarks 17.7.4. We hope to come back to this problem later and prove the motivic decomposition conjecture for all Picard families. The existence of absolute Chow–K¨ unneth decompositions however seems to be out of reach for other examples since vanishing results will hold only for large arithmetic subgroups, i.e., small level. 17.8 Appendix: Algebraic L2 -sub complexes of symmetric powers of the uniformizing bundle of a two–dimensional complex ball quotient X a 2-dim projective variety with a normal crossing divisor D, X = X − D; assume that the coordinates near the divisor are z1 , z2 . Consider the uniformizing bundle of a 2-ball quotient −1/3 −1/3 E = Ω1X (log D) ⊗ KX (log D) ⊕ KX (log D) We consider two cases: 1) D is a smooth divisor (the case we need) and 2) D is a normal crossing divisor. Case 1 Assume that D is defined by z1 = 0. Taking v as the generating −1/3 1 section of KX (log D), dz z1 ⊗ v, dz2 ⊗ v as the generating sections −1/3
of Ω1X (log D) ⊗ KX
(log D), then the Higgs field ϑ : E → E ⊗ Ω1X (log D)
dz1 1 is defined by setting ϑ( dz z1 ⊗ v) = v ⊗ z1 , ϑ(dz2 ⊗ v) = v ⊗ dz2 , and ϑ(v) = 0. dz1 1 Clearly, if ϑ is written as N1 dz z1 + N2 dz2 , then N1 ( z1 ⊗ v) = v,
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1 N1 (dz2 ⊗ v) = 0, N1 (v) = 0, N2 ( dz z1 ⊗ v) = 0, N2 (dz2 ⊗ v) = v, N2 (v) = 0; the kernel of N1 is the subsheaf generated by dz2 ⊗ v and v. Using the usual notation, we then have
dz1 ⊗v z1 Gr0 W (N1 ) = generated by dz2 ⊗ v Gr1 W (N1 ) = generated by
Gr−1 W (N1 ) = generated by v. So with {x} representing the line bundle generated by an element x, one has dz1 ⊗ v} + {dz2 ⊗ v} + {v} z1 KerN1 + z1 E; dz1 dz1 ⊗ (z1 { ⊗ v} + z1 {dz2 ⊗ v} + z1 {v}) z1 z1 dz1 dz2 ⊗ (z1 { ⊗ v} + {dz2 ⊗ v} + {v}) z1 dz1 ⊗ z1 E + dz2 ⊗ (KerN1 + z1 E); z1 dz1 dz1 ∧ dz2 ⊗ (z1 { ⊗ v} + z1 {dz2 ⊗ v} + z1 {v}) z1 z1 dz1 ∧ dz2 ⊗ z1 E. z1
Ω0 (E)(2) = z1 { = Ω1 (E)(2) = + = Ω2 (E)(2) = =
−1/3
Case 2 : As before, taking v as the generating section of KX dz1 dz2 z1 ⊗v, z2 ⊗v
as the generating sections of then the Higgs field
(log D),
1 (log D)⊗K−1/3 (log D), ΩX X
1 ϑ : E → E ⊗ ΩX (log D) 1 is defined by setting ϑ( dz z1 ⊗ v) = v ⊗ ϑ(v) = 0.
dz1 z1 ,
2 ϑ( dz z2 ⊗ v) = v ⊗
dz2 z2 ,
and
dz1 dz2 1 Clearly, if ϑ is written as N1 dz z1 + N2 z2 , then N1 ( z1 ⊗ v) = v, dz1 dz2 2 N1 ( dz z2 ⊗ v) = 0, N1 (v) = 0, N2 ( z1 ⊗ v) = 0, N2 ( z2 ⊗ v) = v, N2 (v) = 0; the kernel of N1 (resp. N2 ) is the subsheaf generated by
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⊗ v (resp.
dz1 z1
⊗ v) and v. We then have
Gr1 W (N1 ) = generated by Gr0 W (N1 ) = generated by Gr−1 W (N1 ) = generated by Gr1 W (N2 ) = generated by Gr0 W (N2 ) = generated by Gr−1 W (N2 ) = generated by Gr1 W (N1 + N2 ) = generated by Gr0 W (N1 + N2 ) = generated by Gr−1 W (N1 + N2 ) = generated by
dz1 ⊗v z1 dz2 ⊗v z2 v dz2 ⊗v z2 dz1 ⊗v z1 v dz1 dz2 + )⊗v ( z1 z2 dz1 dz2 ( − )⊗v z1 z2 v
So, one has dz1 dz2 ⊗ v} + z2 { ⊗ v} + {v} z1 z2 KerN1 ∩ KerN2 + z2 KerN1 + z1 KerN2 ; dz1 dz2 dz1 ⊗ (z1 { ⊗ v} + z1 z2 { ⊗ v} + z1 {v}) z1 z1 z2 dz2 dz2 dz1 ⊗ (z2 { ⊗ v} + z1 z2 { ⊗ v} + z2 {v}) z2 z2 z1 dz1 dz2 ⊗ (z1 KerN2 + z1 z2 KerN1 ) + ⊗ (z2 KerN1 + z1 z2 KerN2 ); z1 z2 dz1 dz2 ∧ ⊗ z1 z2 E, z1 z2
Ω0 (E)(2) = z1 { = Ω1 (E)(2) = + = Ω2 (E)(2) =
For the above two cases, it is to easy to check that ϑ(Ω0 (E)(2) ) ⊂ Ω1 (E)(2) and ϑ(Ω1 (E)(2) ) ⊂ Ω2 (E)(2) . Thus, together ϑ ∧ ϑ = 0, we have the complex ({Ωi (E)(2) }2i=0 , ϑ) 0 → Ω0 (E)(2) → Ω1 (E)(2) → Ω2 (E)(2) → 0 with ϑ as the boundary operator.
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Now we take the 2nd -order symmetric power of (E, ϑ), we obtain a new Higgs bundle S 2 (E, ϑ) (briefly, the Higgs field is still denoted by ϑ) as follows, −2/3 1 (log D) ⊗ KX (log D) ⊕ S 2 (E, ϑ) = S 2 ΩX −2/3
⊕Ω1X (log D) ⊗ KX
−2/3
(log D) ⊕ KX
(log D).
1 (log D) and S 2 (E, ϑ) ⊗ The Higgs field ϑ maps S 2 (E, ϑ) into S 2 (E, ϑ) ⊗ ΩX Ω1X (log D) into S 2 (E, ϑ) ⊗ Ω2X (log D) so that one has a complex with the differentiation ϑ as follows
(∗)
0 → S 2 (E, ϑ) → S 2 (E, ϑ) ⊗ Ω1X (log D) → S 2 (E, ϑ) ⊗ Ω2X (log D) → 0;
more precisely, one has −2/3 −2/3 ϑ S 2 Ω1X (log D) ⊗ KX (log D) ⊂ Ω1X (log D) ⊗ KX (log D) ⊗ Ω1X (log D) −2/3 −2/3 ϑ Ω1X (log D) ⊗ KX (log D) ⊂ KX (log D) ⊗ Ω1X (log D) −2/3 ϑ KX (log D) = 0
and −2/3 1 (log D) ⊗ KX (log D) ⊗ Ω1X (log D) ϑ S 2 ΩX −2/3 ⊂ Ω1X (log D) ⊗ KX (log D) ⊗ Ω2X (log D) −2/3 −2/3 ϑ Ω1X (log D) ⊗ KX (log D) ⊗ Ω1X (log D) ⊂ KX (log D) ⊗ Ω2X (log D) −2/3 ϑ KX (log D) ⊗ Ω1X (log D) = 0
Note: Let V be a SL(2)-module, then S 2 V ⊗ V ' S 3 V ⊕ V ⊗ ∧2 V. In general, one needs to consider the representations of GL(2); in such a case, we can take the determinant of the representation in question, and
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then go back to a representation of SL(2). −2/3 S 2 (E, ϑ) = S 2 Ω1X (log D) ⊗ KX (log D) −2/3
⊕Ω1X (log D) ⊗ KX −2/3
⊕KX S 2 (E, ϑ) ⊗ Ω1X (log D) =
(log D)
(log D)
−2/3 S 2 Ω1X (log D) ⊗ KX (log D) ⊗ Ω1X (log D) −2/3 ⊕ Ω1X (log D) ⊗ KX (log D) ⊗ Ω1X (log D) −2/3
(log D) ⊗ Ω1X (log D) −2/3 S 2 (E, ϑ) ⊗ Ω1X (log D) = S 2 Ω1X (log D) ⊗ KX (log D) ⊗ Ω2X (log D) −2/3 ⊕ Ω1X (log D) ⊗ KX (log D) ⊗ Ω2X (log D) ⊕KX
−2/3
⊕KX
(log D) ⊗ Ω2X (log D)
Assuming that the divisor D is smooth, we next want to consider the L2 holomorphic Dolbeault sub-complex of the above complex (*): 1 0 → (S 2 (E, ϑ))(2) → (S 2 (E, ϑ)⊗ΩX (log D))(2) → (S 2 (E, ϑ)⊗Ω2X (log D))(2) → 0,
and explicitly write down (S 2 (E, ϑ) ⊗ ΩiX (log D))(2) . Note that taking symmetric power for L2 -complex does not have obvious functorial properties in general. We will continue to use the previous notations. For simplicity, we will further 1 set v1 = dz z1 ⊗ v and v2 = dz2 ⊗ v; we also denote e1 ⊗ e2 + e2 ⊗ e1 by e1 e2 , the symmetric product of the vectors e1 and e2 . 1 (log D) ⊗ K−2/3 (log D), as a sheaf, is generated by v v , Thus, S 2 ΩX 1 1 X −2/3
1 (log D) ⊗ K v1 v2 , v2 v2 ; ΩX X
(log D) is generated by v1 v, v2 v; and
−2/3 KX (log D)
is generated by v v. Also, it is easy to check how N1 , N2 act on these generators; as for N1 , we have (Note N1 v1 = v, N1 v2 = 0, N1 v = 0.) N1 (v1 v1 ) = 2v1 v N1 (v1 v2 ) = v2 v N1 (v2 v2 ) = 0 N1 (v1 v) = v v N1 (v2 v) = 0 N1 (v v) = 0. 1 (log D) ⊗K−2/3 (log D) into Ω1 (log D)⊗K−2/3 (log D), Clearly, N1 maps S 2 ΩX X X X
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−2/3
579 −2/3
1 (log D) ⊗ K ΩX (log D) into KX (log D), and then KX (log D) to 0. X So, N1 is of index 3 on the 2nd -order symmetric power S 2 E(as is obvious from the abstract theory since N1 is of index 2 on E); and we then have the following gradings
Gr2 W (N1 ) = generated by v1 v1 Gr1 W (N1 ) = generated by v1 v2 Gr0 W (N1 ) = generated by v1 v, v2 v2 Gr−1 W (N1 ) = generated by v2 v Gr−2 W (N1 ) = generated by v v.
(Note that N1 , acting on E, has two invariant (irreducible) components, one being generated by v1 , v, the other by v2 , so that N1 has three invariant components on S 2 E, as is explicitly showed in the above gradings.) Now we can write down L2 -holomorphic sections of S 2 E, namely the sections generated by v1 v, v2 v2 , v2 v, v v, and z1 S 2 E; in the invariant terms, they should be (S 2 (E, ϑ))(2) = E ImN1 + S 2 (KerN1 ) + z1 S 2 E. 1 (log D)) 2 Now it is easy to also write down (S 2 (E, ϑ) ⊗ ΩX (2) and (S (E, ϑ) ⊗ 2 ΩX (log D))(2) :
dz1 ⊗ (S 2 (ImN1 ) + z1 S 2 E) z1 +dz2 ⊗ (E ImN1 + S 2 (KerN1 ) + z1 S 2 E); dz1 = ∧ dz2 ⊗ (S 2 (ImN1 ) + z1 S 2 E). z1
(S 2 (E, ϑ) ⊗ Ω1X (log D))(2) =
2 (S 2 (E, ϑ) ⊗ ΩX (log D))(2)
Similary, one can determine the algebraic L2 −sub complex of S n (E, ϑ) for any n ∈ N.
17.9 Acknowledgement It is a pleasure to dedicate this work to Jaap Murre who has been so tremendously important for the mathematical community. In particular we want to thank him for his constant support during so many years. We are grateful to Bas Edixhoven, Jan Nagel and Chris Peters for organizing such a wonderful meeting in Leiden. Many thanks go to F. Grunewald, R.-P. Holzapfel, J.-S. Li and J. Schwermer for very helpful discussions.
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References [1] Angel, P. del and S. M¨ uller-Stach: On Chow motives of 3–folds, Transactions of the AMS 352, 1623-1633 (2000). [2] Ash, A., D. Mumford, M. Rapoport and Y. Tai : Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, 1975. [3] Beilinson, A., J. Bernstein and P. Deligne: Faisceaux pervers, analyse et topologie sur les espaces singuliers, Ast´erisque 100, 7–171 (1982). [4] Borel, A. and N. Wallach: Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Math. Studies 94, (1980). [5] Corti, A. and M. Hanamura: Motivic decomposition and intersection Chow groups I, Duke Math. J. 103 (2000), no. 3, 459–522. [6] Deninger, C. and J. Murre: Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math., 422 (1991), 201–219. [7] Esnault, H. and E. Viehweg: Lectures on Vanishing Theorems, freely available under http://www.uni-essen.de/∼mat903. [8] Chai, C. and G. Faltings: Degenerations of Abelian Varieties, Ergebnisse der Mathematik, Band 22, Springer Verlag, 1990. [9] Gordon, B. B. : Canonical models of Picard modular surfaces, in The zeta functions of Picard modular surfaces, Univ. Montr´eal, Montreal, 1992, 1–29. [10] Gordon, B. B.,M. Hanamura and J. P. Murre: Relative Chow-K¨ unneth projectors for modular varieties, J. Reine Angew. Math., 558 (2003), 1–14. [11] Gordon, B. B. , M. Hanamura and J. P. Murre: Chow-K¨ unneth projectors for modular varieties, C. R. Acad. Sci. Paris, Ser. I, 335, 745–750 (2002). [12] Gordon, B. B., M. Hanamura and J. P. Murre: Absolute Chow-K¨ unneth projectors for modular varieties, Crelle Journal 580, 139–155 (2005). [13] Hemperly, J.: The parabolic contribution to the number of linearly independent automorphic forms on a certain bounded domain, Am.Journ.Math 94, 1972. [14] Hirzebuch, F.: Chern numbers of algebraic surfaces- an example, Math. Annalen 266, 351–356 (1984). [15] Holzapfel, R.-P.: Chern numbers of algebraic surfaces - Hirzebruch’s examples are Picard modular surfaces, Math. Nachrichten 126, 255–273 (1986). [16] Holzapfel, R.-P.: Geometry and arithmetic around Euler partial differential equations, Reidel Publ. Dordrecht (1987). [17] Jost, J., Y.-H. Yang and K. Zuo: The cohomology of a variation of polarized Hodge structures over a quasi–compact K¨ahler manifold, preprint math.AG/0312145. ¨nnemann: On the Chow motive of an abelian scheme, in Motives (Seat[18] K. Ku tle, WA, 1991), Providence, RI, 1994, Amer. Math. Soc., 189–205. [19] J.-S. Li and J. Schwermer: On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 no. 1, 141–169 (2004). [20] Looijenga, E.: Compactifications defined by arrangements I: The ball– quotient case, Duke Math. J. 118 (2003), no. 1, 151–187. [21] Matsushima, Y. and G. Shimura: On the cohomology groups attached to certain vector valued differential forms on the product of upper half planes, Ann. of Math. 78, 417–449 (1963). [22] Miller. A.: The moduli of Abelian Threefolds with Complex Multiplication and Compactifications, Preprint. [23] Miller. A.: Families of Abelian Threefolds with CM and their degenerations, Preprint.
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[24] Miyaoka, Y.: Examples of stable Higgs bundles with flat connections, preprint. [25] Mumford, D.: An analytic construction of degenerating abelian varieties over complete rings, Compositio Math. 24, 1972, 239-272. Appeared also as an appendix to [8]. [26] Murre, J. P.: Lectures on Motives, in Proceedings Transcendental Aspects of Algebraic Cycles, Grenoble 2001, LMS Lectures Note Series 313, Cambridge Univ. Press (2004), 123–170. [27] Murre, J. P.: On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990), 190–204. [28] Picard, E.: Sur des fonctions de deux variables ind´ependentes analogues aux fonctions modulaires, Acta. Math. 2, 114–135 (1883). [29] Simpson, C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Am. Math. Soc. 1, No.4, 867– 918 (1988). [30] Vogan, D. and G. Zuckerman: Unitary representations with non–zero cohomology, Compositio Math. 53, 51–90 (1984).
18 The Regulator Map for Complete Intersections Jan Nagel Universit´e Lille 1, Math´ematiques - Bˆ at. M2, F-59655 Villeneuve d’Ascq Cedex, France
[email protected]
To Professor Murre, with great respect.
18.1 Introduction Since the introduction of the theory of infinitesimal variations of Hodge structure, infinitesimal methods have been successfully applied to a number of problems concerning the relationship between algebraic cycles and Hodge theory. One of the common techniques is to study infinitesimal invariants associated to families of algebraic cycles. This approach led to a proof of the infinitesimal Noether–Lefschetz theorem and was further developed by Green and Voisin in their study of the image of the Abel–Jacobi map for hypersurfaces in projective space. This work was reinterpreted and extended by Nori [18]. He proved a connectivity theorem for the universal family XT of complete intersections of multidegree (d0 , . . . , dr ) on a polarised variety (Y, OY (1)) inside the trivial family YT = Y × T . Specifically, he proved that if the fibers of XT → T are n–dimensional then H n+k (YT , XT ; Q) = 0 for all k ≤ n if min(d0 , . . . , dr ) is sufficiently large. In [16] I proved an effective version of Nori’s connectivity theorem; see [23] and [1] for related results in the case Y = PN . For geometric applications of Nori’s theorem one usually does not need the full strength of Nori’s theorem; it often suffices to have H n+k (YT , XT ) = 0 for all k ≤ c, for some integer c ≤ n. The theorems of Noether–Lefschetz and Green–Voisin can be deduced from Nori’s theorem by taking Y = PN and (c, n) = (1, 2m) or (c, n) = (2, 2m − 1). It turns out that in these cases, the degree bounds are sharp. One can also use Nori’s theorem to study the regulator maps on Bloch’s 582
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higher Chow groups, as was noted in [21]. In [23] Voisin considered the extreme case c = n of Nori’s theorem for hypersurfaces in projective space and showed that also in this case the bound is sharp, by constructing interesting higher Chow cycles on hypersurfaces of low degree. One could therefore ask whether the degree bounds computed in [16, Thm. 3.13] are optimal for complete intersections in projective space. In this note we show that this is not the case, by studying the image of the regulator maps defined on the higher Chow groups CHp (X, 1) and CHp (X, 2). We improve the bounds computed in [16, Theorems. 4.4 and 4.6] using two methods: (1) a version of the Jacobi ring introduced in [1]; (2) a correspondence between the cohomology of complete intersections of quadrics and double coverings of projective space, combined with a version of Nori’s theorem for cyclic coverings. These results are treated in sections 2 and 3. We conclude with an improved result on the image of the regulator map (Theorem 18.4.2) which is optimal for CHp (X, 2). Acknowledgment. A part of this paper was prepared during a visit to the Max–Planck Institut f¨ ur Mathematik in Bonn in the spring of 2003. I would like to thank the institute for its hospitality and excellent working conditions.
18.2 Infinitesimal calculations For the definition and basic properties of Bloch’s higher Chow groups CHp (X, q) we refer to [15]. There exist regulator maps 2p−q cp,q : CHp (X, q) → HD (X, Z(p))
that generalise the classical Deligne cycle class map; see [10] for an explicit description of these maps using integration currents. The starting point is the following result; cf. [16, Lemma 4.1] and the references cited there. Q Proposition 18.2.1. Let U ⊂ ri=0 PH 0 (PN , OP (di )) be the open subset parametrising smooth complete intersections of dimension n and multidegree N (d0 , . . . , dr ) in PN , and let XU ⊂ PN U = P × U be the universal family. If k HD (PN × T, XT ) = 0
for all k ≤ 2p − q + 1 and for every smooth morphism T → U , then the image of the regulator map 2p−k cp,q : CHp (X, q) ⊗ Q → HD (X, Q(p)) 2p−q N 2p−q is contained in the image of the restriction map HD (P , Q(p)) → HD (X, Q(p)) if X is very general.
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Using an effective version of Nori’s connectivity theorem, we computed degree bounds for the cases q = 1 and q = 2. Theorem 18.2.2. Put δmin = min(d0 , . . . , dr ), δmax = max(d0 , . . . , dr ). If (n, q) ∈ {(2m, 1), (2m − 1, 2)}, the conclusion of Proposition 18.2.1 holds if the following conditions are satisfied. Pr d + (m − 1)δmin ≥ n + r + 3; (C0 ) Pri=0 i (C1 ) i=0 di + mδmin ≥ n + r + 2 + δmax . Proof See [16, Theorems 4.4 and 4.6]. Corollary 18.2.3. If (n, q) = (2m, 1), the conclusion of Proposition 18.2.1 holds, with the possible exception of the following cases. i) X = V (2) ⊂ P2m+1 , X = V (3) ⊂ P3 , X = V (4) ⊂ P3 , X = V (3) ⊂ P5 ; ii) X = V (d, 2) ⊂ P2m+2 , d ≥ 2; iii) X = V (2, 2, 2) ⊂ P2m+3 . Corollary 18.2.4. If (n, q) = (2m − 1, 2), the conclusion of Proposition 18.2.1 holds, with the possible exception of the following cases. i) X = V (2) ⊂ P2m , X = V (3) ⊂ P2 ; ii) X = V (2, 2) ⊂ P2m+1 , m ≥ 1. Remark 18.2.5. Similar degree bounds can be worked out for q ≥ 3. They coincide with the bounds of Corollary 18.2.3 (q odd) or Corollary 18.2.4 (q even). We have refrained from studying these cases as there is no description of CHp (X, q) using Gersten–Quillen resolutions if q ≥ 3. To see whether the bounds of Corollaries 18.2.3 and 18.2.4 can be improved, we recall the idea of the proof of Theorem 18.2.2. Using mixed Hodge theory, one checks that it suffices to show F m+1 H n+2 (PN T , XT ) = 0
(18.1)
in both cases. Let f : XT → T be the structure morphism, and put p,q Hp,q (XT ) = Rq f∗ ΩpXT /T . Let Hpr (XT ) be the subbundle corresponding to primitive cohomology. By spectral sequence arguments one shows that the condition (18.1) is satisfied if the complex p,n−p p−1,n−p+1 p−2,n−p+2 0 → Hpr (XT ) → Ω1T ⊗ Hpr (XT ) → Hpr (XT )
(18.2)
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is exact for all p ≥ m + 1. Lr ∨ Put E = i=0 OP (di ), P = P(E ) and set ξE = OP (1). Let Σ be the sheaf of differential operators of order ≤ 1 on sections of ξE . Let X be a complete intersection defined by a section s = (f0 , . . . , fr ), and let σ be the corresponding section of H 0 (P, ξE ). Contraction with the 1–jet j 1 (σ) defines a−1 a for all a ≥ 1. Put maps KP ⊗ Σ ⊗ ξE → KP ⊗ ξE a−1 a a J(KP ⊗ ξE ) = Im(H 0 (P, KP ⊗ Σ ⊗ ξE ) → H 0 (P, KP ⊗ ξE )) a a a R(KP ⊗ ξE ) = H 0 (P, KP ⊗ ξE )/J(KP ⊗ ξE ). n−p,p p+r+1 Proposition 18.2.6. We have an isomorphism Hpr (X) ∼ ). = R(KP ⊗ξE
Proof Cf. [5, Section 10.4] and the references cited there. The proof of Theorem 18.2.2 proceeds as follows. By semicontinuity it suffices to check the exactness of (18.2)) pointwise. One can reduce to the case T = H 0 (PN , E) − ∆, where ∆ is the discriminant locus. Hence the tangent space to T is V = H 0 (PN , E). Applying these reductions to the dual of the complex (18.2), we see that it suffices to check the exactness of n−p+2,p−2 n−p+1,p−1 n−p,p Λ2 V ⊗ Hpr (Xt ) → V ⊗ Hpr (Xt ) → Hpr (Xt ) → 0. (18.3)
By Proposition 18.2.6, this complex is isomorphic to p+r+1 p+r p+r−1 ) → 0). ) → R(KP ⊗ξE ) → V ⊗R(KP ⊗ξE R• = (Λ2 V ⊗R(KP ⊗ξE
Let S• (resp. J• ) be the complexes obtained by replacing the terms R(KP ⊗ q q q ξE ) in R• by H 0 (P, KP ⊗ ξE ) (resp. J(KP ⊗ ξE )). The exact sequence of complexes 0 → J• → S• → R• → 0 shows that H1 (R• ) = H0 (R• ) = 0 if H1 (S• ) = H0 (S• ) = 0 (1), H0 (J• ) = 0 (2). Using Castelnuovo–Mumford regularity, one then shows that (C0 ) implies (1) and (C1 ) implies (2). It is possible to improve condition (ii) of Corollary 18.2.3. To this end, Q let U 0 ⊂ ri=1 PH 0 (PN , OP (di )) be the open subset parametrising smooth complete intersections of multidegree (d1 , . . . , dr ), and let YU be the pullQ back of its universal family to U ⊂ ri=0 PH 0 (PN , OP (di )). Given a smooth morphism T → U , the inclusions of pairs N (YT , XT ) ⊂ (PN T , XT ) ⊂ (PT , YT )
induce a long exact sequence k N k k+1 N → H k (PN (PT , XT ) → T , YT ) → H (PT , XT ) → H (YT , XT ) → H
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of cohomology groups. The vanishing of H k (YT , XT ) was investigated by Asakura and S. Saito [1]. For the vanishing of H n+2 (YT , XT ) one introduces the bundles Hp,q (YT , XT ) of relative cohomology and studies exactness of the complex 0 → Hp,n−p+1 (YT , XT ) → Ω1T ⊗Hp−1,n−p+2 (YT , XT ) → Ω2T ⊗Hp−2,n−p+3 (YT , XT ). Exactness of this complex is reduced to exactness of p+r−1 p+r p+r+1 Λ2 V ⊗ R0 (KP ⊗ ξE ) → V ⊗ R0 (KP ⊗ ξE ) → R0 (KP ⊗ ξE )→0 (18.4) 0 where R is the Jacobi ring defined in [1, §1].
