Global Aspects of Complex Geometry

Catanese · Esnault · Huckleberry · Hulek · Peternell (Eds.) Global Aspects of Complex Geometry

Fabrizio Catanese · Hélène Esnault Alan T. Huckleberry · Klaus Hulek Thomas Peternell (Eds.)

Global Aspects of Complex Geometry With 15 Figures

123

Fabrizio Catanese

Hélène Esnault

Lehrstuhl Mathematik VIII Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany e-mail: [email protected]

Mathematik Universität Duisburg-Essen 45117 Essen, Germany e-mail: [email protected]

Alan T. Huckleberry

Klaus Hulek

Institut für Mathematik Universität Bochum Universitätsstraße 150 44801 Bochum, Germany e-mail: [email protected]

FB Mathematik Institut für Mathematik Universität Hannover Welfengarten 1, 30167 Hannover, Germany e-mail: [email protected]

Thomas Peternell Lehrstuhl Mathematik I Mathematisches Institut Universität Bayreuth 95440 Bayreuth, Germany e-mail: [email protected]

Library of Congress Control Number: 2006929537

Mathematics Subject Classification (2000): 14-02, 14F05, 14F42, 14H60, 14J10, 14J32, 14J40, 14J45 32-02, 32M05, 32M25, 32Q15

ISBN-10 3-540-35479-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-35479-6 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Production and Data conversion: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

44/3100/YL

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Preface

Over the period 2000–2006 the Deutsche Forschungsgemeinschaft sponsored a special Schwerpunkt programme, entitled “Global Methods in Complex Geometry”. The articles of this volume grew out of this programme and document some of the scientific activity performed in the realm of the Schwerpunkt. They also aim at giving a broader overview of recent developments in various directions of Complex Geometry such as • Low-dimensional geometry: surfaces of general type, Fano threefolds, Calabi-Yau threefolds; • moduli spaces and families of varieties over curves; • Hodge theory, motivic cohomology and characteristic p-geometry; • moment maps and group actions on flag manifolds; • geometry of singular varieties: vector fields, equisingular families and vector bundles; • geometry of rational curves and pseudo-effective line bundles. The articles are devoted to a broad spectrum of topics, which range from purely algebraic to complex-analytic aspects of our subject. The participants of the Schwerpunkt would like to thank the Deutsche Forschungsgemeinschaft for its generous support. Bayreuth, Essen, Bochum, Hannover, June 2006 Fabrizio Catanese, H´el`ene Esnault, Alan Huckleberry, Klaus Hulek, Thomas Peternell

Contents

Complex Surfaces of General Type: Some Recent Progress Ingrid C. Bauer, Fabrizio Catanese, Roberto Pignatelli . . . . . . . . . . . . . . .

1

Characteristic 0 and p Analogies, and some Motivic Cohomology Manuel Blickle, H´el`ene Esnault, Kay R¨ ulling . . . . . . . . . . . . . . . . . . . . . . . . 59 Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, Gert-Martin Greuel . . . . . 83 Indices of Vector Fields and 1-Forms on Singular Varieties W. Ebeling, S. M. Gusein-Zade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Equisingular Families of Projective Curves Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin . . . . . . . . . . . . . . . 171 Critical Points of the Square of the Momentum Map Peter Heinzner, Henrik St¨ otzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Actions on Flag Manifolds: Related Cycle Spaces Alan Huckleberry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Modularity of Calabi-Yau Varieties Klaus Hulek, Remke Kloosterman, Matthias Sch¨ utt . . . . . . . . . . . . . . . . . . . 271 Some Recent Developments in the Classification Theory of Higher Dimensional Manifolds Priska Jahnke, Thomas Peternell, Ivo Radloff . . . . . . . . . . . . . . . . . . . . . . . 311 Existence of Rational Curves on Algebraic Varieties, Minimal Rational Tangents, and Applications Stefan Kebekus, Luis Sol´ a Conde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

VIII

Contents

Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves Martin M¨ oller, Eckart Viehweg, Kang Zuo . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Hodge Theory and Algebraic Cycles Stefan J. M¨ uller-Stach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 K¨ ahler Geometry of Moduli Spaces of Holomorphic Vector Bundles Georg Schumacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

Complex Surfaces of General Type: Some Recent Progress Ingrid C. Bauer1 , Fabrizio Catanese2 , and Roberto Pignatelli3,
1 2 3

Mathematisches Institut, Lehrstuhl Mathematik VIII, Universit¨ atstraße 30, D-95447 Bayreuth, Germany [email protected] Mathematisches Institut, Lehrstuhl Mathematik VIII, Universit¨ atstraße 30, D-95447 Bayreuth, Germany [email protected] Dipartimento di Matematica, Universit` a di Trento, via Sommarive 14, I-38050 Povo (TN), Italy [email protected]