Theorem 18.2.7 (Asakura–Saito). Put dmax = max(d1 , . . . , dr ). The complex (18.4) is exact if Pr (C0 ) i=0 di + (m − 1)δmin ≥ n + r + 3; P r 0 (C1 ) i=0 di + mδmin ≥ n + r + 2 + dmax . Proof See [1, Thm. 9-3 (ii)]. Corollary 18.2.8. The conclusion of Proposition 18.2.1 holds if (n, q) = (2m, 1) and (d0 , d1 ) = (2, d) if d ≥ 4. Proof If (d0 , d1 ) = (2, d) then YT is a family of odd–dimensional quadrics. As these quadrics have no primitive cohomology, a Leray spectral sequence argument shows that H k (PN T , YT ) = 0 for all k. Hence we obtain isomork (Y , X ) for all k. Using Theorem 18.2.7 we ∼ , X ) H phisms H k (PN = T T T T k N obtain H k (PN T , XT ) = 0 for all k ≤ 2m + 2. Hence HD (PT , XT ) = 0 for all k ≤ 2m + 2.
18.3 Complete intersections of quadrics In this section we show how to exclude the exceptional cases of Corollary 18.2.3 (iii) and Corollary 18.2.4 (ii) using a correspondence between the cohomology of a double covering of projective space and the cohomology of a complete intersection of quadrics, and a version of Nori’s theorem for double coverings (and more generally cyclic coverings) of projective space. We start with Corollary 18.2.3 (iii). Let X = V (Q0 , Q1 , Q2 ) ⊂ P2m+3 be a smooth complete intersection of three quadrics. Given (λ0 : λ1 : λ2 ) ∈ P2 , write Qλ = λ0 Q0 + λ1 Q1 + λ2 Q2 . (By abuse of notation, we use the same
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notation for a quadric Q, its defining equation and its associated symmetric matrix.) Let X = {(x, λ) ∈ P2m+3 × P2 |x ∈ Qλ } be the associated quadric bundle over P2 , and let C = {λ ∈ P2 |corank(Qλ ) ≥ 1} be the discriminant curve. The passage from the complete intersection X to the hypersurface X ⊂ P2m+3 × P2 induces an isomorphism on middle dimensional primitive cohomology. This result is sometimes referred to as the Cayley trick; cf. [7, §6]. Proposition 18.3.1 (Cayley trick). We have an isomorphism of Hodge 2m (X)(−2) ∼ H 2m+4 (X). structures Hpr = pr The isomorphism of the previous Proposition is induced by the correspondence 2 → Γ1 = f −1 (X) =X ×P − f y X.
X
As a smooth quadric in P2m+3 contains two families of (m + 1)–planes, there exists a family Γ2 of (m + 1)–planes contained in the fibers of f : X → P2 . The base of this family is a double covering π : S → P2 that is ramified over the discriminant curve C. Theorem 18.3.2 (O’Grady). The correspondence Γ = Γ2 ◦ t Γ1 induces an 2 (S, Q) ∼ H 2m (X, Q). isomorphism of Hodge structures Hpr = pr Proof See [19]; cf.[11] for a more general result, valid for arbitrary even– dimensional quadric bundles over P2 . The discriminant curve C is defined by the homogeneous polynomial F (λ0 , λ1 , λ2 ) = det(λ0 Q0 + λ1 Q1 + λ2 Q2 ) of degree 2m + 4; if the quadrics Q0 , Q1 and Q2 are general, C is smooth. Consider the vector bundle E = ⊕3 OP (2) on P2m+3 , and the map H : H 0 (P2m+3 , E) → H 0 (P2 , OP (2m + 4))
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that sends a net of quadrics to the equation of its discriminant curve. The map H induces a rational map h : PH 0 (P2m+3 , E) − − > PH 0 (P2 , OP (2m + 4)). Lemma 18.3.3. Let U ⊂ PH 0 (P2 , OP (2m+4)) be the open subset parametrising smooth curves of degree 2m + 4. There exists a Zariski open subset U 0 ⊂ PH 0 (P2m+3 , E) − ∆ such that h : U 0 → U is a smooth morphism. Proof By a classical theorem of Dixon [8], a general smooth plane curve of even degree can be realised as the discriminant curve of a net of quadrics; see [2, Prop. 4.2 and Remark 4.4] for a modern proof. Hence the map h is dominant, and the assertion follows from [9, III, Lemma 10.5)]. In the sequel we shall need a relative version of Theorem 18.3.2. Over U we have the universal family SU → U whose fiber over [F ] ∈ U is the double covering of P2 ramified over the curve V (F ). Let ST = SU ×U T be the pullback of this family to T along the map h : T → U . Over T we have the family of quadric bundles fT : XT → T associated to the family of complete intersections XT → T . We have relative correspondences Γ 1,T y XT
− →
XT
Γ 2,T y ST .
→ XT −
To state the relative version of Theorem 18.3.2 we need some notation. Given a map f : Y → X of topological spaces, let M (f ) be the mapping ∼ cylinder of f . The map f factors as Y ,→ M (f )−→ X where the second map is a homotopy equivalence. Define H k (X, Y ; Z) = H k (M (f ), Z). The groups H k (X, Y ) fit into a long exact sequence f∗
→ H k−1 (Y ) → H k (X, Y ) → H k (X)−−→ H k (Y ) →
(18.5)
of cohomology groups. If f is the inclusion of a subspace, they coincide with the usual relative cohomology of the pair (X, Y ). Since we have isomorphisms 2 2m Hpr (S) ∼ (X) ∼ = H 3 (P2 , S), Hpr = H 2m+1 (P2m+3 , X),
Theorem 18.3.2 can be restated as an isomorphism H 3 (P2 , S) ∼ = H 2m+1 (P2m+3 , X).
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Theorem 18.3.4. The relative correspondence ΓT = Γ2,T ◦ t Γ1,T induces an isomorphism H k+2 (P2T , ST ) ∼ = H 2m+k (PT2m+3 , XT ) for all k ≥ 0. Proof Let fT : XT → T, gT : ST → T, ϕT : PT2m+3 → T, ψT : P2T → T be the projections onto the base T . By the Lefschetz hyperplane theorem and the Barth–Lefschetz theorem for cyclic coverings of projective space [12, Thm. 2.1] we have Rq (fT )∗ Q ∼ = Rq (ϕT )∗ Q, q 6= 2m Rq (gT )∗ Q ∼ 6 2. = Rq (ψT )∗ Q, q = Set V = coker(R2m (ϕT )∗ Q → R2m (fT )∗ Q), W = coker(R2 (ψT )∗ Q → R2 (gT )∗ Q). The correspondence ΓT induces a homomorphism of local systems ΓT,∗ : W → V. By the proper base change theorem and Theorem 18.3.2, Γ(t)∗ is an isomorphism for all t ∈ T . Hence ΓT,∗ : W → V is an isomorphism. Combining the long exact sequence (18.5) with the Lefschetz/Barth–Lefschetz isomorphisms we obtain , XT ) ∼ H 2m+k (P2m+3 = H k−1 (T, V), H k+2 (P2T , ST ) ∼ = H k−1 (T, W), T and the result follows. The vanishing of H ∗ (P2T , ST ) follows from an effective version of Nori’s connectivity theorem for cyclic coverings of projective space, which can be seen as a generalisation of the result of M¨ uller–Stach on the Abel–Jacobi map [13]. An outline of the proof can be found in [17]. Theorem 18.3.5. Let U ⊂ H 0 (Pn , OP (d.e)) be the open subset parametrising cyclic coverings Y → Pn of degree e that ramify over a divisor D ⊂ Pn of degree d.e, and let YU → U be the universal family. Let T → U be a smooth morphism, and c ≤ n an integer. We have F µ H n+k (PnT , YT ) = 0 for all k ≤ c if (µ − c)e + 1)d ≥ n + c. Corollary 18.3.6. The conclusion of Proposition 18.2.1 holds if (n, q) = (2m, 1), r = 2 and (d0 , d1 , d2 ) = (2, 2, 2) if m ≥ 2. Proof Let U 0 be the open subset of P(⊕3 H 0 (P2m+3 , OP (2)) introduced in Lemma 18.3.3, and let T → U 0 be a smooth morphism. By Theorem 18.3.5
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and Theorem 18.3.4 we obtain H 2m+1 (PT2m+3 , XT ) = H 2m+2 (P2m+3 , XT ) = T 0. In a similar way one can remove the exceptions in Corollary 18.2.4 (ii) for m ≥ 2, using the following theorem of M. Reid [20]. Theorem 18.3.7 (Reid). Let X = V (2, 2) ⊂ P2m+1 be a general smooth complete intersection of two quadrics. There is a family of m–planes in the fibers of the associated quadric bundle X → P1 that defines a correspondence Γ between X and a hyperelliptic curve C, branched over a divisor D ⊂ P1 of degree 2m + 2. This correspondence induces an isomorphism Γ∗ : H 1 (C) → H 2m−1 (X).
18.4 Exceptional cases We end with a discussion of the remaining exceptional cases. We start with the case q = 1. There are a number of trivial exceptions coming from the Noether–Lefschetz theorem. Consider the commutative diagram ∗ CHm (X) ⊗C cm+1,1 y
µ
CHm+1 (X, 1) cm+1,1 y
−→ µD
Hdgm (X) ⊗ C∗ −−→
2m+1 (X, Z(m + 1)). HD
The composition of µD with the projection 2m+1 (X, Z(m + 1)) = HD
H 2m+1 (X, C) H m,m (X) → F m+1 H 2m+1 (X, C) + H 2m+1 (X, Z) Hdgm (X)
is an injective map Hdgm (X) ⊗ C∗ =
Hdgm (X) ⊗ C H m,m (X) ,→ . m Hdg (X) Hdgm (X)
Hence µD is injective, and we obtain an injective map from Hdgm pr (X) to the 2m+1 N 2m+1 ∗ cokernel of i : HD (P , Z(m + 1)) → HD (X, Z(m + 1)). This remark covers the cases X = V (2) ⊂ P2m+1 , X = V (3) ⊂ P3 , X = V (2, 2) ⊂ P2m+2 . The cycles that we considered above are decomposable, i.e., belong to the image of the map µ. The other counterexamples in low degree come from indecomposable higher Chow cycles. On K3 surfaces one can produce indecompable higher Chow cycles using rational nodal curves [4]; see [22],
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[14] and [6] for earlier results in this direction. Collino [6] gave examples of indecomposable higher Chow cycles on cubic fourfolds. Remark 18.4.1. Note that both in the case of K3 surfaces and of cubic fourfolds we are dealing with a Hodge structure V of weight 2 with dim V 2,0 = 1. (In the case of a cubic fourfold X, take V = H 4 (X, C)(1).) Hence one might ask whether the existence of indecomposable higher Chow cycles on these varieties is related to the Kuga–Satake construction; cf. [22, 4.4-4.5]. The remaining exceptional case for q = 1 is X = V (3, 2) ⊂ P2m+2 , m ≥ 2 (if m = 1, X is a K3 surface). I do not know what happens in this case. With the notation of section 2, we can show that H0 (S• ) = 0 and H1 (S• ) 6= 0. (The latter result can be seen by decomposing the terms of the complex S• into irreducible SL(V )–modules.) Hence H1 (R• ) = 0 if and only if the map H1 (J• ) → H1 (S• ) is surjective; it seems hard to verify this condition. For q = 2 the situation is much simpler. The only cases to consider are X = V (2) ⊂ P2 , X = V (3) ⊂ P2 , X = V (2, 2) ⊂ P3 . In the first case the conclusion of Proposition 18.2.1 trivially holds, since the target of the regulator map is zero. The remaining two cases are elliptic curves. Bloch [3] showed that the image of 2 c2,2 : CH2 (X, 2) → HD (X, Z(2)) 2 (PN , Z(2)) = 0, the result follows. is nonzero for elliptic curves. Since HD
The results in this note can be summarised as follows. Theorem 18.4.2. Let X be a smooth complete intersection in PN with inclusion map i : X → PN . 1) If dim X = 2m and X is very general, the image of the regulator map 2m+1 N 2m+1 (P , Z(m + 1)) (X, Z(m + 1))/i∗ HD cm+1,1 : CHm+1 (X, 1) → HD
is a torsion group, with the possible exception of the cases i) X = V (2) ⊂ P2m+1 , X = V (d) ⊂ P3 (d ≤ 4), X = V (3) ⊂ P5 ; ii) X = V (2, 2) ⊂ P2m+2 , X = V (3, 2) ⊂ P2m+2 , m ≥ 1; iii) X = V (2, 2, 2) ⊂ P5 . 4) If dim X = 2m − 1 and X is very general, the image of the map 2m+1 2m+1 N cm+1,2 : CHm+1 (X, 2) → HD (X, Z(m + 1))/i∗ HD (P , Z(m + 1))
is a torsion group, unless X is an elliptic curve.
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References [1] Asakura, M. and S. Saito: Generalized Jacobian rings for complete intersections preprint math.AG/0203147 [2] Beauville, A.: Determinantal hypersurfaces. Dedicated to William Fulton on the occasion of his 60th birthday Michigan Math. J. 48 (2000), 39–64. [3] Bloch, S.: Higher regulators, algebraic K-theory, and zeta functions of elliptic curves CRM Monograph Series. 11 Providence, RI: American Mathematical Society (2000) [4] Chen, X. and J. Lewis: The Hodge-D-conjecture for K3 and abelian surfaces J. Algebraic Geom. 14 (2005) 213–240. ¨ller-Stach and C. Peters: Period mappings and period [5] Carlson, J., S. Mu domains Cambridge Studies in Advanced Mathematics 85 Cambridge University Press, Cambridge (2003) [6] Collino, A.: Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds, in Algebraic K-theory and its applications (Trieste, 1997) 370–402, World Sci. Publishing, River Edge, NJ, (1999) [7] Cox, D.: Recent developments in toric geometry, Proc. Sympos. Pure Math., 62, Part 2 Amer. Math. Soc., Providence, RI (1997) 389–436 [8] Dixon, A. C.: Note on the reduction of a ternary quartic to a symmetric determinant Proc. Camb. Phil. Soc. 11 (1902) 350–351 [9] Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics 52 Springer-Verlag, New York-Heidelberg (1977) ¨ller-Stach: The Abel-Jacobi map for higher [10] Kerr, M., J. Lewis and S. Mu Chow groups preprint math.AG/0409116. [11] Laszlo, Y.: Th´eor`eme de Torelli g´en´erique pour les intersections compl`etes de trois quadriques de dimension paire Invent. Math. 98 (1989) 247-264 [12] Lazarsfeld, R.: A Barth-type theorem for branched coverings of projective space Math. Ann. 249 (1980) 153–162 ¨ller–Stach, S.: Syzygies and the Abel-Jacobi map for cyclic coverings [13] Mu Manuscripta Math 82 (1994) 433–443 ¨ller–Stach, S.: Constructing indecomposable motivic cohomology classes [14] Mu on algebraic surfaces J. Algebraic Geom. 6 (1997) 513–543 ¨ller-Stach, S.: Algebraic cycle complexes: Basic properties, in The arith[15] Mu metic and geometry of algebraic cycles Vol. 1 Gordon, B. Brent (ed.) et al., Kluwer Academic Publishers (2000) 285-305 [16] Nagel, J.: Effective bounds for Hodge–theoretic connectivity J. Alg. Geom. 11 (2002) 1–32. [17] Nagel, J.: The image of the regulator map for complete intersections of three quadrics preprint MPI 03-46, 2003 [18] Nori, M.H.: Algebraic cycles and Hodge-theoretic connectivity Invent. Math. 111 (1993) 349–373 [19] nag:O’Grady, K.: The Hodge structure of the intersection of three quadrics in an odd-dimensional projective space Math. Ann. 273 (1986) 277–285. [20] Reid, M.: The intersection of two quadrics, Ph.D. Thesis, Cambridge 1972 (unpublished). Available at www.maths.warwick.ac.uk/ miles/3folds/qu.ps. [21] Voisin, C.: Variations of Hodge structure and algebraic cycles, in Proceedings ICM ’94, Vol. I, Chatterji, S. D. (ed.) Basel: Birkh¨auser (1995) 706-715 [22] Voisin, C.: Remarks on zero-cycles of self-products of varieties, in Moduli of vector bundles Maruyama, Masaki (ed.), New York, NY, Marcel Dekker. Lect. Notes Pure Appl. Math. 179 (1996) 265-285
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[23] Voisin, C.: Nori’s connectivity theorem and higher Chow groups. J. Inst. Math. Jussieu 1 (2002) 307–329.
19 Hodge Number Polynomials for Nearby and Vanishing Cohomology C.A.M. Peters Department of Mathematics, University of Grenoble UMR 5582 CNRS-UJF, 38402-Saint-Martin d’H`eres, France
[email protected]
J.H.M. Steenbrink Institute for Mathematics,Astrophysics and Particle Physics, Radboud University Nijmegen Toernooiveld, NL-6525 ED Nijmegen, The Netherlands
[email protected]
Introduction The behaviour of the cohomology of a degenerating family of complex projective manifolds has been intensively studied in the nineteen-seventies by Clemens, Griffiths, Schmid and others. See [Gr] for a nice overview. Recently, the theory of motivic integration, initiated by Kontsevich and developed by Denef and Loeser, has given a new impetus to this topic. In particular, in the case of a one-parameter degeneration it has produced an object ψf in the Grothendieck group of complex algebraic varieties, called the motivic nearby fibre [B05], which reflects the limit mixed Hodge structure of the family in a certain sense. The purpose of this paper is twofold. First, we prove that the motivic nearby fibre is well-defined without using the theory of motivic integration. Instead we use the Weak Factorization Theorem [AKMW]. Second, we give a survey of formulas containing numerical invariants of the limit mixed Hodge structure, and in particular of the vanishing cohomology of an isolated hypersurface singularity, without using the theory of mixed Hodge structures or of variations of Hodge structure. We hope that in this way this interesting topic becomes accessible to a wider audience.
19.1 Real Hodge structures A real Hodge structure on a finite dimensional real vector space V consists of a direct sum decomposition M VC = V p,q , with V p,q = V q,p p,q∈Z
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on its complexification VC = V ⊗ C. The corresponding Hodge filtration is given by M V r,s . F p (V ) = r≥p
The numbers hp,q (V ) := dim V p,q are the Hodge numbers of the Hodge structure. If for some integer k we have hp,q = 0 for all (p, q) with p + q 6= k the Hodge structure is pure of weight k. Any real Hodge structure is the direct sum of pure Hodge structures. The polynomial X Phn (V ) = hp,q (V )up v q (19.1) p,q∈Z
=
X
hp,k−p (V )up v k−p ∈ Z[u, v, u−1 , v −1 ]
is its associated Hodge number polynomial. † A classical example of a weight k Hodge real structure is furnished by the rank k (singular) cohomology group H k (X) (with R- coefficients) of a compact K¨ahler manifold X. Various multilinear algebra operations can be applied to Hodge structures as we now explain. Suppose that V and W are two real vector spaces with a Hodge structure of weight k and ` respectively. Then: (i) V ⊗ W has a Hodge structure of weight k + ` given by X F p (V ⊗ W )C = F m (VC ) ⊗ F p−m (WC ) ⊂ VC ⊗C WC m
and with Hodge number polynomial given by Phn (V ⊗ W ) = Phn (V )Phn (W ).
(19.2)
(ii) On Hom(V, W ) we have a Hodge structure of weight ` − k : F p Hom(V, W )C = {f : VC → WC | f F n (VC ) ⊂ F n+p (WC ) ∀n} with Hodge number polynomial Phn (Hom(V, W )(u, v) = Phn (V )(u−1 , v −1 )Phn (W )(u, v).
(19.3)
In particular, taking W = R with WC = W 0,0 we get a Hodge structure of weight −k on the dual V ∨ of V with Hodge number polynomial Phn (V ∨ )(u, v) = Phn (V )(u−1 , v −1 ).
(19.4)
† There are other conventions in the literature, for instance, some authors put a sign (−1)p+q in front of the coefficient hpq, (V ) of up v q .
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The category hs of real Hodge structures leads to a ring, the Grothendieck ring K0 (hs) which is is the free group on the isomorphism classes [V ] of real Hodge structures V modulo the subgroup generated by [V ] − [V 0 ] − [V 00 ] where 0 → V 0 → V → V 00 → 0 is an exact sequence of pure Hodge structures and where the complexified maps preserve the Hodge decompositions. Because the Hodge number polynomial(19.1) is clearly additive and by(19.2) behaves well on products the Hodge number polynomial defines a ring homomorphism Phn : K0 (hs) → Z[u, v, u−1 , v −1 ]. As remarked before, pure Hodge structures of weight k in algebraic geometry arise as the (real) cohomology groups H k (X) of smooth complex projective varieties. We combine these as follows: X χHdg (X) := (−1)k [H k (X)] ∈ K0 (hs); (19.5) X eHdg (X) := (−1)k Phn (H k (X)) ∈ Z[u, v, u−1 , v −1 ] (19.6) which we call the Hodge-Grothendieck character and the Hodge-Euler polynomial of X respectively. Let us now recall the definition of the naive Grothendieck group K0 (Var) of (complex) algebraic varieties. It is the quotient of the free abelian group on isomorphism classes [X] of algebraic varieties over C with the so-called scissor relations [X] = [X − Y ] + [Y ] for Y ⊂ X a closed subvariety. The cartesian product is compatible with the scissor relations and induces a product structure on K0 (Var), making it into a ring. There is a nice set of generators and relations for K0 (Var). To explain this we first recall: Lemma 19.1.1. Suppose that X is a smooth projective variety and Y ⊂ X is a smooth closed subvariety. Let π : Z → X be the blowing-up with centre Y and let E = π −1 (Y ) be the exceptional divisor. Then χHdg (X) − χHdg (Y ) = χHdg (Z) − χHdg (E); eHdg (X) − eHdg (Y ) = eHdg (Z) − eHdg (E). Proof By [GH, p. 605] 0 → H k (X) → H k (Z) ⊕ H k (Y ) → H k (E) → 0 is exact.
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Theorem 19.1.2 ([B04, Theorem 3.1]). The group K0 (Var) is isomorphic to the free abelian group generated by the isomorphism classes of smooth complex projective varieties subject to the relations [∅] = 0 and [Z] − [E] = [X] − [Y ] where X, Y, Z, E are as in Lemma 19.1.1. It follows that for every complex algebraic variety X there exist projective smooth varieties X1 , . . . , Xr , Y1 , . . . , Ys such that X X [X] = [Xi ] − [Yj ] in K0 (Var) i
j
and so, using Lemma 19.1.1 we have: Corollary 19.1.3. The Hodge Euler character extends to a ring homomorphism χHdg : K0 (Var) → K0 (hs) and the Hodge number polynomial extends to a ring homomorphism eHdg : K0 (Var) → Z[u, v, u−1 , v −1 ] Remark 19.1.4. By Deligne’s theory [Del71], [Del74] there is a mixed Hodge structure on the real vector spaces H k (X). For our purposes, since we are working with real coefficients, a mixed Hodge structure is just a real Hodge structure, i.e. a direct sum of real Hodge structures of various weights, and so the Hodge character and Hodge number polynomial are defined for any real mixed Hodge structure. However, ordinary cohomology does not behave well with respect to the scissor relation; we need compactly supported cohomology Hck (X; R). But these also carry a Hodge structure and we have the following explicit expression for the above characters. X χHdg (X) = (−1)k [Hck (X))]; X eHdg (X) = (−1)k Phn (Hck (X)). Example 19.1.5. 1) Let U be a smooth, but not necessarily compact complex algebraic manifold. Such a manifold has a good compactification X, i.e. X is a compact complex algebraic manifold and D = X − U is a normal crossing divisor, say D = D1 ∪ · · · DN with Dj smooth and irreducible. We introduce DI
= Di1 ∩ Di2 ∩ · · · ∩ Dim ,
aI
:
DI ,→ X
I = {i1 , . . . , im };
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and we set D(0) = X; ` D(m) = |I|=m DI , m = 1, . . . , N ; ` am = |I|=m aI : D(m) → X. Then each connected component of D(m) is a complex submanifold of X of codimension m. Note that X [U ] = (−1)m [D(m)] ∈ K0 (Var). m
Hence χHdg (U ) =
X
eHdg (U ) =
X
(−1)m χHdg (D(m));
m
(−1)m eHdg (D(m)).
m
2) If X is compact the construction of cubical hyperresolutions (XI )∅6=I⊂A of X from [GNPP] leads to the expression X [X] = (−1)|I|−1 [XI ]. ∅6=I⊂A
and we find: χHdg (X) =
X
(−1)|I|−1 χHdg (XI );
∅6=I⊂A
eHdg (X) =
X
(−1)|I|−1 eHdg (XI ).
∅6=I⊂A
The scissor-relations imply that the inclusion-exclusion principle can be applied to a disjoint union X of locally closed subvarieties X1 , . . . , Xm : χHdg (X) =
m X
χHdg (Xi )
i=1
and a similar expression holds for the Hodge Euler polynomials. Example 19.1.6. Let T n = (C∗ )n be an n- dimensional algebraic torus. Then eHdg (T 1 ) = uv−1 so eHdg (T n ) = (uv−1)n . Consider an n- dimensional toric variety X. It is a disjoint union of T n - orbits.Suppose that X has sk orbits of dimension k. Then eHdg (X) =
n X k=0
sk eHdg (T k ) =
n X k=0
sk (uv − 1)k .
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If X has a pure Hodge structure (e.g. if X is compact and has only quotient singularities) then this formula determines the Hodge numbers of X.
19.2 Nearby and vanishing cohomology In this section we consider a relative situation. We let X be a complex manifold, ∆ ⊂ C the unit disk and f : X → ∆ a holomorphic map which is smooth over the punctured disk ∆∗ . We say that f is a one-parameter S degeneration. Let us assume that E = i∈I Ei = f −1 (0) is a divisor with strict normal crossings on X. We have the specialization diagram k
X∞ −→ ˜ yf e h − →
i
X − ← f y ∆ ← −
E y {0}
where h is the complex upper half plane, e(z) := exp(2πiz) and where X∞ := X ×∆∗ h. We let ei denote the multiplicity of f along Ei and choose a positive integer ˜ → ∆ denote the normalization of the multiple e of all ei . We let f˜ : X pull-back of X under the map µe : ∆ → ∆ given by τ 7→ τ e = t. It fits into a commutative diagram describing the e-th root of f : ρ
˜ X f˜ y
− →
∆
−−→
µe
X f y ∆.
We put Di = ρ−1 (Ei ) , DJ =
\ i∈J
Dj , D(m) =
a
DJ .
|J|=m
Then DJ → EJ is a cyclic cover of degree gcd(ej | j ∈ J). The maps DJ → EJ do not depend on the choice of the integer e and so in particular this is true for the varieties DJ . See e.g. [Ste77] or [B05] for a detailed study ˜ 0 = f˜−1 (0) is now a of the geometry of this situation. The special fibre X complex variety equipped with the action of the cyclic group of order e. Let us introduce the associated Grothendieck-group: Definition 19.2.1. We let Kµ0ˆ (Var) denote the Grothendieck group of complex algebraic varieties with an action of a finite order automorphism modulo the subgroup generated by expressions [P(V )] − [Pn × X] where V is a vector
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bundle of rank n + 1 over X with action which is linear over the action on X. See [B05, Sect. 2.2] for details. As a motivation, we should remark that in the ordinary Grothendieck group the relation [P(V )] = [Pn ×X] follows from the fact that any algebraic vector bundle is trivial over a Zariski-open subset and the fact that the scissor-relations hold. The above relation extend to the case where one has a group action. Definition 19.2.2. Suppose that the fibres of f are projective varieties. Following [B05, Ch. 2] we define the motivic nearby fibre of f by X (−1)m−1 [D(m) × Pm−1 ] ∈ Kµ0ˆ (Var) ψf := m≥1
and the motivic vanishing fibre by φf := ψf − [E] ∈ Kµ0ˆ (Var)
(19.7)
Remark 19.2.3. If we let EI0 be the open subset of EI consisting of points which are exactly on the Ej with j ∈ I, DI0 the corresponding subset of DI ` ` and D0 (i) = |I|=i DI0 . We have DI = J⊃I DJ0 and the scissor relations imply that X X X j [D0 (j) × Pi−1 ] (−1)i−1 [D(i) × Pi−1 ] = (−1)i−1 i i≥1
i≥1
=
X
=
X
j≥i
[D0 (j)] ×
j X i=1
j≥1 j−1
(−1)
(−1)i−1
j [Pi−1 ] i
[D0 (j) × (C∗ )j−1 ].
j≥1
This expression for the motivic nearby fibre has been used in [L]. The motivic nearby fibre turns out to be a relative bimeromorphic invariant: Lemma 19.2.4. Suppose that g : Y → X is a bimeromorphic proper map which is an isomorphism over X − E. Assume that (f ◦ g)−1 (0) is a divisor with strict normal crossings. Then ψf = ψf g . Proof In [B05], the proof relies on the theory of motivic integration [DL]. We give a different proof, based on the weak factorization theorem [AKMW].
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This theorem reduces the problem to the following situation: g is the blowingup of X in a connected submanifold Z ⊂ E, with the following property. Let A ⊂ I be those indices i for which Z ⊂ Ei . Then Z intersects the divisor S S i6∈A Ei transversely, hence Z ∩ i6∈A Ei is a divisor with normal crossings in Z. P i We fix the following notation. We let L := [A1 ] and Pm := [Pm ] = m i=0 L . For J ⊂ I we let j = |J|. So, using the product structure in K0 (Var), we have X ψf = (−1)j−1 [DJ ]Pj−1 . ∅6=J⊂I
S Let E 0 = i∈I 0 Ei0 be the zero fibre of f g. We have I = I 0 ∪ {∗} where E∗0 is the exceptional divisor of g and Ei0 is the proper transform in Y of Ei . Form ρ0 : D0 → E 0 , the associated ramified cyclic covering. For J ⊂ I we let P J 0 = J ∪ {∗}. Note that E∗0 has multiplicity equal to i∈A ei . Without loss of generality we may assume that e is also a multiple of this integer. We have two kinds of j 0 - uple intersections DJ 0 : those which only contain Dj0 , j 6= ∗ and those which contain D∗0 . So, X ψf g = [D∗0 ] + (−1)j−1 [DJ0 ]Pj−1 − [DJ0 0 ]Pj . ∅6=J⊂I
We are going to calculate the difference between ψf g and ψf . Let B ⊂ I − A. Let c = codim(Z, X). Then for all K ⊂ A we have that codim(Z ∩ 0 is the blowing up of DK∪B with centre EK∪B , EK∪B ) = c − k, and DK∪B 0 Z ∩ EK∪B = Z ∩ EB =: ZB and exceptional divisor DK∪B 0 . Hence, if we let −1 0 WB = ρ (ZB ), we have 0 0 [DK∪B ] = [DK∪B ] + [DK∪B 0 ] − [WB ] = [DK∪B ] + [WB ](Pc−k−1 − 1).
Hence ψf g − ψf =
X
cB [WB ]
B
for suitable coefficients cB . Note that Z∅ = Z and that D∗0 = W∅ × Pc−1 and so if B = ∅ we get X c∅ = Pc−1 + (−1)k−1 ((Pc−k−1 − 1)Pk−1 − Pc−k−1 Pk ) ∅6=K⊂A
= Pc−1 +
X
(−1)k (Lk Pc−k−1 + Pk−1 ) = Pc−1
∅6=K⊂A
X
(−1)k = 0.
K⊂A
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In case B 6= ∅ we get X cB = (−1)k+b−1 ((Pc−k−1 − 1)Pk+b−1 − Pc−k−1 Pk+b ) K⊂A
=
X
(−1)k+b (Lk+b Pc−k−1 + Pk+b−1 ) = Pc+b−1
K⊂A
X
(−1)b+k = 0.
K⊂A
Let us now pass to the nearby fibre in the Hodge theoretic sense. In [Schm] and [Ste76] a mixed Hodge structure on H k (X∞ ) was constructed; its weight filtration is the monodromy weight filtration which we now explain. The loop winding once counterclockwise around the origin gives a generator of π(∆∗ , ∗) , ∗ ∈ ∆∗ . Its action on the fibre X∗ over ∗ is well defined up to homotopy and on H k (X∗ ) ' H k (X∞ ) it defines the monodromy automorphism T . Let N = log Tu be the logarithm of the unipotent part in the Jordan decomposition of T . Then W is the unique increasing filtration on H k (X∞ ) W such that N (Wj ) ⊂ Wj−2 and N j : GrW k+j → Grk−j is an isomorphism for all j ≥ 0. In fact, from [Schm, Lemma 6.4] we deduce: Lemma 19.2.5. There is a Lefschetz-type decomposition W
Gr
k
H (X∞ ) =
k M ` M
N r Pk+` ,
`=0 r=0
where Pk+` is pure of weight k + `. The endomorphism N has dim Pk+m−1 Jordan blocs of size m. The Hodge filtration is constructed in [Schm] as a limit of the Hodge filtrations on nearby smooth fibres in a certain sense, and in [Ste76] using the relative logarithmic de Rham complex. With Xt a smooth fibre of f it follows that dim F m H k (Xt ) = dim F m H k (X∞ ) and hence eHdg (X∞ )|v=1 = eHdg (Xt )|v=1 .
(19.8)
We next remark that the nearby cycle sheaf i∗ Rk∗ k ∗ RX can be used to put a Hodge structure on the cohomology groups H k (X∞ ), while the hypercohomology of the vanishing cycle sheaf φf = Cone[i∗ RX → ψf RX ] (it is a complex of sheaves of real vector spaces on E) likewise admits a Hodge structure. In fact, the spectral sequence of [Ste76, Cor. 4.20] shows that χHdg (X∞ ) = χHdg (ψf )
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and then the definition shows that χHdg (X∞ ) − χHdg (E) = χHdg (φf ). In fact this formula motivates the nomenclature “motivic nearby fibre” and “motivic vanishing cycle”. As a concluding remark, the semisimple part Ts of the monodromy is an automorphism of the mixed Hodge structure on H k (X∞ ). We can use these remarks to deduce information about the Hodge numbers on H k (X∞ ) from information about the geometry of the central fibre as we shall illustrate now. Example 19.2.6. 1) Let F, L1 , . . . , Ld ∈ C[X0 , X1 , X2 ] be homogeneous forms with deg F = d and deg Li = 1 for i = 1, . . . , d, such that F · L1 · · · Ld = 0 defines a reduced divisor with normal crossings on P2 (C). We consider the space X = {([x0 , x1 , x2 ], t) ∈ P2 × ∆ |
d Y
Li (x0 , x1 , x2 ) + tF (x0 , x1 , x2 ) = 0}
i=1
where ∆ is a small disk around 0 ∈ C. Then X is smooth and the map f : X → ∆ given by the projection to the second factor has as its zero fibre the union E1 ∪ · · · ∪ Ed of the lines Ei with equation Li = 0. These lines are in general position and have multiplicity one. We obtain d ψf = (d − )[P1 ] 2 so
d−1 eHdg (ψf ) = (1 − )(1 + uv) 2 and substituting v = 1 in this formula we get the formula g = d−1 2 for the genus of a smooth plane curve of degree d. The monodromy on H 1 (X∞ ) has g Jordan blocks of size 2, so is “maximally unipotent”. 2) If we consider a similar example, but replace P2 by P3 and curves by surfaces, lines by planes, then the space X will not be smooth but has ordinary double points at the points of the zerofibre where two of the planes meet the surface F = 0. There are d d2 of such points, d on each line of intersection. If we blow these up, we obtain a family ˜ → ∆ whose zero fibre D = E ∪ F is the union of components f :X Ei , i = 1, . . . , d which are copies of P2 blown up in d(d − 1) points,
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which are copies of P1 × P1 . Thus d 2 2 2 eHdg (D(1)) = d(1 + (d − d + 1)uv + u v ) + d (1 + uv)2 . 2 The double point locus D(2) consists of the d2 lines of intersections of the Ei together with the d2 (d − 1) exceptional lines in the Ei . So and components Fj , j = 1, . . . , d
1 eHdg (D(2)) = d(d − 1)(d + )(1 + uv). 2 Finally D(3) consists of the d3 intersection points of the Ei together with one point on each component Fj , so d d 1 eHdg (D(3)) = +d = d(d − 1)(2d − 1) 3 3 2 We get eHdg (ψf ) =
d−1 1 + 1 (1 + u2 v 2 ) + d(2d2 − 6d + 7)uv 3 3
in accordance with the Hodge numbers for a smooth degree d surface: d−1 1 2,0 0,2 h =h = , h1,1 = d(2d2 − 6d + 7) 3 3 d−1 The monodromy on H 2 (X∞ ) has Jordan blocs of size 3 and 3 1 3 1 2 2 d − d + 2 d + 1 blocks of size 1. 3) Consider a similar smoothing of the union of two transverse quadrics in P3 . The generic fibre is a smooth K3-surface and after blowing up the 16 double points of the total space we obtain the following special fibre: (i) E(1) has two components which are blowings up of P1 × P1 in 16 points, and 16 components isomorphic to P1 × P1 ; hence eHdg (E(1)) = 18(1 + uv)2 + 32uv. (ii) E(2) consists of the 32 exceptional lines together with the strict transform of the intersection of the two quadrics, which is an elliptic curve; hence eHdg (E(2)) = 33(1 + uv) − u − v ; (iii) E(3) consists of 16 points: one point on each exceptional P1 × P1 , so eHdg (E(3)) = 16. We get eHdg (X∞ ) = 1 + u + v + 18uv + u2 v + uv 2 + u2 v 2
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Putting v = 1 we get 2 + 20u + 2u2 , in agreement with the Hodge numbers (1, 20, 1) on the H 2 of a K3-surface. The monodromy has two Jordan blocs of size 2 and 18 blocs of size 1.
19.3 Equivariant Hodge number polynomials We have seen that the mixed Hodge structure on the cohomology of the nearby fibre of a one-parameter degeneration comes with an automorphism of finite order. This leads us to consider the category hsµRˆ of pairs (H, γ) consisting of a real Hodge structure (i.e. direct sum of pure real Hodge structures of possibly different weights) H and an automorphism γ of finite order of this Hodge structure. We are going to consider a kind of tensor product of two such objects, which we call convolution (see [SchS], where this operation was defined for mixed Hodge structures and called join). We will explain this by settling an equivalence of categories between hsµRˆ and a category fhs of so-called fractional Hodge structures. (called Hodge structures with fractional weights in [L]; it is however not the weights which are fractional, but the indices of the Hodge filtration! Definition 19.3.1. (See [L]). A fractional Hodge structure of weight k is a real vector space H of finite dimension, equipped with a decomposition M HC = H a,b a+b=k
where a, b ∈ Q, such that H b,a = H a,b . A fractional Hodge structure is defined as a direct sum of pure fractional Hodge structures of possibly different weights. Lemma 19.3.2. We have an equivalence of categories G : hsµˆ → fhs. Proof Let (H, γ) be an object of hsµˆ pure of weight k. We define Ha = Ker(γ − exp(2πia); HC ) for 0 ≤ a < 1 and for 0 < a < 1 put ˜ p+a,k+1−a−p = Hap,k−p , H
˜ p,k−p = H p,k−p H 0
˜ =: G(H, γ) of fractional Hodge This transforms (H, γ) into a direct sum H structures of weights k + 1 and k respectively. Conversely, for a fractional ˜ of weight k one has a unique automorphism γ of finite Hodge structure H ˜ b,k−b . order which is multiplication by exp(2πib) on H Note that this equivalence of categories does not preserve tensor products! Hence it makes sense to make the following
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Definition 19.3.3. The convolution (H 0 , γ 0 ) ∗ (H”, γ”) of two objects in hsµˆ is the object corresponding to the tensor product of their images in fhs : G (H 0 , γ 0 ) ∗ (H 00 , γ 00 ) = G(H 0 , γ 0 ) ⊗ G(H 00 , γ 00 ). Note that the Hodge number polynomial map Phn extends to a ring homomorphism 1
1
µ ˆ Phn : K0 (fhs) → R := lim Z[u n , v n , u−1 , v −1 ] ←
We denote its composition with the functor G by the same symbol. Hence µ ˆ Phn : K0µˆ (hs) → R
transforms convolutions into products. We equally have an equivariant Hodge-Grothendieck character ˆ : K0µˆ (Var) → K0µˆ (hs) χµHdg
and an equivariant Hodge-Euler characteristic ˆ : K0µˆ (Var) → R. eµHdg
Let f : X → C be a projective morphism where X is smooth, of relative dimension n, with a single isolated critical point x such that f (x) = 0. Construct ψf by replacing the zero fibre X0 by a divisor with normal crossings as above. Then the Milnor fibre of F of f at x has the homotopy type of a wedge of spheres of dimension n. Its cohomology is also equipped with a mixed Hodge structure. Recalling definition 19.7 of the motivic vanishing fibre, it can be shown that µ ˆ ˜n ˆ Phn (H (F )) = (−1)n eµHdg (φf ).
P µ ˆ ˜n (H (F )) = α∈Q, w∈Z m(α, w)uα v w−α . In the literature several Write Phn numerical invariants have been attached to the singularity f : (X, x) → (C, 0). These are all related to the numbers m(α, w) as follows: i) The characteristic pairs [Ste77, Sect. 5]. X Chp(f, x) = m(α, w) · (n − α, w). α,w
ii) The spectral pairs [N-S]: X X Spp(f, x) = m(α, w) · (α, w) + m(α, w) · (α, w + 1). α6∈Z,w
α∈Z,w
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iii) The singularity spectrum in Saito’s sense [Sa]: µ ˆ ˜n SpSa (f, x) = Phn (H (F ))(t, 1).
iv) The singularity spectrum in Varchenko’s sense [Var]: µ ˆ ˜n SpV (f, x) = t−1 Phn (H (F ))(t, 1).
Note that the object φf depends only on the germ of f at the critical point x. Let us as a final remark rephrase the original Thom-Sebastiani theorem (i.e. for the case of isolated singularities): Theorem 19.3.4. Consider holomorphic germs f : (Cn+1 , 0) → (C, 0) and g : (Cm+1 , 0) → (C, 0) with isolated singularity. Then the germ f ⊕ g : (Cn+1 × Cm+1 , (0, 0)) → C with (f ⊕ g)(x, y) := f (x) + g(y) has also an isolated singularity, and ˆ ˆ ˆ (φg ) (φf ) ∗ χµHdg (φf ⊕g ) = −χµHdg χµHdg
so ˆ ˆ ˆ (φg ) ∈ R. (φf ) · eµHdg (φf ⊕g ) = −eµHdg eµHdg
Remark 19.3.5. This theorem has been largely generalized, for functions with arbitrary singularities, and even on the level of motives. Denef and Loeser [DL] defined a convolution product for Chow motives, and Looijenga [L] defined one on Mµˆ = K0µˆ (Var)[L−1 ], both with the property that the Hodge Euler polynomial commutes with convolution and that the ThomSebastiani property holds already on the level of varieties/motives.
References [AKMW] Abramovich, D., K. Karu, K. Matsuki and J. Wlodarczyk: Torification and factorization of birational maps, J. Amer. Math. Soc. 15 531572 (2002). [B04] Bittner, F.: The universal Euler characteristic for varieties of characteristic zero, Comp. Math. 140, 1011-1032 (2004) [B05] Bittner, F.: On motivic zeta functions and the motivic nearby fibre, Math. Z. 249, 63–83 (2005) [Del71] Deligne, P.: Th´eorie de Hodge II, Publ. Math. I.H.E.S, 40, 5–58 (1971) [Del74] Deligne, P.: Th´eorie de Hodge III, Publ. Math., I. H. E. S, 44, 5-77 (1974) [DK] Danilov, V., and A. Khovanski: Newton polyhedra and an algorithm for computing Hodge-Deligne numbers Math., U. S. S. R. Izvestia, 29, 274–298 (1987) [DL] Denef, J. and F. Lœser: Motivic exponential integrals and a motivic ThomSebastiani theorem, Duke Math. J. 99, 285–309 (1999) [Gr] Griffiths, Ph. ed.: Topics in Transcendental Geometry, Annals of Math. Studies 106, Princeton Univ. Press 1984.
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[GH] Griffiths, Ph. and J. Harris: Principles of Algebraic Geometry, Wiley 1978. ´n, F. and V. Navarro Aznar: Sur le th´eor`eme local des cycles [GN] Guille invariants, Duke Math. J., 61, 133–155 (1990) ´n, F., V. Navarro Aznar, P. Pascual-Gainza and F. Puerta: [GNPP] Guille Hyperr´esolutions cubiques et descente cohomologique, Springer Lecture Notes in Math., 1335, (1988) [L] Looijenga, E.: Motivic measures. S´eminaire Bourbaki, 52`eme ann´ee, 19992000, no. 874 [N-S] Nemethi, A. and J. H. M. Steenbrink: Spectral Pairs, Mixed Hodge Modules, and Series of Plane Curve Singularities, New York J. Math. 1 , 149–177 (1995) [Sa] Saito, M.: On the exponents and the geometric genus of an isolated hypersurface singularity. AMS Proc. Symp. Pure Math. 40 Part 2, 465–472 (1983) [Schm] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping, Invent. Math., 22, 211–319 (1973) [SchS] Scherk, J. and J. H. M. Steenbrink: On the Mixed Hodge Structure on the Cohomology of the Milnor Fibre, Mathematische Annalen 271, 641–665 (1985) [Ste76] Steenbrink, J. H. M.: Limits of Hodge structures, Inv.Math., 31, 229–257 (1976) [Ste77] Steenbrink, J. H. M.: Mixed Hodge structures on the vanishing cohomology, in Real and Complex Singularities, Oslo, 1976, Sijthoff-Noordhoff, Alphen a/d Rijn, 525–563 (1977) [Var] Varchenko, A. N.: Asymptotic mixed Hodge structure in the vanishing cohomology. Izv. Akad. Nauk SSSR, Ser. Mat. 45, 540–591 (1981) (in Russian). [English transl.: Math. USSR Izvestija, 18:3, 469–512 (1982)]
20 Direct Image of Logarithmic Complexes and Infinitesimal Invariants of Cycles Morihiko Saito RIMS Kyoto University, Kyoto 606-8502 Japan
[email protected]
Abstract We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered D-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered logarithmic complexes with coefficients, and this is essentially equivalent to the cohomology class of the cycle.