Introduction In this article we shall give an overview of some recent developments in the theory of complex algebraic surfaces of general type. After the rough or Enriques-Kodaira classification of complex (algebraic) surfaces, dividing compact complex surfaces in four classes according to their Kodaira dimension −∞, 0, 1, 2, the first three classes nowadays are quite well understood, whereas even after decades of very active research on the third class, the class of surfaces of general type, there is still a huge number of very hard questions left open. Of course, we made some selection, which is based on the research interest of the authors and we claim in no way completeness of our treatment. We apologize in advance for omitting various very interesting and active areas in the theory of surfaces of general type as well as for not being able to mention all the results and developments which are important in the topics we have chosen. Complex surfaces of general type come up with certain (topological, birational) invariants, topological as for example the topological Euler number e and the self intersection number of the canonical divisor K 2 of a minimal surface, which are linked by several (in-) equalities. In the first chapter we will summarize the classically known inequalities, which force surfaces of general type in a certain region of the plane having K 2 and e as coordinates, and we shall briefly comment on the so-called geography problem, whether,


The present work was performed in the realm of the SCHWERPUNKT “Globale Methoden in der komplexen Geometrie”, and was also supported by a VIGONIDAAD Program. A first draft of this article took origin from the lectures by the second author at the G.A.C. Luminy Meeting, october 2005: thanks to the organizers!

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

given numerical invariants lying in the admissible range, i.e., fulfilling the required inequalities, does there exist a surfaces having these invariants. We shall however more broadly consider the three classical invariants K 2 , pg , q, which determine the other invariants χ := 1 − q + pg , e = 12χ − K 2 . An important new inequality, which Severi tried without success to establish, and which has been attacked for many years with partial results by several authors, asserts that a surface of maximal Albanese dimension satisfies the inequality K 2 ≥ 4χ. We will report on Pardini’s surprisingly simple proof of this so-called Severi’s conjecture (cf. [Par05]). The study of the pluricanonical maps is an essential technique in the classification of surface of general type. The main results concerning the m-canonical maps with m ≥ 3 go back to an earlier period and we refer to [Cat87b] for a report on them. We will report in the second chapter on recent developments concerning the bicanonical map; we would like to mention Ciliberto’s survey (cf. [Cil97]) on this topic for the state of art ten years ago. Here instead, we combine a discussion of this topic with the closely intertwined problem of classification of surfaces with low values of the numerical invariants. In the third chapter we report on surfaces of general type with geometric genus pg equal to four, a class of surfaces whose investigation was started by Federigo Enriques (cf. chapter VIII of his book ’Le superficie algebriche’, [Enr49]). By Gieseker’s theorem we know that for fixed K 2 and χ there exists a quasi projective coarse moduli space MK 2 ,χ for the birational equivalence classes of surfaces of general type. It is a very challenging problem to understand the geometry of these moduli spaces even for low values of the invariants. The case pg = 4 is studied via the behaviour of the canonical map. While it is still possible to divide the moduli space into various locally closed strata according to the behaviour of the canonical map, it is very hard to decide how these strata patch together. Using certain presentations of Gorenstein rings of codimension 4 introduced by M. Reid and D. Dicks, which arrange the defining equations as Pfaffians of certain matrices with many symmetries in such a way that these equations behave well under deformation, it is possible to exhibit explicit deformations, which allow to “connect” certain irreducible components of the moduli space. Inspired by a construction of A. Beauville of a surface with K 2 = 8, pg = q = 0, the second author defined Beauville surfaces as surfaces which are rigid and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2. In this case the moduli space of surfaces orientedly homeomorphic to a given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces.