Introduction Let X, S be complex manifolds or smooth algebraic varieties over a field of characteristic zero. Let f : X → S be a projective morphism, and D be a divisor on S such that f is smooth over S − D. We have a filtered locally free O-module (V i , F ) on S − D underlying a variation of Hodge structure whose fiber Vsi at s ∈ S − D is the cohomology of the fiber H i (Xs , C). If D is a divisor with normal crossings on S, let Ve i denote the Deligne extension [7] of V i such that the the eigenvalues of the residue of the connection are contained in [0, 1). The Hodge filtration F is naturally extended to Ve i by [25]. We have the logarithmic de Rham complex DRlog (Ve i ) = Ω•S (log D) ⊗O Ve i , which has the Hodge filtration F p defined by ΩjS (log D) ⊗O F p−j Ve i . In general, V i can be extended to a regular holonomic DS -module M i on which a local defining equation of D acts bijectively. By [23], M i and hence the de Rham complex DR(M i ) have the Hodge filtration F . If Y := f ∗ D is a divisor with normal crossings on X, then Ω•X (log Y ) has the Hodge filtration F de• fined by the truncation σ (see [8]) as usual, i.e. F p Ω•X (log Y ) = ΩX≥p (log Y ). 609
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Theorem 1. Assume Y = f ∗ D is a divisor with normal crossings. There is an increasing split filtration L on the filtered complex Rf∗ (Ω•X (log Y ), F ) such that we have noncanonical and canonical isomorphisms in the filtered derived category: L Rf∗ (Ω•X (log Y ), F ) ' i∈Z (DR(M i ), F )[−i], • i GrL i Rf∗ (ΩX (log Y ), F ) = (DR(M ), F )[−i].
If D is a divisor with normal crossings, we have also L Rf∗ (Ω•X (log Y ), F ) ' i∈Z (DRlog (Ve i ), F )[−i], GrL Rf∗ (Ω• (log Y ), F ) = (DRlog (Ve i ), F )[−i]. i
X
This follows from the decomposition theorem (see [2]) extended to the case of the direct image of (OX , F ) as a filtered D-module, see [22]. Note that Hodge modules do not appear in the last statement if D is a divisor with normal crossings. The assertion becomes more complicated in the non logarithmic case, see Remark (i) in 20.2.5. A splitting of the filtration L is given by choosing the first noncanonical isomorphism in the filtered decomposition theorem, see (20.12). A canonical choice of the splitting is given by choosing an relatively ample class, see [9]. Let CHp (X − Y, n) be Bloch’s higher Chow group, see [3]. In the analytic case, we assume for simplicity that f : (X, Y ) → (S, D) is the base change of a projective morphism of smooth complex algebraic varieties f 0 : (X 0 , Y 0 ) → 0 , and an (S 0 , D0 ) by an open embedding of complex manifolds S → San p p element of CH (X − Y, n) is the restriction of an element of CH (X 0 − Y 0 , n) to X − Y . If n = 0, we may assume that it is the restriction of an analytic cycle of codimension p on X. From Theorem 1, we can deduce Corollary 1. With the above notation and assumption, let ξ ∈ CHp (X − Y, n). Then, choosing a splitting of the filtration L in Theorem 1 (or more precisely, choosing the first noncanonical isomorphism in the filtered decomposition theorem (20.12), we have the total infinitesimal invariant L i δS,D (ξ) = (δS,D (ξ)) ∈ i≥0 Hi (S, F p DR(M 2p−n−i )), L i (resp. δ S,D (ξ) = (δ S,D (ξ)) ∈ i≥0 Hi (S, GrpF DR(M 2p−n−i )), ) i
i (ξ) (respectively. δ where δS,D S,D (ξ)) is independent of the choice of a splitj
j ting if the δS,D (ξ) (respectively. δ S,D (ξ)) vanish for j < i. In the case D is a divisor with normal crossings, the assertion holds with DR(M 2p−n−i ) replaced by DRlog (Ve 2p−n−i ).
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This shows that the infinitesimal invariants in [14, 13, 27, 5, 1, 24] can be defined naturally in the cohomology of filtered logarithmic complexes with coefficients in variations of Hodge structures if D is a divisor with normal crossings, see 20.2.4 for the compatibility with [1]. Note that if S is Stein or affine, then Hi (S, F p DRlog (Ve q )) is the i-th cohomology group of the complex whose j-th component is Γ(S, ΩjS (log D)⊗O F p−j Ve q ). If D is empty, then an i (ξ), δ i inductive definition of δS,D S,D (ξ) was given by Shuji Saito [24] using the filtered Leray spectral sequence together with the E2 -degeneration argument in [6]. He also showed that the infinitesimal invariants depend only on the cohomology class of the cycle. If S is projective, then it follows from [8] that i (ξ)) is equivalent to the cycle class of ξ the total infinitesimal invariant (δS,D 2p−n in HDR (X−Y ) by the strictness of the Hodge filtration, and the filtration L comes from the Leray filtration on the cohomology of X −Y , see Remark (iii) in 20.2.5. Corollary 1 is useful to study the behavior of the infinitesimal invariants i (ξ). near the boundary of the variety. If D is empty, let δSi (ξ) denote δS,D p i We can define δDR,S (ξ) as in [19] by omitting F before DR in Corollary 1 where D = ∅. Corollary 2. Assume S is projective. Let U = S − D. Then for each i ≥ 0, i (ξ), δ i i i δS,D S,D (ξ), δU (ξ) and δDR,U (ξ) are equivalent to each other, i.e. one of them vanishes if and only if the others do. i (ξ)). Morei (ξ)) is determined by (δUi (ξ)), and (δUi (ξ)) by (δS,D Indeed, (δDR,U i (ξ)) is equivalent to (δ i over, (δS,D DR,U (ξ)) by the strictness of the Hodge filtration [8] applied to (X, Y ) together with Theorem 1, see § 20.2.3. For the i relation with δ S,D (ξ), see § 20.2.1. Note that the equivalence between δUi (ξ) i and δDR,U (ξ) in the case of algebraic cycles (i.e. n = 0) was first found by J.D. Lewis and Shuji Saito in [19] (assuming a conjecture of Brylinski and Zucker and the Hodge conjecture and using an L2 -argument). The above arguments seem to be closely related with their question, see also Remark (i) in §20.2.5 below. As another corollary of Theorem 1 we have Corollary 3. Assume f induces an isomorphism over S − D, and Y = f ∗ D is a divisor with normal crossings on X. Then Ri f∗ ΩpX (log Y ) = 0
if i + p > dim X.
This follows immediately from Theorem 1 since M i = 0 for i 6= 0. Corollary 3 is an analogue of the vanishing theorem of Kodaira-Nakano. However,
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this does not hold for a non logarithmic complex (e.g. if f is a blow-up with a point center). This corollary was inspired by a question of A. Dimca. I would like to thank Dimca, Lewis and Shuji Saito for good questions and useful suggestions. In Section 20.1 , we prove Theorem 1 after reviewing some basic facts on filtered differential complexes. In Section 20.2 we explain the application of Theorem 1 to the infinitesimal invariants of (higher) cycles. In Section 20.3 we give some examples using Lefschetz pencils.
20.1 Direct image of logarithmic complexes 20.1.1 Filtered differential complexes Let X be a complex manifold or a smooth algebraic variety over a field of characteristic zero. Let Db F (DX ) (respectively. Db F (DX )r ) be the bounded derived category of filtered left (respectively. right) DX -modules. Let Db F (OX , Diff) be the bounded derived category of filtered differential complexes (L, F ) where F is exhaustive and locally bounded below (i.e. Fp = 0 for p 0 locally on X), see [22, 2.2], We have an equivalence of categories DR−1 : Db F (OX , Diff) → Db F (DX )r ,
(20.1)
whose quasi-inverse is given by the de Rham functor DRr for right Dmodules, see 20.1.2 below. Recall that, for a filtered OX -module (L, F ), the associated filtered right D-module DR−1 (L, F ) is defined by DR−1 (L, F ) = (L, F ) ⊗O (D, F ),
(20.2)
and the morphisms (L, F ) → (L0 , F ) in M F (OX , Diff) correspond bijectively to the morphisms of filtered D-modules DR−1 (L, F ) → DR−1 (L0 , F ). More precisely, the condition on (L, F ) → (L0 , F ) is that the composition Fp L → L → L0 → L0 /Fq L0 is a differential operator of order ≤ p − q − 1. The proof of (20.1) can be reduced to the canonical filtered quasi-isomorphism for a filtered right D-module (M, F ) DR−1 ◦DRr (M, F ) → (M, F ), which follows from a calculation of a Koszul complex. Note that the direct image f∗ of filtered differential complexes is defined by the sheaf-theoretic direct image Rf∗ , and this direct image is compatible
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with the direct image f∗ of filtered D-modules via (20.1), see [22, 2.3.], So we get Rf∗ = DRr ◦f∗ ◦DR−1 : Db F (OX , Diff) → Db F (OS , Diff),
(20.3)
where we use DRr for right D-modules (otherwise there is a shift of complex). 20.1.2 De Rham complex The de Rham complex DRr (M, F ) of a filtered right D-module (M, F ) is defined by V (DRr (M, F ))i = −i ΘX ⊗O (M, F [−i]) for i ≤ 0. (20.4) Here (F [−i])p = Fp+i in a compatible way with (F [−i])p = F p−i and Fp = F −p . Recall that the filtered right D-module associated with a filtered left D-module (M, F ) is defined by X (M, F )r := (Ωdim , F ) ⊗O (M, F ), X
(20.5)
X = 0 for p 6= − dim X. This induces an equivalence of catwhere GrFp Ωdim X egories between the left and right D-modules. The usual de Rham complex DR(M, F ) for a left D-module is defined by
(DR(M, F ))i = ΩiX ⊗O (M, F [−i])
for i ≥ 0,
(20.6)
and this is compatible with (20.4) via (20.5) up to a shift of complex, i.e. DR(M, F ) = DRr (M, F )r [− dim X].
(20.7)
20.1.3 Logarithmic complex Let X be as in § 20.1.1, and Y be a divisor with normal crossings on X. Let (V, F ) be a filtered locally free O-module underlying a polarizable variation of Hodge structure on X−Y . Let (Ve , F ) be the Deligne extension of (V, F ) to X such that the eigenvalues of the residue of the connection are contained in [0, 1). Then we have the filtered logarithmic de Rham complex DRlog (Ve , F ) such that F p of its i-th component is ΩiX (log Y ) ⊗ F p−i Ve . If (M, F ) = (OX , F ) with GrFp OX = 0 for p 6= 0, then DRlog (OX , F ) = (Ω•X (log X), F ). Let Ve (∗Y ) be the localization of Ve by a local defining equation of Y . This is a regular holonomic left DX -module underlying a mixed Hodge module,
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and has the Hodge filtration F which is generated by the Hodge filtration F on Ve , i.e. P Fp Ve (∗Y ) = ν ∂ ν F −p+|ν| Ve , Q where Fp = F −p and ∂ ν = i ∂iνi with ∂i = ∂/∂xi . Here (x1 , . . . , xn ) is a local coordinate system such that Y is contained in {x1 · · · xn = 0}. By [23, 3.11] we have a filtered quasi-isomorphism ∼ DRlog (Ve , F ) −→ DR(Ve (∗Y ), F ).
(20.8)
This generalizes the filtered quasi-isomorphism in [7] ∼
(Ω•X (log Y ), F ) −→ DR(OX (∗Y ), F ).
(20.9)
Note that the direct image of the filtered DX -module (Ve (∗Y ), F ) by X → pt in the case X projective (or proper algebraic) is given by the cohomology group of the de Rham complex DR(Ve (∗Y ), F ) (up to a shift of complex) by definition, and the Hodge filtration F on the direct image is strict by the theory of Hodge modules. So we get F p Hi (X − Y, DR(V )) := F p Hi (X, DR(Ve (∗Y ))) = Hi (X, F p DR(Ve (∗Y ))) = Hi (X, F p DRlog (Ve )).
(20.10)
20.1.4 Decomposition theorem Let f : X → S be a projective morphism of complex manifolds or smooth algebraic varieties over a field of characteristic zero. Then the decomposition theorem of Beilinson, Bernstein and Deligne [2] is extended to the case of Hodge modules ([22, 23]), and we have noncanonical and canonical isomorphisms ) L f∗ (OX , F ) ' j∈Z Hj f∗ (OX , F )[−j] in Db F (DS ), (20.11) L Hj f∗ (OX , F ) = Z⊂S (MZj , F ) in M F (DS ), where Z are irreducible closed analytic or algebraic subsets of S, and (MZj , F ) are filtered DS -modules underlying a pure Hodge module of weight j+dim X and with strict support Z, i.e. MZj has no nontrivial sub nor quotient module whose support is strictly smaller than Z. (Here M F (DS ) denotes the category of filtered left DS -modules.) Indeed, the second canonical isomorphism follows from the strict support decomposition which is part of the definition of pure Hodge modules, see [22, 5.1.6.]. The first noncanonical isomorphism follows from the strictness of the Hodge filtration and the relative hard Lefschetz theorem for the direct image (see [22, 5.3.1]) using the
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E2 -degeneration argument in [6] together with the equivalence of categories L Db F (DS ) ' Db G(BS ). Here BS = i∈N Fi DS and Db G(BS ) is the derived category of bounded complexes of graded left BS -modules M•• such that Mij = 0 for i 0 or |j| 0, see [22, 2.1.12]. We need a derived category associated to some abelian category in order to apply the argument in [6] (see also [9]). In the algebraic case, we can also apply [6] to the derived category of mixed Hodge modules on S and it is also possible to use [23, 4.5.4] to show the first noncanonical isomorphism. If f is smooth over the complement of a divisor D ⊂ S and Y := f ∗ D is a divisor with normal crossings, then the filtered direct image f∗ (OX (∗Y ), F ) is strict (see [23, 2.15]), and we have noncanonical and canonical isomorphisms ) L f∗ (OX (∗Y ), F ) ' j∈Z Hj f∗ (OX (∗Y ), F )[−j] in Db F (DS ), (20.12) Hj f∗ (OX (∗Y ), F ) = (MSj (∗D), F ) in M F (DS ). Here (MSj (∗D), F ) is the ‘localization’ of (MSj , F ) along D which is the direct image of (MSj , F )|U by the open embedding U := S − D → S in the category of filtered D-modules underlying mixed Hodge modules. (By the Riemann-Hilbert correspondence, this gives the direct image in the category of complexes with constructible cohomology because D is a divisor.) The Hodge filtration F on the direct image is determined by using the V -filtration of Kashiwara and Malgrange, and (MSj (∗D), F ) is the unique extension of (MSj , F )|U which underlies a mixed Hodge module on S and whose underlying DS -module is the direct image in the category of regular holonomic DS -modules, see [23, 2.11]. So the second canonical isomorphism follows because the left-hand side satisfies these conditions. (Note that (MZj , F ) for Z 6= S vanishes by the localization, because Z ⊂ D if (MZj , F ) 6= 0 .) The first noncanonical isomorphism follows from the strictness of the Hodge filtration and the relative hard Lefschetz theorem by the same argument as above.
20.1.5 Proof of Theorem 1 Let r = dim X − dim S. By (20.3)(20.9) and (20.12), we have isomorphisms Rf∗ (Ω•X (log Y ), F ) = DRr ◦f∗ ◦DR−1 (Ω•X (log Y ), F ) = DRr ◦f∗ (OX (∗Y ), F )[− dim X] (20.13) L i ' i∈Z DR(MS (∗D), F )[−r − i], where the shift of complex by r follows from the difference of the de Rham complex for left and right D-modules. Furthermore, letting L be the filtra-
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tion induced by τ on the complex of filtered DS -modules f∗ (OX (∗Y ), F )[−r], we have a canonical isomorphism i−r GrL i f∗ (OX (∗Y ), F )[−r] = (MS (∗D), F )[−i],
(20.14)
and the first assertion follows by setting M i = MSi−r (∗D). The second assertion follows from the first by (20.8). This completes the proof of Theorem 1. 20.2 Infinitesimal invariants of cycles 20.2.1 Cycle classes Let X be a complex manifold, and C•,• denote the double complex of vector spaces of currents on X. The associated single complex is denoted by C• . Let F be the Hodge filtration by the first index of C•,• (using the truncation σ in [8]). Let ξ be an analytic cycle of codimension p on X. Then it is well known that ξ defines a closed current in F p C2p by integrating the restrictions of C ∞ forms with compact supports on X to the smooth part of the support of ξ (and using a triangulation or a resolution of singularities of the cycle). So we have a cycle class of ξ in H 2p (X, F p Ω•X ). Assume X is a smooth algebraic variety over a field k of characteristic zero. Then the last assertion still holds (where Ω•X means Ω•X/k ), see [11]. Moreover, for the higher Chow groups, we have the cycle map (see [4, 10, 12, 15, 16]) 2p−n cl : CHp (X, n) → F p HDR (X),
where the Hodge filtration F is defined by using a smooth compactification of X whose complement is a divisor with normal crossings, see [8]. This cycle 2p−n (X) because we map is essentially equivalent to the cycle map to GrpF HDR can reduce to the case k = C where we have the cycle map cl : CHp (X, n) → HomMHS (Q, H 2p−n (X, Q)(p)), and morphisms of mixed Hodge structures are strictly compatible with the Hodge filtration F . 20.2.2 Proof of Corollary 2 By § 20.2.1 the cycle class of ξ belongs to H 2p−n (X, F p Ω•X (log Y )). By theorem 1, this gives the total infinitesimal invariant L 2p−n−i δS,D (ξ) = (δS,D (ξ)) ∈ i∈Z H2p−n−i (S, F p DR(M i )),
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and similarly for δ S,D (ξ). So the assertion follows. 20.2.3 Proof of Corollary 2 Choosing the first noncanonical isomorphism in the filtered decomposition theorem (20.12), we get canonical morphisms compatible with the direct sum decompositions L L i p q−i )) → i≥0 Hi (S − D, F p DR(M q−i )) i≥0 H (S, F DR(M L → i≥0 Hi (S − D, DR(M q−i )), and these are identified with the canonical morphisms Hq (X, F p Ω•X (log Y )) → Hq (X − Y, F p Ω•X−Y ) → Hq (X − Y, Ω•X−Y ). By Deligne [8], the composition of the last two morphisms is injective because of the strictness of the Hodge filtration, see also § 20.1.4. So we get i i (ξ), δ i (ξ), δ i the equivalence of δS,D U DR,U (ξ). The equivalence with δ S,D (ξ) follows from § 20.2.1.
20.2.4 Compatibility with the definition in [1] When D is empty, the infinitesimal invariants are defined in [1] by using the extension groups of filtered D-modules together with the forgetful functor from the category of mixed Hodge modules to that of filtered D-modules. Its compatibility with the definition in this paper follows from the equivalence of categories (20.1) and the compatibility of the direct image functors (20.3). Note that for (L, F ) ∈ Db F (OX , Diff) in the notation of (1.1), we have a canonical isomorphism Exti ((Ω•X , F ), (L, F )) = Hi (X, F0 L),
(20.15)
where the extension group is taken in Db F (OX , Diff). Indeed, the left-hand side is canonically isomorphic to Exti (DR−1 (Ω•X , F ), DR−1 (L, F )) = Hi (X, F0 HomD (DR(DX , F ), DR−1 (L, F ))), = Hi (X, F0 DRr DR−1 (L)), and the last group is isomorphic to the right-hand side of (20.15) which is independent of a representative of (L, F ). If X is projective, then this assertion follows also from the adjoint relation for filtered D-modules.
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If X is smooth projective and Y is a divisor with normal crossings, then the cycle class can be defined in Ext2p ((Ω•X , F ), Ω•X (log Y ), F [p])) = H2p (X, F p Ω•X (log Y )) = F p H2p (X, Ω•X (log Y )). 20.2.5 Remarks i) If we use (20.11) instead of (20.12) we get an analogue of Theorem 1 for non logarithmic complexes. However, the assertion becomes more complicated, and we get noncanonical and canonical isomorphisms ) L Rf∗ (Ω•X , F ) ' i∈Z,Z⊂S (DR(MZi−r ), F )[−i]. (20.16) L • i−r GrL i Rf∗ (ΩX , F ) = Z⊂S (DR(MZ ), F )[−i]. This implies an analogue of Corollary 1. If D is a divisor with normal crossings, we have a filtered quasi-isomorphism for Z = S ∼
fi−r ), F ) −→ (DR(M i−r ), F ), (DRlog (M S S
(20.17)
fi−r ) is the intersection of DR(M i−r ) with DRlog (Ve i ). where DRlog (M S S S This seems to be related with a question of Lewis and Shuji Saito, see also [19]. ii) If dim S = 1, we can inductively define the infinitesimal invariants in Corollary 1 by an argument similar to [24] using [26]. iii) Assume S is projective and D is a divisor with normal crossings. Then the Leray filtration for X → S → pt is given by the truncation τ on the complex of filtered DS -modules f∗ (OX (∗Y ), F ), and gives the Leray filtration on the cohomology of X − Y (induced by the truncation τ as in [8]). Indeed, the graded pieces Hj f∗ (OX (∗Y ), F ) of the filtration τ on S coincide with (Ve j+r (∗D), F ), and give the open direct images by U → S of the graded pieces (V j+r , F ) of the filtration τ on U as filtered D-modules underlying mixed Hodge modules. Note that the morphism U → S is open affine so that the direct image preserves regular holonomic D-modules. 20.3 Examples 20.3.1 Lefschetz pencils Let Y be a smooth irreducible projective variety of dimension n embedded in a projective space P over C. We assume that Y 6= P and Y is not contained in a hyperplane of P so that the hyperplane sections of Y are parametrized
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by the dual projective spaces P∨ . Let D ⊂ P∨ denote the discriminant. This is the image of a projective bundle over Y (consisting of hyperplanes tangent to Y ), and hence D is irreducible. At a smooth point of D, the corresponding hyperplane section of Y has only one ordinary double point. We assume that the associated vanishing cycle is not zero in the cohomology of general hyperplane section X. This is equivalent to the non surjectivity of H n−1 (Y ) → H n−1 (X). A Lefschetz pencil of Y is a line P1 in P intersecting the discriminant D at smooth points of D (corresponding to hyperplane sections having only one ordinary double point). We have a projective morphism π : Y˜ → P1 such that Y˜t := π −1 (t) is the hyperplane section corresponding t ∈ P1 ⊂ P and Y˜ is the blow-up of Y along a smooth closed subvariety Z of codimension 2 which is the intersection of Y˜t for any (or two of) t ∈ P1 . A Lefschetz pencil of hypersurface sections of degree d is defined by replacing the embedding of Y using OY (d) so that a hyperplane section corresponds to a hypersurface section of degree d. Here OY (d) for an integer d denote the invertible sheaf induced by that on P as usual.
20.3.2 Hypersurfaces containing a subvariety Let Y, P be as in § 20.3.1. Let E ⊂ Y be a closed subvariety (which is not necessarily irreducible nor reduced). Let E{i} = {x ∈ E : dim Tx E = i}. Let IE be the ideal sheaf of E in Y . Let δ be a positive integer such that IE (δ) is generated by global sections. By [18, 20] (or [21]) we have the following ) If dim Y > max{dim E{i} + i} and d ≥ δ, then there is a (20.18) smooth hypersurface section of degree d containing E. We have furthermore ) If dim Y > max{dim E{i} + i} + 1 and d ≥ δ + 1, then there is a Lefschetz pencil of hypersurface sections of degree d containing E. (20.19) Indeed, we have a pencil such that Y˜t has at most isolated singularities, because Y˜t is smooth near the center Z which is the intersection of generic two hypersurfaces sections containing E, and hence is smooth, see [18, 20] (or [21]). Note that a local equation of Y˜t near Z is given by f − tg if t is
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identified with an appropriate affine coordinate of P1 where f, g are global sections of IE (d) corresponding to smooth hypersurface sections. To get only ordinary double points, note first that the parameter space of the hypersurfaces containing E is a linear subspace of P∨ . So it is enough to show that this linear subspace contains a point of the discriminant D corresponding to an ordinary double point. Thus we have to show that an isolated singularity can be deformed to ordinary double points by replacing the corP responding section h ∈ Γ(Y, IE (d)) with h + i ti gi where gi ∈ Γ(Y, IE (d)) and the ti ∈ C are general with sufficiently small absolute values. Since d ≥ δ + 1, we see that Γ(Y, IE (d)) generates the 1-jets at each point of the complement of E. So the assertion follows from the fact that for a function P with an isolated singularity f , the singularities of {f + i ti xi = 0} are ordinary double points if t1 , . . . , tn are general, where x1 , . . . , xn are local coordinates. (Note that f has an ordinary double point if and only if the morphism defined by (∂f /∂x1 , . . . , ∂f /∂xn ) is locally biholomorphic at this point.)
20.3.3 Construction For Y, P be as in § 20.3.1, let iY,P : Y → P denote the inclusion. Assume i∗Y,P : H j (P) → H j (Y ) is surjective for any j 6= dim Y,
(20.20)
where cohomology has coefficients in any field of characteristic zero. This condition is satisfied if Y is a complete intersection. Let E1 , E2 be m-dimensional irreducible closed subvarieties of Y such that E1 ∩ E2 = ∅,
deg E1 = deg E2 .
Here dim Y = n = 2m + s + 1 with m ≥ 0, s ≥ 1. Let E = E1 ∪ E2 . With the notation of (3.2), assume d > δ, i∗X (j) ,Y
:
H n−j (Y
dim Y > max{dim E{i} + i} + s, )→
H n−j (X (j) )
(20.21)
is not surjective for j ≤ s, (20.22)
where X (j) is a general complete intersection of multi degree (d, . . . , d) and of codimension j in Y . (This is equivalent to the condition that the vanishing cycles for a hypersurface X (j) of X (j−1) are nonzero.) Let X be a general hypersurface of degree d in Y containing E, see (20.18). Let L denote the intersection of X with a general linear subspace of codimension m + s in the projective space. Then [Ea ] (a = 1, 2) and c[L ∩ X] have the same cohomology class in H 2m+2s (X) for some c ∈ Q, because
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dim H 2m+2s (X) = 1 by the weak and hard Lefschetz theorems together with (20.20). Let ξa = [Ea ] − c[L ∩ X] ∈ CHm+s (X)Q (a = 1, 2). These are homologous to zero. It may be expected that one of them is non torsion, generalizing an assertion in [24]. More precisely, let S be a smooth affine rational variety defined over a finitely generated subfield k of C and parametrizing the smooth hypersurfaces of degree d containing E as above so that there is the universal family X → S defined over k (see [2, 28]). Assume X corresponds to a geometric generic point of S with respect to k, i.e. X is the geometric generic fiber for some embedding k(S) → C. Let ξa,X = [Ea ×k S] − c[L]X ∈ CHm+s (X)Q , where [L]X is the pull-back of [L] by X → Y . Since the local system {H 2m+2s−j (Xs )} on S is constant for j < s and S is smooth affine rational, we see that δSj (ξa,X ) = 0 for j < s. Then it may be expected that δSs (ξa,X ) 6= 0 for one of a, where S can be replaced by any non empty open subvariety. We can show this for s = 1 as follows. (For s > 1, it may be necessary to assume further conditions on d, etc.)
20.3.4
The case s = 1
Consider a Lefschetz pencil π : Y˜ → P1 such that Y˜t := π −1 (t) for t ∈ P1 is a hypersurface of degree d in Y containing E. Here Y˜ is the blow-up of Y along a smooth closed subvariety Z, and Z is the intersection of Y˜t for any t ∈ P1 . Note that Y˜t has an ordinary double point for t ∈ Λ ⊂ P1 , where Λ denotes the discriminant, see (20.19). Since Z has codimension 2 in Y , we have the isomorphism H n (Y˜ ) = H n (Y ) ⊕ H n−2 (Z),
(20.23)
so that the cycle class of [Ea × P1 ] − c[L]Y˜ ∈ CHm+1 (Y˜ )Q in H n (Y˜ ) is identified with the difference of the cycle class clZ (Ea ) ∈ H n−2 (Z) and the cycle class of L in H n (Y ). Indeed, the injection H n−2 (Z) → H n (Y˜ ) in the above direct sum decomposition is defined by using the projection Z × P1 → Z and the closed embedding Z × P1 → Y˜ , and the injection H n (Y ) → H n (Y˜ ) is the pull-back by Y˜ → Y , see [17]. By assumption, one of the clZ (Ea ) is not contained in the non primitive part, i.e. not a multiple of the cohomology class of the intersection of general hyperplane sections. Indeed, if both are contained in the non primitive part,
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then clZ (E1 ) = clZ (E2 ) and this implies the vanishing of the self intersection number Ea · Ea in Z. We will show that the cycle class of [Ea × P1 ] − c[L]Y˜ does not vanish in the cohomology of π −1 (U ) for any non empty open subvariety of P1 , in other L words, it does not belong to the image of t∈Λ0 HYn˜ (Y˜ ) where Λ0 is any finite t subset of P1 containing Λ. (Note that the condition for the Lefschetz pencil is generic, and for any proper closed subvariety of the parameter space, there is a Lefschetz pencil whose corresponding line is not contained in this subvariety.) Thus the assertion is reduced to that dim HYn˜ (Y˜ ) is independent of t ∈ P1 t because this implies that the image of H n (Y˜ ) → H n (Y˜ ) is independent of t. Y˜t
(Note that the Gysin morphism H n−2 (Y˜t ) → H n (Y˜ ) for a general t can be identified with the direct sum of the Gysin morphism H n−2 (Y˜t ) → H n (Y ) and the restriction morphism H n−2 (Y˜t ) → H n−2 (Z) up to a sign, and the image of the last morphism is the non primitive part by the weak Lefschetz theorem.) By duality, this is equivalent to that Rn π∗ QY˜ is a local system on P1 . Then it follows from the assumption that the vanishing cycles are nonzero, see (20.22).
References [1] Asakura, M.: Motives and algebraic de Rham cohomology, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000, pp. 133–154. [2] Beilinson, A., Bernstein, J. and Deligne, P.: Faisceaux pervers, Ast´erisque, vol. 100, Soc. Math. France, Paris, 1982. [3] Bloch, S.: Algebraic cycles and higher K-theory, Advances in Math., 61 (1986), 267–304. [4] Bloch, S.: Algebraic cycles and the Beilinson conjectures, Contemporary Math. 58 (1) (1986), 65–79. [5] Collino, A.: Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Alg. Geom. 6 (1997), 393–415. [6] Deligne, P.: Th´eor`eme de Lefschetz et crit`eres de d´eg´en´erescence de suites spectrales, Inst. Hautes Etudes Sci. Publ. Math. 35 (1968), 259–278. [7] Deligne, P.: Equation diff´erentielle `a points singuliers r´eguliers, Lect. Notes in Math. vol. 163, Springer, Berlin, 1970. [8] Deligne, P.: Th´eorie de Hodge I, Actes Congr`es Intern. Math., 1970, vol. 1, 425-430; II, Publ. Math. IHES, 40 (1971), 5–57; III, ibid., 44 (1974), 5–77. [9] Deligne, P.: D´ecompositions dans la cat´egorie d´eriv´ee, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994, pp. 115–128. [10] Deninger, C. and A. Scholl: The Beilinson conjectures, in Proceedings Cambridge Math. Soc. (eds. Coats and Taylor) 153 (1992), 173–209.
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[11] El Zein, F.: Complexe dualisant et applications `a la classe fondamentale d’un cycle, Bull. Soc. Math. France M´em. No. 58 (1978) [12] Esnault, H. and E. Viehweg: Deligne-Beilinson cohomology, in Beilinson’s conjectures on Special Values of L-functions, Academic Press, Boston, 1988, pp. 43–92. [13] Green, M.L.: Griffiths’ infinitesimal invariant and the Abel-Jacobi map, J. Differential Geom. 29 (1989), 545–555. [14] Griffiths, P.A.: Infinitesimal variations of Hodge structure, III, Determinantal varieties and the infinitesimal invariant of normal functions, Compositio Math. 50 (1983), 267–324. [15] Jannsen, U.: Deligne homology, Hodge-D-conjecture, and motives, in Beilinson’s conjectures on Special Values of L-functions, Academic Press, Boston, 1988, pp. 305–372. [16] Jannsen, U.: Mixed motives and algebraic K-theory, Lect. Notes in Math., vol. 1400, Springer, Berlin, 1990. [17] Katz, N.: Etude cohomologique des pinceaux de Lefschetz, in Lect. Notes in Math., vol. 340, Springer Berlin, 1973, pp. 254–327. [18] Kleiman, S. andA. Altman: Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra 7 (1979), 775–790. [19] Lewis, J.D. and S. Saito, preprint. [20] Otwinowska, A: Monodromie d’une famille d’hypersurfaces, preprint (math.AG/0403151). [21] Otwinowska, A. and M. Saito, M.: Monodromy of a family of hypersurfaces containing a given subvariety, preprint (math.AG/0404469). [22] Saito, M.: Modules de Hodge polarisables, Publ. RIMS, Kyoto Univ. 24 (1988), 849–995. [23] Saito, M.: Mixed Hodge Modules, Publ. RIMS Kyoto Univ. 26 (1990), 221– 333. [24] Saito, S.: Higher normal functions and Griffiths groups, J. Algebraic Geom. 11 (2002), 161–201. [25] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping, Inv. Math. 22 (1973), 211–319. [26] Steenbrink, J.H.M.: Limits of Hodge structures, Inv. Math. 31 (1975/76), no. 3, 229–257. [27] Voisin, C.: Variations de structure de Hodge et z´ero-cycles sur les surfaces g´en´erales, Math. Ann. 299 (1994), 77–103. [28] Voisin, C.: Transcendental methods in the study of algebraic cycles, in Lect. Notes in Math. vol. 1594, pp. 153–222.
21 Correspondence of Elliptic Curves and Mordell-Weil Lattices of Certain Elliptic K3’s Tetsuji Shioda Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan.
[email protected]
To Jacob Murre
Abstract We study the Mordell-Weil lattice of certain elliptic K3 surfaces, related to the Kummer surface of a product abelian surface. Our aim is first to determine the precise structure of such a lattice, and second to give some explicit generators, in the case beyond rational elliptic surfaces.
21.1 Introduction The main purpose of this paper is to study certain elliptic fibrations on the Kummer surface of a product abelian surface, both geometrically (equationfree) and with the use of equations, and to identify the elements of the Mordell-Weil lattice coming from algebraic cycles on the Kummer surface, especially from the correspondences of the factor elliptic curves. We state here the main results in terms of Weierstrass equations, which should show some new feature of Mordell-Weil lattices for elliptic K3 surfaces, different from the well-studied case of rational elliptic surfaces. The background will be explained after the statements. Theorem 21.1.1. Let C1 , C2 be two elliptic curves with the absolute invariant j1 , j2 , defined over an algebraically closed field k of characteristic 6= 2, 3. Let F (1) denote the elliptic curve over the rational function field k(T ) y 2 = x3 − 3αx + (T + where 624
1 − 2β), T
(21.1)
Mordell-Weil Lattices of Certain Elliptic K3’s
α=
p 3
j1 j2 , β =
p
(1 − j1 )(1 − j2 ).
625
(21.2)
Assume that j1 6= j2 (i.e. C1 , C2 are not isomorphic to each other). Then there is a natural isomorphism of Hom(C1 , C2 ) to the Mordell-Weil lattice F (1) (k(T )), ϕ 7→ Rϕ , such that the height of Rϕ hRϕ , Rϕ i is equal to 2 deg(ϕ). In other words, there is a natural isomorphism of lattices: Hom(C1 , C2 )[2] ' F (1) (k(T )).
(21.3)
Theorem 21.1.2. Let F (2) denote the elliptic curve over k(t), obtained from F (1) by the base change T = t2 . Assume that j1 6= j2 . Then the Mordell-Weil lattice F (2) (k(t)) contains a sublattice of finite index 2h (h = rk Hom(C1 , C2 )) which is naturally isomorphic to the direct sum of lattices Hom(C1 , C2 )[4] ⊕ A∗2 [2]⊕2
(21.4)
where A∗2 denotes the dual lattice of the root lattice A2 (of rank 2). N. B. (1) The absolute invariant is normalized so that j = 1 for y 2 = x3 − x. (2) Given a lattice L, we denote by L[n] the lattice structure on L with the norm (or pairing) multiplied by n. (3) For the root lattices, we refer to [4]. Here we briefly mention some background of the above results; more details will be given later in § 21.2 and § 21.3. Let S (n) be the associated elliptic surface to the elliptic curve F (n) (n = 1, 2). Then they are both K3 surfaces, and S (2) is isomorphic to the Kummer surface S = Km(C1 × C2 ) of the product of two elliptic curves. This elliptic fibration on the Kummer surface and S (1) are discovered by Inose ([5]) in search for the notion of isogeny between singular K3 surfaces ([6]). More recently, Kuwata ([8]) has made a nice observation on Inose’s results; he introduces elliptic K3 surfaces corresponding to F (n) (n ≤ 6) defined by the base change T = tn , and shows that their Mordell-Weil rank can become as large as 18, the maximum in case of char(k) = 0. Inspired by the work of Inose and Kuwata, we ([15]) have made a preliminary study on these elliptic K3 surfaces from the viewpoint of Mordell-Weil lattices. Now, given any ϕ ∈ Hom(C1 , C2 ), the image of its graph under the rational map from C1 ×C2 to S (and to S (1) ) gives a curve on S (or S (1) ), and this image curve determines a rational point Pϕ ∈ F (2) (k(t)) (or Rϕ ∈ F (1) (k(T ))) by the formalism of Mordell-Weil lattices (see § 21.2; compare [17]). The
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correspondence ϕ 7→ Rϕ in Theorem 21.1.1 is “natural” in the sense that it is defined in this way. Theorems 21.1.1 and 21.1.2 are the refinement of some results announced in [15], covering also the case of arbitrary characteristic 6= 2, 3. As an application to the case of the higher rank, we mention the following: Example 21.1.3. Assume j1 6= 0, j2 = 0 (C2 : y 2 = x3 − 1). Then the Mordell-Weil lattice F (6) (k(t)) of the elliptic curve over k(t) y 2 = x3 + (t12 − 2βt6 + 1),
β = 1 − j12
(21.5)
is of rank r = h + 16 and it contains a finite index sublattice isomorphic to the direct sum L0 ⊕ E8 [2] ⊕ D4∗ [4]⊕2 ,
rk L0 = h
(21.6)
where E8 , D4 , A2 are root lattices and ∗ means the dual lattice. If char(k) = 0, then we have (h = 2 or 0) L0 = A2 [6d0 ] or 0
(21.7)
d0 (≥ 2) being the degree of minimal isogeny C1 → C2 (cf. §21.8). The generators of k(t)-rational points of this sublattice can be given explicitly in case h = 0 or if d0 is small. This paper is organized as follows. In §21.2, we review the formalism of Mordell-Weil lattices which is our main tool. In the next sections, we study the elliptic fibrations on the Kummer surface of a product abelian surface. After reviewing the so-called double Kummer pencils (§21.3), we study the Inose’s pencil, first by a geometric method (§21.4) and second by introducing equations (§21.5). With these preparations, we prove our main results (and Theorems 21.1.1 and 21.1.2) in §21.6. Some comments for the case j1 = j2 (§21.7) and examples (§21.8) are given. We hope to come back to the higher rank case in some other occasion. It is my pleasure to dedicate this paper to Professor Jacob Murre on the occasion of his 75th birthday. It was reported at the workshop on Algebraic Cycles held at Lorentz Center, Leiden, in his honor. The paper has been prepared partly during my stay at the Max-Planck-Institut f¨ ur Mathematik, Bonn, in the summer of 2004. I would like to thank the MPI for the hospitality and excellent atmosphere, and my special thanks go to Professor Hirzebruch for everything he has done for me. Finally I thank the referee for his/her careful reading of the manuscripts and for useful suggestions.
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21.2 Review of the MWL-formalism Let us make a review of the basic formalism of Mordell-Weil lattices, fixing some notation (cf. [14]). Let E/K be an elliptic curve over the function field K = k(C) of a smooth projective curve C/k. The base field k is an algebraically closed field of arbitrary characteristic (later we assume char(k) 6= 2, 3). Let f : S −→ C be the associated elliptic surface (the Kodaira-N´eron model of E/K); S is a smooth projective surface over k and E is the generic fibre of f . The set of K-rational points of E, E(K), is in a natural bijective correspondence with the set of the sections of f . For P ∈ E(K), we use the same symbol P to denote the corresponding section P : C → S and the symbol (P ) to denote the image curve in S; thus for example (O) denotes the image of the zero-section in S. Let Sing(f ) (resp. Red(f )) denote the set of singular fibres (reducible singular fibres) of f . It is known that, if Sing(f ) 6= ∅, then E(K) is finitely generated (Mordell-Weil theorem). Let NS(S) be the N´eron-Severi group of S which is defined as the group of divisors on the surface S modulo algebraic equivalence; the class of a divisor D is denoted by [D] (or simply by D if no confusion will arise). Let T = T (f ) denote the subgroup generated by the classes of the zero-section (O), any fibre F and all the irreducible components of reducible fibres which are disjoint from (O). Then we have a natural isomorphism E(K) ' NS(S)/T.
(21.8)
The correspondence is given by P 7→ [(P )] mod T , and the inverse correspondence is induced by the following map of the divisor group of S to E(K): µ(D) = µf (D) = sum(D|E ) ∈ E(K)
(21.9)
Namely, given a divisor D on S, restrict it to the generic fibre E and take ¯ K ¯ being the algebraic closure of the summation of its components (∈ E(K), K) with respect to the group law of E. Now NS(S) forms an indefinite integral lattice with respect to the intersection pairing, and T forms a sublattice, called the trivial sublattice, which has P an orthogonal decomposition T = U ⊕ v∈Red(f ) Tv where U is the unimodular rank 2 lattice generated by (O), F , and Tv is the lattice of rank mv − 1 spanned by the irreducible components away from (O) of the reducible fibre f −1 (v). Each Tv is a root lattice of type A, D, E, up to sign, by Kodaira [7]. The key idea of Mordell-Weil lattices is to define the lattice structure on
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T. Shioda
Table 21.1. Values of local contribution Tv− type of Fv
A1 III
E7 III
A2 ∗
E6
IV
IV
∗
Ab−1
Db+4 Ib∗ (b
Ib (b ≥ 2)
contrv (P )
1/2
3/2
2/3
4/3
contrv (P, Q) (i < j)
−−
1/3
2/3
1 (i = 1) 1 + b/4 (i > 1)
i(b − i)/b i(b − j)/b
≥ 0)
1/2 (i = 1) (2 + b)/4 (i > 1)
the Mordell-Weil group via the intersection theory on the surface as follows. There is a unique homomorphism ν : E(K) −→ NQ = NS(S) ⊗ Q
(21.10)
satisfying the condition: for every P ∈ E(K), ν(P ) ⊥ T,
ν(P ) ≡ [(P )]
mod TQ .
(21.11)
Then E(K) modulo torsion is embedded into the orthogonal complement of T in NQ , which is negative-definite by the Hodge index theorem. Therefore, by defining the height pairing on E(K) by the formula: hP, Qi := −(ν(P ) · ν(Q)),
(21.12)
one obtains the structure of a positive-definite lattice on E(K)/E(K)tor . It is called the Mordell-Weil lattice (abbreviated from now on as MWL) of the elliptic curve E/K or the elliptic surface f : S → C. The height formula takes the following explicit form: X hP, Qi = χ + (P O) + (QO) − (P Q) − contrv (P, Q) (21.13) v∈Red(f )
where χ is the arithmetic genus of S, (P Q) denotes the intersection number of the sections (P ) and (Q), and contrv (P, Q) is a local contribution at v. For later use, we copy the table from [14, (8.16)], Table 21.2, in which i, j are defined so that the section (P ) (or (Q)) intersects the i-th (or j-th) irreducible component of the singular fibre f −1 (v) under suitable numbering. The determinant of N´eron-Severi lattice and that of MWL are related by the formula: det NS(S) = det(E(K)/E(K)tor ) · det T /|E(K)tor |2 .
(21.14)
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629
Remarks. 1) Given the information of the trivial lattice T , it is easy to write down the explicit formula for ν(P ) = (P )+· · · satisfying the Linear Algebra condition (21.11). Indeed this is how the height formula (21.13) is derived in general. On the other hand, it is also possible to compute the height by applying the original definition (21.12), especially when P is given as P = µ(D) for some divisor D. This method gives an algorithm suited for computer calculation which can be used for checking theoretical computation. 2) The structure of MWL is clarified in the case where S is a rational elliptic surface. In this case, the lattices in question form a hierarchy dominated by the root lattice E8 , the unique positive-definite even unimodular lattice of rank 8 (cf. [11]). Also it is easy in this case to give the generators of rational points, for example, and there are many interesting applications. Beyond this case, not much is known even in the next simplest case of elliptic K3 surfaces.
21.3 The Kummer pencils In the subsequent sections, we study certain elliptic K3 surfaces related to the Kummer surface of a product abelian surface. In general, let A be an abelian surface and let ιA denote the inversion automorphism of A. We assume that char(k) 6= 2. The Kummer surface S = Km(A) is a smooth K3 surface obtained from the quotient surface A/ιA by resolving the 16 singular points corresponding to the points of order 2 on A. The Picard number is given by ρ(S) = ρ(A) + 16. Now consider the case of a product abelian surface, i.e. A = C1 × C2 where C1 , C2 are elliptic curves. If we denote by h the rank of the free module Hom(C1 , C2 ) of homomorphisms of C1 to C2 , then ρ(S) = h + 18,
h := rk Hom(C1 , C2 )
(21.15)
since we have ρ(A) = 2 + h. It is known that H = Hom(C1 , C2 ) has the structure of a positive-definite lattice such that the norm of ϕ ∈ H is deg(ϕ), the degree of the homomorphism (see the Remark below). First we look at the Kummer pencil (cf. [6, §2]), i.e. the elliptic fibration π1 : S = Km(C1 × C2 ) → P1
(21.16)
induced from the projection of A to C1 . It has the 4 singular fibres of type I0∗ :
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T. Shioda
π1−1 (v¯i ) = 2Fi +
X
Aij .
(21.17)
j∈I
Here we use the following notation. Let I = {0, 1, 2, 3} and let {vi |i ∈ I} ⊂ C1 be the 2-torsion points (take v0 = the origin); similarly for {vj0 |j ∈ I} ⊂ C2 . We denote by v¯i the image point of vi under C1 → C1 /ι1 = P1 . The curves Fi , Gj ⊂ S(i, j ∈ I) are the image of vi × C2 , C1 × vj0 under the rational map of degree two A → S. Further Aij denotes the exceptional curve corresponding to vi × vj0 . All the 24 curves {Fi , Gj , Aij } on S are smooth rational curves with self-intersection number −2 (i.e. −2-curves). The intersection numbers among these curves are given as follows:
(Fi · Fj ) = −2δij , (Gi · Gj ) = −2δij , (Fi · Gj ) = 0, (Aij · Akl ) = −2δik δjl , (Fi · Akl ) = δik , (Gi · Akl ) = δil .