Complex Surfaces of General Type: Some Recent Progress

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These surfaces, and the more general surfaces isogenous to a product, not only provide cheap counterexamples to the Friedman-Morgan speculation (which will be treated more extensively in the sixth section of this article), but provide also a wide class of surfaces quite manageable in order to test conjectures, and offer also counterexamples to various problems. The ease with which one can handle these surfaces is based on the fact that these surfaces are determined by “discrete” combinatorial data. Beauville surfaces, their relations to group theory and to Grothendieck’s theory of ’Dessins d’enfants’ will be discussed in the fourth chapter. It is a very difficult and very intriguing problem to decide whether two algebraic surfaces, which are not deformation equivalent, are in fact diffeomorphic. The theory of Lefschetz fibrations provides an algebraic tool to prove that two surfaces are diffeomorphic. By a theorem of Kas (which holds also in the symplectic context) two Lefschetz fibrations are diffeomorphic if and only if their corresponding factorizations of the identity in the mapping class group are equivalent under the equivalence relation generated by Hurwitz moves and by simultaneous conjugation. We outline the theory, which was used with success in [CW04] in chapter five, which we end with a brief report on the status of two very old conjectures by Chisini concerning cuspidal curves and algebraic braids. As already mentioned before, one of the fundamental problems in the theory of surfaces of general type is to understand their moduli spaces, in particular the connected components which parametrize the deformation equivalence classes of minimal surfaces of general type. By a classical result of Ehresmann, two deformation equivalent algebraic varieties are diffeomorphic. The other direction, i.e., whether two diffeomorphic minimal surfaces of general type are indeed in the same connected component of the moduli space, was an open problem since the eighties. We discuss in the last chapter the various counterexamples to the Friedman-Morgan speculation, who expected a positive answer to the question (unlike the second author, cf. [Kat83]). Moreover, we briefly report on another equivalence relation introduced by the second author, the so-called quasi ´etale-deformation (Q.E.D.) equivalence relation, i.e., the equivalence relation generated by birational equivalence, by quasi ´etale morphisms and by deformation equivalence. For curves and surfaces of special type two varieties are Q.E.D. equivalent if and only if they have the same Kodaira dimension, whereas there are infinitely many surfaces of general type, which are pairwise not Q.E.D. equivalent.

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1 Old and New Inequalities 1.1 Invariants of Surfaces n Let X be a compact complex manifold and let ΩX be its canonical bundle, i.e., the line bundle of holomorphic n−forms (usually denoted by ωX , since it is a dualizing sheaf in the sense of Serre duality). A corresponding canonical divisor is usually denoted by KX . To X one associates its canonical ring ⊗m R(X) := ⊕m≥0 H 0 (ωX ).

The trascendency degree over C of this ring leads to • the Kodaira dimension κ(X) := tr(R(X)) − 1, if R(X) = C, otherwise κ(X) := −∞. The Kodaira dimension is invariant under deformation (by Siu’s theorem [Siu02], generalizing Iitaka’s theorem for surfaces) and can assume the values −∞, 0, . . . , n = dim X . Definition 1. X is said to be of general type if the Kodaira dimension is maximal, κ(X) = dim X. We are interested in the case of surfaces, i.e., of manifolds of dimension 2, of general type. The three principal invariants under deformations for the study of these surfaces are • the self intersection of the canonical class KS2 of a minimal model, • the geometric genus pg := h0 (ωX ) and • the irregularity q := h1 (OS ) = h0 (ΩS1 ). The equality h1 (OS ) = h0 (ΩS1 ) follows by Hodge theory since every algebraic surface is projective. The invariants we have introduced, with the exception of KS2 , are not only deformation invariants but also birational invariants. Definition 2. A smooth surface S is called minimal (or a minimal model) iff it does not contain any exceptional curve E of the first kind (i.e. E ∼ = P1 , E 2 = −1). Every surface can be obtained by a minimal one (its “minimal model”) after a finite sequence of blowing ups of smooth points; this model is moreover unique if κ(S) ≥ 0 (see III.4.4, III.4.5 and III.4.6 of [BHPV04]). Thus, every birational class of surfaces of general type contains exactly one minimal surface, and one classifies surfaces of general type by studying their minimal models. To each minimal surface of general type we will associate its numerical • type (KS2 , pg , q),

Complex Surfaces of General Type: Some Recent Progress

5

a triple of integers given by the three invariants introduced above. In fact these determine all other classical invariants, as • the Euler-Poincar´e characteristic of the trivial sheaf χ(OS ) = 1 − q + pg ; 2 • the topological Euler characteristic e(S) = c2 (S) S ) − KS ;  12χ(O
m= ⊗m 0 2 • the plurigenera Pm (S) := h (ωX ) = χ(OS ) + 2 KS . The epression for c2 is a classical theorem of M. Noether, and the expression for the plurigenera follows by Riemann-Roch and by Mumford’s vanishing theorem. By the theorems on pluricanonical maps (cf. [Bom73]), minimal surfaces S of general type with fixed invariants are birationally mapped to normal surfaces X in a fixed projective space of dimension P5 (S) − 1. X is uniquely determined, is called the canonical model of S, and is obtained contracting to points all the (-2)-curves of S (curves E ∼ = P1 , with E 2 = −2). Let us recall Gieseker’s theorem Theorem 1 (Gieseker [Gie77]). There exists a quasi-projective coarse moduli scheme for canonical models of surfaces of general type S with fixed KS2 and c2 (S). In particular, we can consider the subscheme MKS2 ,pg ,q corresponding to minimal surfaces of general type of type (KS2 , pg , q). By the above theorem, it is a quasi projective scheme, in particular, it has finitely many irreducible components. It is a dream ever since to completely describe MKS2 ,pg ,q for as many types as possible. 1.2 Classical Inequalities and Geography Obviously the first question is: for which values of (KS2 , pg , q) is MKS2 ,pg ,q non empty? For example, it is clear that pg (S) and q(S) are always nonnegative, since they are dimensions of vector spaces. In fact much more is known. In the following table we collect the well known classical inequalities holding among the invariants of minimal surfaces of general type: KS2 (N ) KS2 (D) if q (M Y ) KS2