(21.18)
Note that each of the 4 curves Gj gives a section of π1 since it intersects the fibre (21.17) with intersection multiplicity 1. Take G0 = (O) as the zero-section. Then the other sections Gj are of order 2. The generic fibre of π1 is isomorphic to the constant elliptic curve C2 over k(C1 ) ⊃ k(P1 ), but of course not over k(P1 ). Proposition 21.3.1. Let E denote the generic fibre of π1 . Then we have E(k(P1 )) ' Hom(C1 , C2 ) ⊕ (Z/2Z)2
(21.19)
i.e. the Mordell-Weil lattice E(k(P1 ))/(tor) is isomorphic to the lattice H := Hom(C1 , C2 ) with norm ϕ 7→ deg(ϕ). Proof This should be well known if we ignore the lattice structure, but for the sake of completeness, let us first check the isomorphism of both side as groups. Take any P ∈ E(k(P1 )), and regard it as a section σ : P1 → S. Its pullback to C1 , σ ˜ : C1 → A = C1 × C2 , is of the form σ ˜ (u) = (u, α(u)) (u ∈ C1 ), where α : C1 → C2 is a morphism such that α(−u) = −α(u). Hence we have α(u) = ϕ(u) + v 0 for some homomorphism ϕ ∈ Hom(C1 , C2 ) and a 2-torsion point v 0 ∈ C2 . This establishes the bijection of both sides. For any nonzero ϕ ∈ Hom(C1 , C2 ), consider the image Γ = Γϕ of its graph under the rational map A → S. Let Qϕ = µ(Γϕ ) ∈ E(k(P1 )), with µ = µf for f = π1 defined by (2.2) and (3.2). It is easy to see that ϕ 7→ Qϕ is a homomorphism. We must prove that the height hQϕ , Qϕ i is equal to deg(ϕ). To prove this, we use the height formula (21.13) for P = Qϕ ; note that
Mordell-Weil Lattices of Certain Elliptic K3’s
631
χ = 2 (for a K3): hP, P i = 4 + 2(P O) −
X
contrv (P ).
For a moment, admit Lemma 21.3.2 below. Then the term (P O) = (Γ · G0 ) is given by (21.20). On the other hand, Table 21.2 shows that we have contrv (P ) = 1 iff the section (P ) = Γ meets a non-identity component of I0∗ -fibre. Thus the sum of local contribution is 3 (or 2 or 0) according to the case (a) (or (b) or (c)) of the Lemma. This proves hP, P i = deg(ϕ). Lemma 21.3.2. Given ϕ ∈ Hom(C1 , C2 ), ϕ 6= 0, let d = deg(ϕ) and set 0 (i ∈ I); let n = n(ϕ) denote the number of distinct j(i). Then ϕ(vi ) = vj(i) the curve Γ = Γϕ is a (−2)-curve on S, and it satisfies Γ · Ai,j = δi,j(i) and Γ · Fi = 0 for all i ∈ I. The intersection number of Γ with Gi is described in the table 21.20 below according to the three cases a) n = 4, b n = 2 or c) n = 1, which can be also characterized by the following properties: a) d is odd, c) ϕ = 2ϕ1 for some ϕ1 ∈ Hom(C1 , C2 ), b) otherwise. (In case b), we change ordering vj0 so that {j(i)|i ∈ I} = {0, 1}.) Then we have Γ · G0 Γ · G1 Γ · G2 Γ · G3
a) b) c) (d − 1)/2 (d − 2)/2 (d − 4)/2 (d − 1)/2 (d − 2)/2 d/2 (d − 1)/2 d/2 d/2 (d − 1)/2 d/2 d/2
(21.20)
˜ be the graph of ϕ in A = C1 × C2 . Then a 2-torsion point of Proof Let Γ 0 ˜ A lies on Γ if and only if it is of the form vi × vj(i) for some i ∈ I. Under ˜ is mapped to Γ. Thus we have the rational map A → S of degree two, Γ Γ ∩ Ai,j 6= ∅ if and only if j = j(i), in which case the intersection number is one. Hence the first assertion. As for the intersection of Γ with Gj , for ˜ · (C1 × v 0 ) has degree d = deg(ϕ), of which example with G0 , we note that Γ 0 0 n = 4/n simple intersection occur at the 2-torsion points. Hence we have Γ · G0 = (d − n0 )/2 on S, as asserted. We can argue similarly for other Gj . Theorem 21.3.3. The N´eron-Severi lattice NS(S) of the Kummer surface S = Km(C1 × C2 ) is generated by the 24 curves Fi , Gj , Aij together with Γϕ (ϕ ∈ Hom(C1 , C2 )). Its determinant is given by det NS(S) = 24 · det Hom(C1 , C2 ).
(21.21)
Proof The first assertion follows from (21.8) (applied to E = E) and (21.19),
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T. Shioda
since both trivial lattice T and the torsion part (Z/2Z)2 are generated by curves belonging to the 24 curves. For (21.21), apply the formula (21.14), where we have det T = 44 (as T = U ⊕ D4⊕4 ) and |E(K)tor | = 22 . Remark. The prototype of the above arguments is the well known fact: NS(A) = T0 ⊕ T0⊥ ,
(T0⊥ )[−1] ' Hom(C1 , C2 )[2]
(21.22)
where A = C1 × C2 and T0 is the sublattice of NS(A) generated by C1 × v00 and v0 × C2 . It relates the correspondence theory of curves to geometry of surfaces, and its most remarkable application is Weil’s proof of the Riemann hypothesis for curves over a finite field ([19]). At any rate, it shows that Hom(C1 , C2 )[2] (with 2 deg(ϕ) as the norm of ϕ) is a (positive-definite) integral lattice and det NS(A) = 2h det Hom(C1 , C2 ). Note that Hom(C1 , C2 ) itself is not necessarily an integral lattice. Corollary 21.3.4. Let N0 denote the sublattice of NS(S) generated by the 24 curves {Fi , Gj , Aij }. Then (i) N0 is an indefinite lattice of rank 18 and det 24 . (ii) Assume that C1 and C2 are not isogeneous to each other. Then we have NS(S) = N0 , namely NS(S) is generated by the 24 curves Fi , Gj , Aij . This is obvious from Theorem 21.3.3. It is classically well known (e.g.[12]). Proposition 21.3.5. The map ϕ 7→ Γϕ induces a surjective homomorphism from Hom(C1 , C2 ) to N S(S)/N0 . Proof It suffices to show that Γϕ+ψ ≡ Γϕ + Γψ
mod N0 ,
(21.23)
since the surjectivity follows from Theorem 21.3.3. Consider the divisor D = Γϕ+ψ − (Γϕ + Γψ ) + Γ0 on S. The restriction of D to the generic fibre E of the Kummer pencil π1 gives µ(D) = Qϕ+ψ − (Qϕ + Qψ ) = O, µ being the map (21.9). Hence D is algebraically equivalent to a sum of irreducible components of fibres of π1 , which proves (21.23). 21.4 Inose’s pencil Next we define Inose’s pencil on S = Km(C1 × C2 ). Take the following divisors on S: Φ1 = G1 + G2 + G3 + 2(A01 + A02 + A03 ) + 3F0 , (21.24) Φ2 = F1 + F2 + F3 + 2(A10 + A20 + A30 ) + 3G0 , where Fi , Gj , Aij are the −2-curves used in §3. They are disjoint and they have the same type as a singular fibre of type IV ∗ . Recall that a divisor on a
Mordell-Weil Lattices of Certain Elliptic K3’s
633
K3 surface X is a fibre of some elliptic fibration on X if it has the same type as one of the Kodaira’s list of singular fibres (cf.[7], [12]). Therefore there is an elliptic fibration, say f : S → P1 , such that f −1 (0) = Φ1 and f −1 (∞) = Φ2 . We call it Inose’s pencil on the Kummer surface S = Km(C1 × C2 ). Note that the divisors Φ1 , Φ2 are interchanged when the order of the factors C1 , C2 is changed. Let E be the generic fibre of f ; it will be identified with the elliptic curve F (2) /k(t) of Theorem 21.1.2 later. Each of the 9 curves Aij (i, j 6= 0) defines a section of f ; for instance, A33 intersects the fibre Φ1 transversally at the simple component G3 . Let us choose A33 = (O) as the zero-section. To avoid confusion, we let Qij ∈ E(k(t)) denote the section such that (Qij ) = Aij . Throughout § 21.4, we make the assumption: (#) f has no other reducible fibres than Φ1 , Φ2 . Lemma 21.4.1. Under (#), the Mordell-Weil lattice L = E(k(t)) of the Inose’s pencil has rank 4 + h, and the 9 sections Qij form a sublattice L1 of rank 4 isomorphic to A∗2 [2]⊕2 . Proof Under the assumption (#), the trivial lattice T is isomorphic to U ⊕E6⊕2 , of rank 14, and hence the Mordell-Weil rank is equal to ρ(S)−14 = 4 + h by (21.8), (21.15). Also E(k(t)) is torsion-free by the height formula. Let us compute the height of Q = Qij (i, j 6= 0) by (21.13). The curve (Q) = Aij hits the singular fibre Φ1 (of type IV ∗ ) at a non-identity component iff i = 1, 2. By Table 21.2, the local contribution is equal to contrv (Q) = 4/3 for i = 1, 2, and = 0 for i = 3. By replacing i by j, we get the corresponding value at the fibre Φ2 . Hence we have hQ, Qi = 4 − 4/3 − 4/3 = 4/3. Similarly we can compute the height pairing hQ, Q0 i for Q 6= Q0 . Thus we see that both {Q11 , Q22 } and {Q12 , Q21 } span a mutually orthogonal sublattice (isomorphic to A∗2 [2]) with 4/3 2/3 the Gram matrix . 2/3 4/3 Now we turn our attention to the curves Γϕ (ϕ ∈ H) to study the remaining rank h part of the Mordell-Weil lattice of f . Let Pϕ = µf (Γϕ ) ∈ E(k(t)),
(21.25)
with µf as in (21.9). To compute the height hPϕ , Pϕ i, we cannot directly apply the height formula (21.13) as before, because we do not know the local contribution terms. Thus we need to go back to the original definition (21.12).
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T. Shioda
Lemma 21.4.2. Given ϕ ∈ Hom(C1 , C2 ) with hPϕ , Pϕ i has the following value: 2 2d − d + 1 hPϕ , Pϕ i = 2d2 − 2d + 4/3 2 2d + d
d = deg(ϕ), the height (a) (b) (c)
(21.26)
in the respective case a), b), c) for ϕ stated in Lemma 21.3.2. Proof Set P = Pϕ . Consider the case (c) where ϕ is divisible by 2 in H; in particular d is divisible by 4. Solving the linear algebra condition (21.11) for (P ) = Γϕ , we find by a direct computation that ν(P ) = Γϕ − 3dΦ −
3d 2 (O)
+ 3d 2 (A01
+ d(F1 + F2 + G1 + G2 ) + 2dF0 + (3d − 1)G0
+ A02 ) + dA03 + 2d(A10 + A20 ) +
3d 2 A30
(21.27)
where Φ denotes a fibre class of f . Then by (21.13), we compute hP, P i = −(ν(P )2 ) using Lemma 21.3.2 and verify that it is equal to 2d2 + d, as asserted. The other cases (a), (b) can be verified in the same way. It follows from above that Pϕ 6= 0 for ϕ 6= 0, but we cannot say that the map ϕ → Pϕ is an injective map from H = Hom(C1 , C2 ) to the Mordell-Weil lattice, since we only know (Prop. 21.3.5 that the map ϕ → Γϕ induces a group homomorphism H → NS(S)/N0 . To remedy this situation, let us proceed as follows. Consider the orthogonal complement of L1 in L (with the notation of Lemma 21.4.1): L01 = L⊥ 1.
(21.28)
Obviously the Q-vector space V = L ⊗ Q is an orthogonal direct sum of V1 = L1 ⊗ Q and V10 = L01 ⊗ Q, although the lattice L itself is not in general equal to the direct sum L1 ⊕ L01 . Decompose Pϕ ∈ V as a sum of the V1 -component and V10 -component: Pϕ = [Pϕ ]+ + [Pϕ ]− ,
[Pϕ ]+ ∈ V1 , [Pϕ ]− ∈ V10 .
(21.29)
Lemma 21.4.3. The V1 -component of Pϕ ∈ L is represented by the following element: d d−1 2 (Q12 + Q21 ) + 2 (Q11 + Q22 ) (a) d−1 [Pϕ ]+ = (21.30) (Q12 + Q21 + Q22 ) + d2 Q11 (b) d2 (Q + Q + Q + Q ) (c) 11 12 21 22 2 In particular, it is an element of L1 if and only if ϕ is in the case (c), i.e. ϕ = 2ϕ1 for some ϕ1 ∈ Hom(C1 , C2 ).
Mordell-Weil Lattices of Certain Elliptic K3’s
635
Proof The first part is verified by a linear algebra computation. For the second part, note that in case (c), d = deg(ϕ) = 4 deg(ϕ1 ) is divisible by 4. Thus d/2 is an integer, and [Pϕ ]+ ∈ L1 . In case (a), d is odd, and we can easily see that (Q12 + Q21 ) is not divisible by 2 in L. The case (b) can be treated in a similar way. Lemma 21.4.4. Depending on the case of ϕ ∈ H, we have 2 (a) 2d − 2d + 1 h[Pϕ ]+ , [Pϕ ]+ i = 2d2 − 3d + 4/3 (b) 2 2d (c)
(21.31)
and, for any ϕ ∈ H, h[Pϕ ]− , [Pϕ ]− i = d.
(21.32)
Proof Using Lemma 21.4.1, check first that both (Q12 +Q21 ) and (Q11 +Q22 ) have height 4 and they are orthogonal. By Lemma 21.4.3, we see for instance in case (a) that the “height” of [Pϕ ]+ is equal to d d−1 2 ( )2 · 4 + ( ) · 4 = 2d2 − 2d + 1. 2 2 Other cases are similar, and this proves (21.31). Now it follows from (21.29) that h[Pϕ ]− , [Pϕ ]− i = hPϕ , Pϕ i − h[Pϕ ]+ , [Pϕ ]+ i Comparing (21.26) and (21.31), we conclude (21.32) that the “height” of [Pϕ ]− is equal to d in all cases. Proposition 21.4.5. Let Rϕ := 2[Pϕ ]− ∈ L01 .
(21.33)
Then the map ϕ 7→ Rϕ gives an imbedding of the lattice H[4] into L01 . Proof Let N = NS(S). By (21.8), we have L ∼ = N/T . Under this iso∼ morphism, we have L1 = N0 /T , since N0 is generated by the 24 curves (cf. Cor.21.3.4) of which 15 (resp. 9) give generators of T (resp. L1 ). It follows that L/L1 ∼ = N/N0 , and the map ϕ 7→ Pϕ induces a group homomorphism H → L/L1 by Proposition 21.3.5. On the other hand, the orthogonal projection V → V10 induces the homomorphism L/L1 → 12 L01 sending Pϕ mod L1 to [Pϕ ]− . By composing the two maps, we obtain a homomorphism 1 H → L01 , 2
ϕ 7→ [Pϕ ]−
(21.34)
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T. Shioda
preserving the norm (or height) by (21.32). In other words, the map ϕ 7→ Rϕ gives an injective homomorphism H → L01 such that hRϕ , Rϕ i = 4 deg(ϕ).
(21.35)
This proves the assertion.
21.5 Defining equation of Inose’s pencil Now we introduce the equations to make more detailed analysis. Suppose that the elliptic curve Cl (l = 1, 2) is given by the Weierstrass equation: Cl : yl2 = fl (xl ) = x3l + · · ·
=
3 Y
(xl − al,k ).
(21.36)
k=1
The 2-torsion points of C1 , C2 are given by vi = (a1,i , 0), vj0 = (a2,j , 0). The function t = y2 /y1
(21.37)
on A = C1 × C2 is invariant under ιA , and it defines an elliptic fibration on the Kummer surface S whose generic fibre is isomorphic to the plane cubic curve over k(t) defined by f1 (x1 )t2 = f2 (x2 ).
(21.38)
The following result is essentially due to Inose [5], for which we give a simplified proof bellow (cf. [15]): Proposition 21.5.1. The elliptic fibration on the Kummer surface S induced by t = y2 /y1 is isomorphic to the Inose’s pencil. The Weierstrass form of the cubic curve (21.38) is given by E (2) : y 2 = x3 − 3αt4 x + t4 (t4 − 2βt2 + 1) (21.39) p √ where α, β are defined by (1.2), i.e. α = 3 j1 j2 , β = (1 − j1 )(1 − j2 ). There are two singular fibres of type IV ∗ at t = 0 and ∞, and the other singular fibres are given, in an abridged form, as follows: i) I1 × 8 if j1 6= j2 , j1 j2 6= 0, ii) II × 4 if j1 6= j2 , j1 j2 = 0, iii) I2 × 2 + I1 × 4 if j1 = j2 6= 0, 1, iv) I2 × 4 if j1 = j2 = 1, v) IV × 2 if j1 = j2 = 0.
Mordell-Weil Lattices of Certain Elliptic K3’s
637
Proof To prove the first assertion, we claim that the divisor of the function t on S is equal to (t) = Φ1 − Φ2
(21.40)
where Φ1 , Φ2 are the divisors defined by (21.24). Indeed, by (21.36) and (21.37), we have (t2 ) = (f2 (x2 )/f1 (x1 )) =
3 X
(x2 − a2,k ) −
3 X
(x1 − a1,i ).
(21.41)
i=1
k=1
Since the function x1 defines the first Kummer pencil π1 (3.2), we have X X (x1 − a1,i ) = π1−1 (vi ) − π1−1 (v0 ) = 2Fi + Ai,j − (2F0 + A0,j ) (21.42) j∈I
j∈I
by (3.3). Writing down the corresponding fact for the the second Kummer pencil π2 , we have X X Ai,j − (2G0 + Ai,0 ). (21.43) (x2 − a2,j ) = π2−1 (vj0 ) − π2−1 (v00 ) = 2Gj + i∈I
i∈I
Then, by (21.24), (21.41), (21.42) and (21.43), we can easily check that (t2 ) = 2(Φ1 − Φ2 ).
(21.44)
This implies our claim (21.40), proving that the function t defines Inose’s pencil. Next, setting T = t2 , we consider the linear pencil of plane cubic curves f1 (x1 )T = f2 (x2 ).
(21.45)
The base points (x1 , x2 ) = (vk , vl0 ) define nine k(T )-rational points of the generic member, which can be transformed to a Weierstrass form over k(T ) such that one of the points, say (v3 , v30 ), is mapped to the point at infinity (cf. [2]). By carrying out the computation, one obtains an equation is of the form: E (1) : y 2 = x3 + AT 2 x + T 2 B(T )
(21.46)
where A is a constant and B(T ) is a quadratic polynomial which depend on the coefficients of f1 , f2 . By replacing x, y, T by suitable constant multiples, they can be normalized so that A = −3α, B(T ) = T 2 − 2βT + 1, (21.47) p √ with α = 3 j1 j2 , β = (1 − j1 )(1 − j2 ) as in (21.2). (Note that the choice of
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the cube root or square root is irrelevant, since they give rise to isomorphic Weierstrass equations.) Going back to (21.38), we see that the Weierstrass form of this plane cubic is given by E (2) = E (1) |T =t2 defined by (21.39). The singular fibres are easily determined by using [7] or [18]. (Also it is a simple consequence of the following lemma, since the map t 7→ T = t2 is a double cover ramified only at t = 0 and ∞. ) Lemma 21.5.2. The elliptic surface corresponding to E (1) is a rational surface. It has two singular fibres of type IV at T = 0 and ∞, and other singular fibres are given as follows: i) I1 × 4 if j1 6= j2 , j1 j2 6= 0, ii) II × 2 if j1 6= j2 , j1 j2 = 0, iii) I2 + I1 × 2 if j1 = j2 6= 0, 1, iv) I2 × 2 if j1 = j2 = 1, (v) IV if j1 = j2 = 0. Proof The discriminant ∆(E (1) ) is given by T 4 (B(T )2 − 4α3 T 2 ) up to constant. Then the verification is a simple exercise using [7] or [18]. Proposition 21.5.3. The Mordell-Weil lattice E (1) (k(T )) is isomorphic to (A∗2 )⊕2 if j1 6= j2 , and to A∗2 ⊕ h1/6i, or h1/6i⊕2 or A∗2 ⊕ Z/3Z if j1 = j2 , in the respective case (iii) or (iv) or (v). E (1) (k(T )) is generated by the rational points of the form x = aT, y = T (cT + d). If j1 6= j2 , there are 12 such points which are the 12 minimal vectors of height (or norm) 2/3 in (A∗2 )⊕2 . Proof Assume j1 6= j2 . By the height formula (21.13), a point P = (x, y) ∈ E (1) (k(T )) has the minimal norm 2/3 if and only if (P O) = 0 and (P ) passes through the non-identity component of each of the two reducible fibres of type IV ∗ . The first condition (P O) = 0 implies that the coordinates x, y of P are polynomial of degree ≤ 2 or 3 (cf.[14, §10]) and the second condition implies that their constant terms as well as the highest terms should vanish. Hence the result follows. The same method can be applied for the case j1 = j2 . We note some consequence of Proposition 21.5.1: Corollary 21.5.4. The Mordell-Weil rank r(2) = rk E (2) (k(t)) is equal to 4 + h if j1 6= j2 , and to 2 + h (resp. h = 2) in case (iii) (resp. (iv) or (v)). Proof The rank r(2) is equal to the Picard number of S minus the rank of the trivial lattice (cf. (21.8)), so the verification is immediate.
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Corollary 21.5.5. The torsion subgroup of E (2) (k(t)) is trivial in case (i)(iv) and Z/3Z in case (v). Proof This follows from the classification results of Shimada [13] (cf. also [10], [9]). Corollary 21.5.6. The condition (#) in § 21.4 holds if and only if j1 6= j2 , i.e. C1 and C2 are not isomorphic to each other.
21.6 MWL of Inose’s pencil We keep the notation from the previous sections. We assume the condition j1 6= j2 in this section. The Mordell-Weil lattice L = E (2) (k(t)) obviously contains E (1) (k(T )) with T = t2 , which can be identified with the sublattice L1 of Lemma 21.4.1. Recall that the height of a point gets multiplied by the degree of the base change (see [14, Prop. 8.12]). Let σ : t 7→ −t be the non-trivial automorphism of the quadratic extension k(t)/k(T ). It acts naturally on L and we have L1 = E (1) (k(T )) = {P ∈ L|P σ = P }.
(21.48)
On the other hand, letting F (1) denote the quadratic twist of E (1) /k(T ) with respect to k(t)/k(T ) (t2 = T ): F (1) : y 2 = x3 − 3αT 4 x + T 5 (T 2 − 2βT + 1),
(21.49)
F (1) (k(T ))→ ˜ {P ∈ E (2) (k(t))|P σ = −P } =: L001 ⊂ L
(21.50)
we have
where Q = (x(T ), y(T )) ∈ F (1) (k(T )) corresponds to P = (x(t2 )/t2 , y(t2 )/t3 ). Note that L001 is orthogonal to L1 (and L001 ∩L1 = 0) since there is no 2-torsion (in fact, L is torsion free under the assumption). Moreover, as is standard in this situation, we have for any P ∈ L 2P = (P + P σ ) + (P − P σ ),
(P + P σ ) ∈ L1 ,
(P − P σ ) ∈ L001 . (21.51)
It follows that L1 + L001 ⊃ 2L and L001 = L01 is the orthogonal complement of L1 , (4.5). Also V1 (or V10 ) in §4 is the eigenspace with eigenvalue 1 (or −1) for the action of σ on V = L ⊗ Q. Theorem 21.6.1. Assume j1 6= j2 . Then the Mordell-Weil lattice L =
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E (2) (k(t)) contains the sublattice L1 ⊕ L01 with finite index I = 2h where 24 (21.52) L1 = E (1) (k(T ))[2] ∼ = A∗2 [2]⊕2 , rk L1 = 4, det L1 = 2 , 3 L01 ∼ = F (1) (k(T ))[2] ∼ = H[4], rk L01 = h, det L01 = 22h · δ (21.53) in which h (or δ) denotes as before the rank (or det) of H = Hom(C1 , C2 ). Proof The only facts yet to be proven in the the above statements are the following: i) I = 2h , ii) L01 ∼ (21.54) = H[4] Letting ν be the index of H[4] in L01 in Proposition 21.4.5, we have det L01 = 4h · δ/ν 2 .
(21.55)
det L = det(L1 ⊕ L01 )/I 2 = 24 /32 · 4h δ/(ν 2 I 2 )
(21.56)
Hence we have
On the other hand, by applying (21.14) to E = E (2) , S = Km(C1 × C2 ) and T = U ⊕ E62 , we have det N S(S) = det L · 32 , which gives by Theorem 21.3.3 det L = 24 · δ/32 .
(21.57)
By comparing (21.56) and (21.57), we have I · ν = 2h .
(21.58)
The next lemma shows I = 2h , and hence ν = 1 by (21.58), which is equivalent to the claim (ii) in (21.54) . This proves both claims of (21.54). Lemma 21.6.2. The map ϕ 7→ Pϕ induces an isomorphism H/2H ∼ = L/(L1 + L01 ).
(21.59)
In particular, the index I = [L : L1 + L01 ] is equal to 2h . Proof The map in question induces a surjective homomorphism of H to L/L1 (as shown in the proof of Proposition 21.4.5), and hence H → L/(L1 + L01 ) is also a surjection. By (6.4), the latter map induces a surjection H/2H → L/(L1 + L01 ), which is also an injection by Lemma 21.4.3. Theorem 21.6.3. Let S (1) denote the elliptic surface associated with F (1) /k(T ). Then it is a K3 surface, and it has two singular fibres of type II ∗ at T = 0, ∞, and the other singular fibres are given as follows:
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i) I1 × 4 if j1 6= j2 , j1 j2 6= 0, ii) II × 2 if j1 6= j2 , j1 j2 = 0, (iii) I2 + I1 × 2 if j1 = j2 6= 0, 1, iii) I2 × 2 if j1 = j2 = 1, (v) IV if j1 = j2 = 0. The Mordell-Weil rank r(1) = rk F (1) (k(T )) is equal to h if j1 6= j2 , and to h − 1 (resp. h − 2 = 0) in case iii) (resp. iv) or v)). Assume j1 6= j2 . Then the Mordell-Weil lattice F (1) (k(T )) is isomorphic to H[2] = Hom(C1 , C2 )[2]. Proof The singular fibres are checked in the same way as in Lemma 21.5.2 for E (1) /k(T ) which is the twist of F (1) /k(T ), and it shows that S (1) is a K3 surface since the Euler number (or the order of the discriminant) is 24. As for the rank formula, it follows from Corollary 21.5.4 and r(1) = rk E (2) (k(t)) − rk E (1) (k(T )).
(21.60)
The final assertion is just a restatement of the fact L01 ∼ = H[4] proven in Theorem 21.6.1, (21.53), in view of the height behavior under the base change (here, of degree two). Now Theorem 21.1.1 or 21.1.2 in the Introduction (§21.1) follow from the above Theorem 21.6.3 or 21.6.1. (Note that F (1) /k(T ) in §(21.1) and §21.6 are the same up to simple coordinate change. Also F (2) /k(t) and E (2) /k(t) are isomorphic.) They are formulated in terms of elliptic curves only, without reference to a K3 or Kummer surface, but the latter is essential for the proof as seen above.
21.7 Comments on the case j1 = j2 We have excluded the case j1 = j2 for the sake of simplicity in some of the above discussion. In this case, we have “extra” reducible fibres in Inose’s pencil (see Proposition21.5.1) and the Mordell-Weil rank drops. We can clarify this situation by the use of the curve Γϕ (§21.3) for the “ isomorphism correspondence”. Proposition 21.7.1. Assume j1 = j2 , i.e. C1 and C2 are isomorphic elliptic curves. Let ϕ : C1 → C2 be any isomorphism. Then the curve Γϕ (the image of the graph of ϕ in A = C1 × C2 under the rational map A → S) is an irreducible component of an extra reducible fibre. Proof We can assume that C1 = C2 and it is defined by (5.1). Recall that the elliptic fibration f : S → P1 is given by the function(21.37): t = y2 /y1 .
(21.61)
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Suppose ϕ : (x1 , y1 ) 7→ (x2 , y2 ) is an automorphism of C1 . i) If ϕ = id is the identity, then we have t = y2 /y1 = 1 on its graph. Hence the curve Γid is contained in the fibre over t = 1, f −1 (1). Similarly, if ϕ = −id is the inversion, we have y2 = −y1 so that t = −1. Hence Γ−id ⊂ f −1 (−1). In this way, we get two reducible fibres of type I2 at t = 1, −1 for general value of j1 , namely for j1 6= 0, 1. ii) Let C1 : y12 = x31 − x1 (j1 = √ 1) and suppose ϕ : (x1 , y1 ) 7→ (−x1 , ±iy1 ). Then we have t = ±i (i = −1). In this case, we get four reducible fibres of type I2 at t = 1, −1, i, −i. iii) Let C1 : y12 = x31 − 1 (j1 = 0) and suppose ϕ : (x1 , y1 ) 7→ (ωx1 , ±y1 ) (ω 3 = 1). Then we have t = ±1. In this case, we get two singular fibres of type IV at t = 1, −1. The three curves Γϕ for three values of ω give the three irreducible components for a type IV -fibre. This completes the proof.
21.8 Examples First, the general case of Theorem 21.6.1 and 21.6.3 is very simple. Example 21.8.1. Assume that C1 , C2 are non-isogeneous elliptic curves. Then F (2) (k(t)) ∼ (21.62) = A∗2 [2], F (1) (k(T )) = {0} The generators of k(t)-rational points are given by Proposition 21.5.3 by setting T = t2 . Next a special case of Theorem 21.6.3 implies: Example 21.8.2. Suppose that C2 : y 2 = x3 − 1 (j2 = 0) and j1 = j is √ arbitrary. We have α = 0, β = 1 − j, and F (1) has the equation: Fj = F (1) : y 2 = x3 + T 5 (T 2 − 2βT + 1),
j = 1 − β2.
(21.63)
Then 1) Fj has a k(T )-rational point (6= O) if and only if j = j(C1 ) for some elliptic curve C1 isogeneous (but not isomorphic) to C2 , and thus 2) Fj (k(T )) 6= {O} holds only for countably many values of j ∈ k. Let us write down further properties of the elliptic curve Fj /k(T ). For simplicity, assume char(k) = 0 and let Fj (k(T )) 6= {O}. 3) The Mordell-Weil lattice Fj (k(T )) is isomorphic to A2 [d0 ] where d0 = deg(ϕ0 ) is the minimal isogeny ϕ0 : C1 → C2 . (A2 is the root lattice.) 4) The minimal height is 2d0 ≥ 4.
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5) There is a rational point P of height hP, P i = 4 if and only if C1 has degree 2 isogeny to C2 . In this case, P = (ξ, η) is an “integral point” with both ξ, η ∈ k[T ] with deg(ξ) = 4, deg(η) = 6. (This follows from the height formula.) 6) Such C1 is unique up to isomorphism and one has j = j(C1 ) = 125/4. Then Fj is given by √ Fj : y 2 = x3 + T 5 (T 2 − 11 −1T + 1). (21.64) which is equivalent (up to coordinate change) to the following equation: y 2 = x3 + T 5 (T 2 − 11T − 1).
(21.65)
7) There are three different methods to find an integral point P = (ξ, η) of height 4 of the form in (5). [N.B. The resulting integral point is essentially the same by the uniqueness in (5) or (6).] (i) straightforward computer search ([3]), (ii) explicit computation of Pϕ = µ(Γϕ ) for the degree two isogeny ϕ, explained in the present paper (the result was announced in [15]), (iii) use the idea of “Shafarevich partner” ([16]). We outline the third method below, because this simple device can be useful in more general situation. 8) Take a rational elliptic surface with 4 singular fibres I1 × 2, I5 × 2 (see e.g. [1]). Assume that the fibres of type I5 are at T = 0, ∞. Write down the minimal Weierstrass equation of the generic fibre as follows: Y 2 = X 3 − 3ξX − 2η,
ξ, η ∈ k[T ]
(21.66)
Then the discriminant is equal to ξ 3 − η 2 up to constant, but at the same time it should take the form cT 5 (T − v1 )(T − v2 ) by the data of singular fibres. This shows that P = (ξ, η) gives rise to an integral point of required form. Note that this argument can be reversed to prove the existence and uniqueness of the rational elliptic surface with singular fibres I1 × 2, I5 × 2, from the knowledge of such an integral point of height 4.
References [1] Beauville, A.: Les familles stables de courbes elliptiques sur P1 , C. R. Acad. Sci. Paris, 294 (I), 657–660 (1982). [2] Cassels, J.W.S.: Lectures on Elliptic Curves Cambridge Univ. Press (1991). [3] Chahal, J., M. Meijer and J. Top: Sections on certain j = 0 elliptic surfaces, Comment. Math. Univ. St. Pauli 49 (2000).
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[4] Conway, J. and N. Sloane: Sphere Packings, Lattices and Groups, SpringerVerlag (1988); 2nd ed.(1993); 3rd ed.(1999). [5] Inose, H. : Defining equations of singular K3 surfaces and a notion of isogeny in Intl. Symp. on Algebraic Geometry/ Kyoto 1977 Kinokuniya, 495–502 (1978). [6] Inose, H. and T. Shioda: On singular K3 surfaces, in: Complex Analysis and Algebraic Geometry, Iwanami Shoten and Cambridge Univ. Press, 119–136 (1977). [7] Kodaira, K.: On compact analytic surfaces II-III, Ann. of Math. 77, 563626(1963); 78, 1–40(1963); Collected Works, III, 1269–1372, Iwanami and Princeton Univ. Press (1975). [8] Kuwata, M.: Elliptic K3 surfaces with given Mordell-Weil rank, Comment. Math. Univ. St. Pauli 49 (2000). [9] Nishiyama, K.: On Jacobian fibrations on some K3 surfaces and their MordellWeil groups, Japan J. Math. 22, 293–347 (1996). [10] Oguiso, K.: On Jacobian fibrations on the Kummer surfaces of product of non-isogeneous elliptic curves, J. Math. Soc. Japan 41, 651–680 (1989). [11] Oguiso, K. and T. Shioda: The Mordell–Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli 40, 83–99 (1991). [12] Piateckii-Shapiro, I. and I. R. Shafarevich: A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5, 547–587 (1971). [13] Shimada, I.: On elliptic K3 surfaces, Michigan Math. J. 47 (2000), 423–446. [14] Shioda, T.: On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39, 211–240 (1990). [15] — : A note on K3 surfaces and sphere packings, Proc.Japan Acad. 76A, 68–72 (2000). [16] — : Elliptic surfaces and Davenport-Stothers triples, Comment. Math. Univ. St. Pauli (to appear). [17] — : Classical Kummer surfaces and Mordell-Weil Lattices, Proc. KIAS Conf. Algebraic Geometry 2004 (to appear). [18] Tate, J: Algorithm for determining the type of a singular fiber in an elliptic pencil, SLN 476, 33-52 (1975). [19] Weil, A.: Vari´et´es ab´eliennes et courbes alg´ebriques, Hermann, Paris (1948/1973).
22 Motives from Diffraction Jan Stienstra Mathematisch Instituut, Universiteit Utrecht, the Netherlands
[email protected]
Dedicated to Jaap Murre and Spencer Bloch
Abstract We look at geometrical and arithmetical patterns created from a finite subset of Zn by diffracting waves and bipartite graphs. We hope that this can make a link between Motives and the Melting Crystals/Dimer models in String Theory.
22.1 Introduction Why is it that, occasionally, mathematicians studying Motives and physicists searching for a Theory of Everything seem to be looking at the same examples, just from different angles? Should the Theory of Everything include properties of Numbers? Does Physics yield realizations of Motives which have not been considered before in the cohomological set-up of motivic theory? Calabi-Yau varieties of dimensions 1 and 2, being elliptic curves and K3-surfaces, have a long and rich history in number theory and geometry. Calabi-Yau varieties of dimension 3 have played an important role in many developments in String Theory. The discovery of Mirror Symmetry attracted the attention of physicists and mathematicians to Calabi-Yau’s near the large complex structure limit[13, 20]. Some analogies between String Theory and Arithmetic Algebraic Geometry near this limit were discussed in[16, 17, 18]. Recently new models appeared, called Melting Crystals and Dimers [14, 11], which led to interesting new insights in String Theory, without going near the large complex structure limit. The present paper is an attempt to find motivic aspects of these new models. We look at geometrical and enumerative patterns associated with a finite subset A of Zn . The 645
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geometry comes from waves diffracting on A and from a periodic weighted bipartite graph generated by A. The latter is related to the dimers(although here we can not say more about this relation). Since the tori involved in these models are naturally dual to each other there seems to be some sort of mirror symmetry between the diffraction and the graph pictures. The enumerative patterns count lattice points on the diffraction pattern, points on varieties over finite fields and paths on the graph. They are expressed through a sequence of polynomials BN (z) with coefficients in Z and via a limit for z ∈ C: Q(z) = lim |BN (z)|−N N →∞
−n
.
(22.1)
Limit formulas like (22.1) appear frequently and in very diverse contexts in the literature, e.g. for entropy in algebraic dynamical systems in [7, Theorem 4.9], for partition function per fundamental domain in dimer models, in [11, Theorem 3.5] for integrated density of states in [9, p. 206]. Moreover, Q(z) appears as Mahler measure in [4], as the exponential of a period in Deligne cohomology in [6, 15], and in instanton counts in [18]; see the remark at the end of Section 22.5. With some additional restrictions A provides the toric data fora family of Calabi-Yau varieties and various well-known results about Calabi-Yau varieties near the large complex structure limit can be derived from the Taylor series expansion of log Q(z) near z = ∞; see the Remark at the end of Section 22.6. In the present paper we are not so much interested in the large complex structure limit. Instead we focus on the polynomials BN (z) and the limit formula (22.1). This does not require conditions of ‘Calabi-Yau type’. When waves are diffracted at some finite set A of points in a plane, the diffraction pattern observed in a plane at large distance is, according to the Frauenhofer model, the absolute value squared of the Fourier transform of A. There is no mathematical reason to restrict this model to dimension 2. Also the points may have weights ≥ 1. So, we take a finite subset A of Zn and positive integers ca (a ∈ A). These data can be summarized as a distribution P D = a∈A ca δa , where δa denotes the Dirac delta distribution, evaluating test functions at the point a. The Fourier transform of D is the function P b D(t) = a∈A ca e−2πiht,ai on Rn ; here h, i is the standard inner product on Rn . The diffraction pattern consists of the level sets of the function X X 2 b |D(t)| = ca cb e2πiht,a−bi = ca cb cos(2πht, a − bi) . a,b∈A
a,b∈A
This function is periodic with period lattice Λ∨ dual to the lattice λ spanned over Z by the differences a−b with a, b ∈ A. Throughout this note we assume
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that λ has rank n. Looking at the intersections of the diffraction pattern 100
90
80
70
60
50
40
30
20
10
10
20
30
40
50
60
70
80
90
100
Fig. 22.1. Diffraction pattern for A = {(1, 0), (0, 1), (−1, −1)} ⊂ Z2 , all ca = 1.
with the lattices
1 ∨ NΛ
multN (r) := ]{t ∈
we introduce the enumerative data
1 ∨ ∨ NΛ /Λ
2 b | |D(t)| = r}
for N ∈ N, r ∈ R ,
(22.2)
and use these to define polynomials BN (z) as follows: Definition 22.1.1. BN (z) :=
Y
(z − r)multN (r) .
(22.3)
r∈R
P n One could also introduce the generating function F(z, T ) := N ∈N BN (z) T N , but except for the classical number theory of the case n = 1 (see Section 22.7.1), and the observation that Formula (22.1) gives Q(z) as the radius of convergence of F(z, T ) as a complex power series in T , we do not yet have appealing results about F(z, T ). For the graph model we start from the same data: the finite set A ⊂ Zn , the weights ca (a ∈ A) and the lattice λ spanned by the differences a − b with a, b ∈ A. We must now assume that A ∩ Λ = ∅. One can then construct a weighted bipartite graph Γ as follows. Bipartite graphs have two kinds of vertices, often called black and white. The set of black vertices of Γ is λ. The set of white vertices of Γ is A + λ. Note
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d
t
t
d
d @ d t
d
t d d t @ d t d @ t
d
@
t
d
d t
d
@
t
@ @
t
t
d t
t
d
d t
t
d @ @ t d
d
t
d @ t
t d
d t
t d
Fig. 22.2. A fundamental parallelogram of the lattice 3λ and a piece of the bipartite graph Γ for A = {(1, 0), (0, 1), (−1, −1)} ⊂ Z2 , and all ca = 1. By identifying opposite sides of the parallelogram one obtains the graph Γ3 .
that A + λ is just one single coset of λ in Zn . In Γ there is an (oriented) edge from vertex v1 to vertex v2 if and only if v1 is black, v2 is white and v2 − v1 ∈ A. If v2 − v1 = a ∈ A the edge is said to be of type a and gets weight ca . The graph Γ is λ-periodic and for every N ∈ N one has the finite graph ΓN := Γ/ N Λ, which is naturally embedded in the torus Rn/ N Λ. By a closed path of length 2k on Γ or ΓN we mean a sequence of edges (e1 , e2 , . . . , e2k−1 , e2k ) such that for i = 1, . . . , k the intersection e2i−1 ∩ e2i contains a white vertex and e2i ∩ e2i+1 contains a black vertex; here e2k+1 = e1 . By the weight of such a path we mean the product of the weights of the edges e1 , e2 , . . . , e2k−1 , e2k . We denote the set of closed paths of length 2k on ΓN by ΓN (2k). Enumerating the closed paths on ΓN according to length and weight we prove in Section 22.4 that this leads to a new interpretation of the polynomials BN (z): Theorem 22.1.2.
n
BN (z) = z N exp −
X
X
weight(γ)z −k .
(22.4)
k≥1 γ∈ΓN (2k)
Formulas (22.3) and (22.4) transfer the enumerative data between the two models. We pass to algebraic geometry with the Laurent polynomial X ±1 W (x1 , . . . , xn ) = ca cb xa−b ∈ Z[x±1 (22.5) 1 , . . . , xn ] , a,b∈A
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Qn λj 2 = W (e2πit1 , . . . , e2πitn ); here xλ := b which satisfies |D(t)| j=1 xj if λ = (λ1 , . . . , λn ) ∈ λ. For N ∈ N let µN denote the group of N -th roots of unity and let µΛ N := Hom(Λ, µN ) be the group of homomorphisms from the lattice λ to µN . Thus the defining formula (22.3)) can be rewritten as:
BN (z) =
Y
(z − W (x)) .
(22.6)
x∈µΛ N
Written in the form (22.6)) the polynomials BN (z) appear as direct generalizations of quantities introduced by Lehmer [12] for a 1-variable (i. e. n = 1) polynomial W (x). Using (22.6) one can easily show (Proposition 22.5.1) that the polynomials BN (z) have integer coefficients and that BN 0 (z) divides BN (z) in Z[z] if N 0 divides N in Z. Thus for h ∈ Z also BN (h) is an integer. Lehmer was particularly interested in the prime factorization of these integers in case n = 1 [12, 7, 15]. Also for general n ≥ 1 these prime factorizations must be interesting, for instance because they relate to counting points on varieties over finite fields; see Section 22.5 for details. Thus prime factorization gives a third occurrence of BN (z) in enumerative problems, related to counting points on varieties over finite fields. Q(z) appears in [4, 6, 15] as Mahler measure with ties to special values of l-functions. It would be nice if the limit formula (22.1))together with the prime factorization of the numbers BN (z) (with z ∈ Z) could shed new light on these very intriguing ties. In Section 22.2 we study the density distribution of the level sets in the diffraction pattern. Passing from measures to complex functions with the Hilbert transform we find one interpretation of Q(z), BN (z) and (22.1)). In Section 22.3 we briefly discuss another interpretation in connection with the spectrum of a discretized Laplace operator. In Section 22.4 we prove Theorem 22.1.2. In Section 22.5 we pass to toric geometry, where the diffraction pattern reappears as the intersection of a real torus with a family of hypersurfaces in a complex torus and where log Q(z) becomes a period integral, while the prime factorization of BN (z) for z ∈ Z somehow relates to counting points on those hypersurfaces over finite fields. In Section 22.6 we discuss sequences of integers which appear as moments of measures, path counts on graphs and coefficients in Taylor expansions of solutions of Picard-Fuchs differential equations. Finally, in Section 22.7 we present some concrete examples.
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22.2 The diffraction pattern 2 is periodic with period lattice Λ∨ dual to the lattice λ: b The function |D(t)|
Λ∨ := {t ∈ Rn | ht, a − bi ∈ Z , ∀a, b ∈ A} , Λ := Z−Span{a − b | a, b ∈ A}. Throughout this note we assume that the lattices λ and Λ∨ have rank n. 2 descends to a function on Rn/ ∨ . Defining b Because of this periodicity |D(t)| Λ n n 2πiht,vi for t ∈ R the function et : R → C by et (v) = e we obtain an isomorphism of real tori
Rn/ Λ∨ ' UΛ ,
t 7→ et ,
(22.7)
where UΛ := Hom(Λ, U) is the torus of group homomorphisms from the lattice λ to the unit circle U := {x ∈ C | |x| = 1 }. Recall that a group homomorphism ψ : Λ → U induces an algebra homomorphism ψ∗ from the group algebra C[Λ] to C. Thus C[Λ] is the natural algebra of functions on UΛ and ψ∗ evaluates functions at the point ψ of UΛ . The inclusion Λ ⊂ Zn identifies C[Λ]with the subalgebra of the algebra of Laurent polynomials ±1 C[x±1 1 , . . . , xn ], which consists of C-linear combinations of the monomials Q λ xλ := nj=1 xj j with λ = (λ1 , . . . , λn ) ∈ λ. Thus, via (22.7)), the func2 coincides with the Laurent polynomial W (x , . . . , x ) defined b tion |D(t)| 1 n 2 b in (22.5)). Positivity of the coefficients ca implies that the function |D(t)| attains its maximum exactly at the points t ∈ Λ∨ . In terms of the torus UΛ and the function W : UΛ → R this means that W attains its maximum exactly at the origin 1 of the torus group UΛ : X ∀x ∈ UΛ − {1} : W (x) < W (1) = C 2 with C := ca . a∈A
Some important aspects of the density distribution in the diffraction pattern are captured by the function V : R → R,
V (r) :=
2 ≤ r} b volume{t ∈ Rn/ Λ∨ | |D(t)| . n volume( R / Λ∨ )
(22.8)
We view the derivative dV (r) of V as a measure on R . In our analysis it will be important that the measure dV (r) is also the push forward of the standard 2 . Another insight into b measure dt1 dt2 . . . dtn on Rn by the function |D(t)| the diffraction pattern comes from its intersection with the torsion subgroup of UΛ . For N ∈ N let µN ⊂ U denote the group of N -th roots of unity. Then
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the group of N -torsion points in UΛ is µΛ N := Hom(Λ, µN ) and (22.2))can be rewritten as −1 multN (r) = ](µΛ (r)) N ∩ W
N ∈ N, r ∈ R .
for
Moreover we set, in analogy with (22.8)), VN (r) :=
1 ]{x ∈ µΛ N | W (x) ≤ r} Nn
for N ∈ N, r ∈ R .
The derivative of the step function VN : R → R is the distribution X multN (r) δr , dVN (r) = N −n
(22.9)
r
which assigns to a continuous function f on R the value Z X X f (r)dVN (r) := N −n multN (r) f (r) = N −n f (W (x)) . R
r
(22.10)
x∈µΛ N
One thus finds a limit of distributions lim dVN (r) = dV (r) ;
N →∞
(22.11)
by definition, this means that for every continuous function f on R Z Z lim f (r)dVN (r) = f (r)dV (r) . (22.12) N →∞ R
R
Measure theory is connected with complex function theory by the Hilbert transform. The Hilbert transform of the measure dV (r) is the function − π1 H(z) defined by Z 1 H(z) := dV (r) for z ∈ C − I ; (22.13) z − r R here I := {r ∈ R | 0 < V (r) < 1} ⊂ [0, C 2 ] is the support of the measure dV (r). The measure can be recovered from its Hilbert transform because for r0 ∈ R dV 1 (r0 ) = lim (H(r0 + i) − H(r0 − i)) . dr 2πi ∈R,↓0 Another way of writing the connection between dV (r) and H(z) is I Z I Z 1 1 f (z) f (z)H(z)dz = dz dV (r) = f (r)dV (r) 2πi γ R 2πi γ z − r R (22.14)
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for holomorphic functions f defined on some open neighborhood U of the interval I in C and closed paths γ in U −I encircling I once counter clockwise. Next we consider the function Z Q : C − I −→ R>0 , Q(z) := exp − log |z − r|dV (r) . (22.15) R d This function satisfies dz log Q(z) = −H(z) and thus (22.14)) can be rewritten as Z I −1 f (r)dV (r) . f (z)d log Q(z) = 2πi γ R
This means that, at least intuitively, the functions Q(z) and e−V (r) correspond to each other via some kind of comparison isomorphism. In order to find the analogue of (22.11)) in terms of functions on C − I we apply (22.10))to the function f (r) = log |z − r| on R with fixed z ∈ C − I: Z log |z − r|dVN (r) = N −n log |BN (z)| (22.16) R
Q where BN (z) = r∈I (z − r)multN (r) = x∈µΛ (z − W (x)) as in (22.3)) and N (22.6)). Combining (22.12)), (22.15)) and (22.16)) we find the limit announced in (22.1)): Q
Q(z) = lim |BN (z)|−N
Proposition 22.2.1.
N →∞
−n
for every z ∈ C − I.
22.3 The Laplacian perspective Convolution with the distribution D gives the operator Df (v) :=
X
ca f (v − a)
a∈A
on the space of C-valued functions on Rn . Let Df (v) :=
X
ca f (v + a) and
a∈A
∆ := DD ,
∆f (v) =
X
ca cb f (v + a − b) .
a,b∈A
For a sufficiently differentiable function f on Rn the Taylor expansion ∆f (v) = C 2 f (v) +
1 2
n X X i,j=1 a,b∈A
ca cb (ai − bi )(aj − bj )
∂2f (v) + . . . ∂vi ∂vj
shows that the difference operator ∆ − C 2 is a discrete approximation of the 2 at its b Laplace operator corresponding to the Hessian of the function |D(t)| maximum.
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Remark. In [9] Gieseker, Kn¨orrer and Trubowitz investigate Schr¨odinger equations in solid state physics via a discrete approximation of the Laplacian. In their situation the Schr¨odinger operator is the discretized Laplacian plus a periodic potential function. So from the perspective of [9]the present note deals only with the (simple) case of zero potential. On the other hand we consider more general discretization schemes and possibly higher dimensions. We now turn to the spectrum of ∆. For t ∈ Rn the function et : Rn → C 2: b given by et (v) = e2πiht,vi is an eigenfunction for ∆ with eigenvalue |D(t)| X 2 b ∆et (v) = ca cb e2πiht,v+a−bi = |D(t)| et (v) . a,b∈A
Take a positive integer N . The space of C-valued C ∞ -functions on Rn which are periodic for the sublattice N λ of λ is spanned by the functions et with t in the dual lattice N1 Λ∨ . The characteristic polynomial of the restriction of ∆ to this space is therefore (see (22.3) and (22.6) Y Y Y 2 b (z−|D(t)| ) = (z−W (x)) = (z−r)multN (r) = BN (z) . t∈
1 ∨ ∨ NΛ /Λ
x∈µΛ N
r∈R
With (22.9)) and (22.11)) the measure dV (r) can now be interpreted as the density of the eigenvalues of ∆ on the space of C-valued C ∞ -functions on Rn which are periodic for some sublattice N λ of λ. 22.4 Enumeration of paths on a periodic weighted bipartite graph In this section we prove Theorem 22.1.2. Recall from the Introduction just before Theorem 22.1.2 the various ingredients: the finite set A ⊂ Zn , the weights ca , the lattice λ and the graphs Γ and ΓN . Recall also the closed paths on ΓN , their lengths and weights, and the set ΓN (2k) of closed paths of length 2k on ΓN . Consider a path (e1 , e2 , . . . , e2k−1 , e2k ) on Γ with edge ei going from black to white if i is odd, respectively from white to black if i is even. Let s denote the starting point of the path (i. e. the black vertex of edge e1 ). Let for j = 1, . . . , k edge e2j−1 be of type aj and edge e2j of type bj . Then the end point of the path (i. e. the black vertex of e2k ) i s P Q s + kj=1 (aj − bj ). The weight of the path i s kj=1 caj cbj . The path closes P on ΓN if and only if kj=1 (aj − bj ) ∈ N λ. Next recall from (22.5)) that P a−b and set W (x) = a,b∈A ca cb x X (N ) mk := N −n W (x)k . (22.17) x∈µΛ N
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J. Stienstra (N )
So mk is the sum of the coefficients those monomials in W (x)k with expo(N ) nent in N λ. In view of the above considerations mk is therefore equal to the sum of the weights of the paths on Γ which start at s, have length 2k and close in ΓN . Since on ΓN there are N n black vertices and on a path of length 2k there are k black vertices we conclude N n (N ) m = k k
X
weight(γ) .
(22.18)
γ∈ΓN (2k)
From (22.6) one sees for |z| > C 2 N −n log BN (z) = log z + N −n
X x∈µΛ N
X m(N ) k log(1−W (x)z −1 ) = log z− z −k . k k≥1
(22.19) Combining (22.18) and (22.19) we find X X n BN (z) = z N exp −
weight(γ) z −k .
k≥1 γ∈ΓN (2k)
This finishes the proof of Theorem 22.1.2. (N )
Remark. In the Laplacian perspective N n mk is the trace of the operator ∆k on the space of C-valued C ∞ -functions on Rn which are periodic for the sublattice N λ of λ. The polynomial BN (z) is the characteristic polynomial of ∆ on this space. Formula (22.19)) gives the well-known relation between the characteristic polynomial of an operator and the traces of its powers. Remark. One may refine the above enumerations by keeping track of the homology class to which the closed path belongs. That means that instead of (22.17) one extracts from the polynomial W (x)k the sub polynomial consisting of terms with exponent in N λ. Such a refinement of the enumerations with homology data appears also in the theory of dimer models (cf. [11]), but its meaning for the diffraction pattern is not clear.
22.5 Algebraic geometry. Q The polynomial BN (z) = x∈µΛ (z − W (x)) has coefficients in the ring of N integers of the cyclotomic field Q(µN ) and is clearly invariant under the Galois group of Q(µN ) over Q. Consequently, the coefficients of BN (z) lie in Z. The same argument applies to the polynomial BN (z)BN 0 (z)−1 = Q 0 x∈µΛ −µΛ 0 (z − W (x)) if N divides N . Thus we have proved N
N
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Proposition 22.5.1. For every N ∈ N the coefficients of BN (z) lie in Z. If N 0 divides N in Z, then BN 0 (z) divide s BN (z) in Z[z]. Fix a prime number p and a positive integer ν ∈ Z>0 . Let W(Fpν ) denote the ring of Witt vectors of the finite field Fpν (see e. g. [3]). So, W(Fpν ) is a complete discrete valuation ring with maximal ideal pW(Fpν ) and residue field Fpν . The Teichm¨ uller lifting is a map τ : Fpν −→ W(Fpν ) such that x ≡ τ (x) mod p ,
τ (xy) = τ (x)τ (y)
∀x, y ∈ Fpν .
Every non-zero x ∈ Fpν satisfies ν −1
xp
= 1.
Thus there is an isomorphism µpν −1 ' F∗pν . Such an isomorphism composed with the Teichm¨ uller lifting gives an embedding j : µpν −1 ,→ W(Fpν ). Thus for x ∈ µΛ we get pν −1 W (j(x)) ∈ W(Fpν ) . Recall the p-adic valuation on Z: for k ∈ Z, k 6= 0: vp (k) := max{v ∈ Z | pv divides k} . Proposition 22.5.2. For p, ν as above and for z ∈ Z the p-adic valuation of the integer Bpν −1 (z) satisfies vp (Bpν −1 (z)) ≥ ]{ξ ∈ (F∗pν )n | W (ξ) = z in Fpν } .
(22.20)
Proof From (22.6) we obtain the product decomposition, with factors in W(Fpν ), Y BN (z) = (z − W (τ (ξ))) . ξ∈(F∗pν )n
The result of the proposition now follows because z − W (τ (ξ)) ∈ pW(Fpν )
⇔
W (ξ) = z in Fpν
Remark. In (22.30)) we give an example showing that in (22.20)) we may have a strict inequality. Remark about the relation with Mahler measure and L-functions. The logarithmic Mahler measure m(F ) and the Mahler measure M(F ) of a
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Laurent polynomial F (x1 , . . . , xn ) with complex coefficients are: II dx1 dxn 1 log |F (x1 , . . . , xn )| · ... · , m(F ) := n (2πi) x1 xn |x1 |=...=|xn |=1 M(F ) := exp(m(F )) . Boyd [4] gives a survey of many (two-variable) Laurent polynomials for which m(F ) equals (numerically to many decimal places) a ‘simple’ non-zero rational number times the derivative at 0 of the L-function of the projective plane curve Zf defined by the vanishing of f : m(F ) · Q∗ = L0 (ZF , 0) · Q∗ .
(22.21)
Deninger [6] and Rodriguez Villegas [15] showed that the experimentally observed relations (22.21)) agree with predictions from the Bloch-Beilinson conjectures. Rodriguez Villegas [15] provided actual proofs for a few special examples. Since the measure dV (r) is the push forward of the mea2 , one can rewrite Formula b sure dt1 dt2 . . . dtn on Rn by the function |D(t)| (22.15)) as: Z 1 dx1 dx2 dxn − log Q(z) = log |z − W (x1 , . . . , xn )| ... . (22.22) n (2πi) Un x1 x2 xn On the right hand side of (22.22)) we now recognize the logarithmic Mahler ±1 measure of the Laurent polynomial z − W (x1 , . . . , xn ) ∈ C[x±1 1 , . . . , xn ]. For fixed z ∈ Z formulas (22.1) and (22.20) provide a link between Q(z) and counting points over finite fields on the variety with equation W (x1 , . . . , xn ) = z. It may be an interesting challenge to further extend these ideas to a proof of a result like (22.21).
22.6 Moments. Important invariants of the measure dV (r) are its moments mk (k ∈ Z≥0 ): R1 R R1 b 1 , . . . , tn )|2k dt1 . . . dtn mk := R rk dV (r) = 0 . . . 0 |D(t b 1 , . . . , tn )|2k = constant term of Fourier series |D(t = constant term of Laurent polynomial W (x1 , . . . , xn )k . (22.23) The relation between the moments and the functions H(z), Q(z) defined in (22.13) and (22.15) is: for z ∈ R, z > C 2 , X X mk H(z) = mk z −k−1 , Q(z) = z −1 exp z −k . (22.24) k k≥0
k≥1
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It is clear that the moments mk of dV (r) are non-negative integers. They satisfy all kinds of arithmetical relations. There are, for instance, recurrences like (22.28)) and congruences like the following Lemma 22.6.1. mkpα+1 ≡ mkpα mod pα+1 for every prime number p and k, α ∈ Z≥0 . Proof The Laurent polynomial W (x1 , . . . , xn ) has coefficients in Z. Therefore α+1
W (x1 , . . . , xn )kp
α
±1 ≡ W (xp1 , . . . , xpn )kp mod pα+1 Z[x±1 1 , . . . , xn ] .
The lemma follows by taking constant terms. Theorems 1. 1, 1. 2, 1. 3 in [1] together with the above lemma immediately yield the following integrality result for series and product expansions: Corollary 22.6.2. For z ∈ R, z > C 2 Y X X mk (1 − z −k )−bk z −k = z −1 + Ak z −k−1 = z −1 z −1 exp k k≥1
k≥1
k≥1
(22.25) with Ak , bk ∈ Z for all k ≥ 1. Remark. In [1, 8] the result of Corollary 22.6.2 is used to interpret zQ(z) as the Artin-Mazur zeta function of a dynamical system, provided the integers bk are not negative. We have not yet found such a dynamical system within the present framework. For N ∈ N the moments of the measure dVN (r) are, by definition, Z X (N ) mk := rk dVN (r) = N −n W (x)k . R
x∈µN
These are the same numbers as in (22.17)). Proposition 22.6.3. With the above notations we have (N )
mk ≥ mk ≥ 0 (N ) mk = mk if (N )
for all N, k , N > k maxa,b∈A max1≤j≤n |aj − bj | .
Proof N n (mk − mk ) is the sum of the coefficients of all non-constant monomials in the Laurent polynomial W (x1 , . . . , xn )k with exponents divisible by N . Since all coefficients of W (x1 , . . . , xn ) are positive, this shows (N ) mk ≥ mk ≥ 0. Assume N > k maxa,b∈A, 1≤j≤n |aj −bj |. Then all exponents in the monomials of the Laurent polynomial W (x1 , . . . , xn )k are > −N and < N . So only the exponent of the constant term is divisible by N . Therefore (N ) mk = mk .
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Note the natural interpretation (and proof) of this proposition in terms of closed paths on the graph ΓN : closed paths on ΓN which are too short are in fact projections of closed paths on Γ. Corollary 22.6.4. For N > ` max max |aj − bj | and |z| > C 2 : a,b∈A 1≤j≤n
N −n
Q(z) · |BN (z)|
! X mk − m(N ) −k k = exp z . k k>`
This not only gives an estimate for the rate of convergence of (22.1)) with respect to the usual absolute value on C, but it also yields the following congruence of power series in z −1 : BN (z)−N
−n
≡ z −1 +
X
Ak z −k−1 mod z −`−1
k≥1
with Ak as in (22.25)) Remark about the relation with the large complex structure limit. Since the measure dV (r) is the push forward of the measure dt1 dt2 . . . dtn 2 , one can rewrite (22.13) as b on Rn by the function |D(t)| 1 H(z) = (2πi)n
Z Un
1 dx1 dx2 dxn ... z − W (x1 , . . . , xn ) x1 x2 xn
for z ∈ C − I. From this (and the residue theorem) one sees that H(z) is a period of some differential form of degree n − 1along some (n − 1)-cycle on the hypersurface in (C∗ )n given by the equation W (x1 , . . . , xn ) = z. As z varies we get a 1-parameter family of hypersurfaces. The function H(z) is a solution of the Picard-Fuchs differential equation associated with (that (n − 1)-form on) this family of hypersurfaces. The Picard-Fuchs equation is equivalent with a recurrence relation for the coefficients mk in the power series expansion (22.24) of H(z) near z = ∞. All this is standard knowledge about Calabi-Yau varieties near the large complex structure limit and there is an equally standard algorithm to derive from the Picard-Fuchs differential equation enumerative information about numbers of instantons (or rational curves); see for instance [20, 13, 18]. On the other hand, we have the enu(N ) merative data mk of the present paper. In the limit for N → ∞ these yield the moments mk and, hence, the Picard-Fuchs differential equation and eventually the instanton numbers.
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22.7 Examples 22.7.1 n = 1 Mahler measures of one variable polynomials have a long history with many interesting results; see the introductory sections of [4, 7, 15]. We limit our discussion to one example, without a claim of new results. This simple, yet non-trivial, example has n = 1, A = {−1, 1} ⊂ Z, c−1 = c1 = 1 and hence W (x) = (x + x−1 )2 .
2 b |D(t)| = 2 + 2 cos(4πt) ,
The moments are mk = constant term of (x + x−1 )2k =
2k k
and hence by (22.24)): for z ∈ R, z > 4, X 2k 1 z −k−1 = p H(z) = , k z(z − 4) k≥0 ! Z p dz 1 Q(z) = exp − p = z − 2 − z(z − 4) . 2 z(z − 4) Applying Formula (22.17)) to the present example we find X 2k (N ) mk = , j j≡k mod N
which nicely illustrates Proposition 22.6.3. Setting z = 2 + u + u−1 one finds for the polynomials BN (z) defined in (22.3): Y Y BN (z) = (z − 2 − x2 − x−2 ) = u−N (u − x2 )(u − x−2 ) x2 ∈µN
x2 ∈µN
= uN + u−N − 2 N N p p 1 1 z − 2 − z(z − 4) + z − 2 + z(z − 4) −2 = 2 2 X N 1−N = −2 + 2 z j (z − 4)j (z − 2)N −2j . 2j j
ˇ sev polynoSo, BN (z) is up to some shift and normalization the N -th Cebyˇ N mial. The above computation also shows BN (z) = Q(z) + Q(z)−N − 2 and thus, in agreement with (22.1), lim BN (z)−N
N →∞
−1
= Q(z) .
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For actual computation of BN (z) in case z ∈ Z one can use the generating series identity: X T TN . = − log 1 − (z − 4) ] BN (z) N (1 − T )2 N ≥1
For z = 6 one finds (using PARI) X BN (6)T N = 2T + 12T 2 + 50T 3 + 192T 4 + 722T 5 + 2700T 6 + 10082T 7 +37632T 8 + 140450T 9 + 524172T 10 + 1956242T 11 +7300800T 12 + 27246962T 13 + 101687052T 14 +379501250T 15 + 1416317952T 16 + 5285770562T 17 + . . . For primes p in the displayed range the number Bp−1 (6) is divisible by p2 for p ≡ ±1 mod 12 and is not divisible by p for p ≡ ±5 mod 12 and is exactly divisible by p if p = 2, 3. We also checked 52 | B24 (6) and 72 | B48 (6). This −1 agrees with the number of solutions of the equation p u + u = 4 in Fp and 1 Fp2 . If z ∈ Z, z > 4, then Q(z) = 2 ( z − 2 − z(z − 4)) is a unit in the p real quadratic field Q( z(z − 4)). According to Dirichlet’s class number formula it relates to the l-function of this real quadratic field: √ D log(Q(z)) = L(1, χ) 2h where D, h, χ are the discriminant, class number, character, respectively, of p the real quadratic field Q( z(z − 4)) (see e. g. [5]). The relations between Mahler measures and values of l-functions, which have been observed for some curves, are perfect analogues of the above class number formula (see [15]).
22.7.2 The honeycomb pattern For a nice two-dimensional example we take A = {(1, 0), (0, 1), (−1, −1)} ⊂ Z2 , c(1,0) = c(0,1) = c(−1,−1) = 1 and hence b 1 , t2 )|2=3 + 2 cos(2π(t1 − t2 )) + 2 cos(2π(2t1 + t2 )) + 2 cos(2π(t1 + 2t2 )), |D(t −1 −1 −1 W (x1 , x2 )= (x1 + x2 + x−1 1 x2 )(x1 + x2 + x1 x2 ) −1 −2 −1 −1 −2 2 2 =x1 x−1 2 + x1 x2 + x1 x2 + x1 x2 + x1 x2 + x1 x2 + 3 .
As a basis for the lattice λ we take (2, 1) and (−1, −2). This leads to coor−2 Λ dinates u1 = x21 x2 and u2 = x−1 1 x2 on the torus U . In these coordinates
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the function W reads −1 −1 −1 W (u1 , u2 ) = u1 + u−1 1 + u2 + u2 + u1 u2 + u1 u2 + 3 (22.26) −1 = (u1 + u2 + 1)(u−1 1 + u2 + 1) .
Figure 22.2 shows a piece of the graph Γ. Figure 22.1 shows some level sets b 1 , t2 )|2 . The dual lattice Λ∨ is spanned by (1, 0) and of the function |D(t b 1 , t2 )|2 equals 9 and is attained ( 13 , 13 ). The maximum of the function |D(t b 1 , t2 )|2 equals 0 and at the points of Λ∨ . The minimum of the function |D(t 1 1 ∨ is attained at the points of (0, − 3 ) + Λ and ( 3 , 0) + Λ∨ . There are saddle points with critical value 1 at (− 61 , 13 ) + Λ∨ , ( 13 , − 61 ) + Λ∨ and ( 16 , 16 ) + Λ∨ . In terms of the coordinates u1 , u2 the maximum lies at (u1 , u2 ) = (1, 1), the minima at (e2πi/3 , e4πi/3 ), (e4πi/3 , e2πi/3 ) and the saddle points at (1, −1), (−1, 1), (−1, −1). The algebraic geometry of this example concerns the 1-parameter family of elliptic curves with equation z − W (u1 , u2 ) = 0. In homogeneous coordinates (U0 : U1 : U2 ) on the projective plane P2 , with u1 = U1 U0−1 , u2 = U2 U0−1 , this becomes a homogeneous equation of degree 3: (U0 U1 + U0 U2 + U1 U2 )(U0 + U1 + U2 ) − zU0 U1 U2 = 0 .
(22.27)
Beauville [2] showed that there are exactly six semi-stable families of elliptic curves over P1 with four singular fibres. The pencil (22.27) is one of these six. It has singular fibres at z = 0, 1, 9, ∞ with Kodaira types I2 , I3 , I1 , I6 , respectively. Note that the first three match the critical points and levels in the diffraction pattern. After blowing up the points (1, 0, 0), (0, 1, 0), (0, 0, 1) of P2 one gets the Del Pezzo surface dP3 . The elliptic pencil (22.27) naturally lives on dP3 . It has six base points, corresponding to six sections of the pencil. Since the base points have a zero coordinate, these sections do not intersect the real torus UΛ . Equations (22.26) and (22.27) also appear in the literature in connection with the string theory of dP3 . Formula (22.23) and some manipulations of binomials give the moments: 2 k X k 2j mk = . j j j=0
These numbers satisfy the recurrence relation (see [19, Table 7]) (k + 1)2 mk+1 = (10k 2 + 10k + 3)mk − 9k 2 mk−1 .
(22.28)
We refer to [19, Example C] and to [18, Example ]6] for relations of these numbers to modular forms and instanton counts. Golyshev [10] can derive the recurrence(22.28) from the quantum cohomology of dP3 . The numerical
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evidence for the relation (22.21) between Mahler measure and L-function in this example is given in [4, Table 2]. With Formulas (22.17) and (22.26) one easily calculates X k k − i1 k k − i2 (N ) . mk = i1 j1 i2 j2 i1 ≡i2 mod N, j1 ≡j2 mod N
(N )
Note that these formulas confirm mk = mk for k < N . Equation (22.27) is clearly invariant under permutations of U0 , U1 , U2 . Therefore the diffraction pattern has this S3 -symmetry too. Since only the critical points have a nontrivial stabilizer in S3 the multiplicities multN (r) in this example satisfy multN (9) multN (0) multN (1) multN (r)
= = ≡ ≡
1 2 3 mod 6 0 mod 6
∀N if 3|N if 2|N if r 6= 0, 1, 9,
(22.29) ∀N.
We have computed the numbers multN (r) for some values of N . We found for instance B6 (z) = z 2 (z − 1)15 (z − 3)6 (z − 4)6 (z − 7)6 (z − 9) . −1 ∗ We computed W (u1 , u2 ) = (u1 + u2 + 1)(u−1 1 + u2 + 1) for u1 , u2 ∈ F7 :the (i, j)-entry of the following 6 × 6-matrix is W (i, j) mod 7: 2 3 0 3 0 1 3 3 4 0 1 1 0 4 0 1 4 1 . 3 0 1 3 4 1 0 1 4 4 0 1 1 1 1 1 1 1
This yields the following count of points over F7 : z mod 7 : 0 1 2 3 4 5 6 ]{ξ ∈ (F∗7 )2 | W (ξ) = z in F7 } : 8 15 1 6 6 0 0 Thus we see that the inequality in (22.20) can be strict: v7 (B6 (53)) = 12 > 6 = ]{ξ ∈ (F∗7 )2 | W (ξ) = 53 in F7 } .
(22.30)
Acknowledgment. It is my pleasure to dedicate this paper to Jaap Murre and Spencer Bloch on the occasion of their 75th , respectively, 60th birthdays. Both have been very important for my formation as a mathematician, from PhD-student time till present.
Motives from Diffraction
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