≥1 χ≥1 ≥ 2pg − 4 or the weaker KS2 ≥ 2χ(OS ) − 6 > 0, KS2 ≥ 2pg or the weaker if q > 0, KS2 ≥ 2χ(OS ) ≤ 9χ

We have labeled by (N)= Noether, (D) = Debarre, (MY) = Miyaoka-Yau the rows, corresponding to the names of the inequalities ([Deb82], [Deb83], [Miy77], [Yau78], see also [BHPV04], chap. 7).

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

Fig. 1. The geography of minimal surfaces of general type

In figure 1 we have drawn the limit lines (i.e., where equality holds) of the various inequalities in the (χ, KS2 )-plane. The above listed inequalities show that the pair of invariants χ, KS2 of a minimal surface of general type gives a point with integral coordinates in the convex region limited by the “bold” piecewise linear curve. Moreover, if q > 0 this point cannot be at the “right” of the line D. We drew one more line in our picture, labeled by S. This is the Severi line K 2 = 4χ, i.e., the equality case of the Severi inequality K 2 ≥ 4χ ⇔ K 2 ≥ 12 e, which will be discussed in detail at the end of this section. 1.3 Surfaces Fibred over a Curve An important method for the study of surfaces of general type is to consider relatively minimal fibrations of surfaces over curves f : S → B. Definition 3. A fibration f : S → B is a surjective morphism with connected fibres. We are interested in the case of fibrations of surfaces to curves, meaning that in this paper S and B will always be smooth compact complex manifolds of respective dimensions 2 and 1. The fibration is said to be relatively minimal if f does not contract any rational curve of self intersection −1 to a point.

Complex Surfaces of General Type: Some Recent Progress

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One denotes • by b the genus of the base curve B; • by g the genus of a general fibre. To avoid confusion, let us point out that a fibration is called rational or irrational according to the genus b of the base being 0 or > 0. On the other hand, the genus of the fibration is the genus g of the fibre. For example, if we say f is a genus 2 rational fibration, we intend that g = 2 and b = 0. The classical way of saying: a genus b pencil of curves of genus g is however still the most convenient way to describe a fibration. To a relatively minimal fibration f one associates ∨ • its relative canonical bundle ωS|B := ωS ⊗ f ∗ (ωB ) and ⊗n • the sheaves (∀n ≥ 0) Vn := f∗ (ωS|B ).

The sheaves Vn are vector bundles (i.e., locally free sheaves) with very nice properties. Theorem 2 (Fujita [Fuj78a], [Fuj78b])). The vector bundles Vn are semipositive, i.e., every locally free quotient of it has nonnegative degree. To be more precise, V1 is a direct sum of an ample vector bundle with q(S) − b copies of the trivial bundle and with some undecomposable stable degree 0 vector bundle without global sections. Zucconi [Zuc97] proved moreover that if one of those stable bundles has rank 1, then it is a torsion line bundle. For n ≥ 2 we have: Theorem 3 (Esnault-Viehweg [EV90]). ∀n ≥ 2 the vector bundle Vn is ample unless f has constant moduli, which means that all the smooth fibres are isomorphic. ⊗n = 0 ∀n ≥ 2 by Since R1 f∗ ωS|B = OB by relative duality, and R1 f∗ ωS|B the assumption of relative minimality, one can compute the Euler characteristic of Vn by Riemann-Roch, and consequently its degree. We introduce the following invariants of the fibration f :

• the self intersection of the relative canonical divisor Kf2 := ωS|B · ωS|B = KS2 − 8(g − 1)(b − 1), • the Euler characteristic of the relative canonical divisor χf = χ(ωS|B ) = χ(OS ) − (g − 1)(b − 1), • its slope λ(f ) := Kf2 /χf .

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

The slope is clearly defined only for χf = 0, or equivalently (as we will see soon) if the fibration is not a holomorphic bundle. The above mentioned computation gives deg Vn = χf +

n(n − 1) 2 Kf 2

and since by Fujita’s theorem these numbers are nonnegative this gives the two inequalities Kf2 ≥ 0 and χf ≥ 0 respectively known as Arakelov’s inequality (cf. [Ara71]) and Beauville’s inequality (cf. [Bea82]). In fact, we have the following list of inequalities (A) (B) (ZS) (N N ) (X)

Kf2 ≥ 0 , i.e., KS2 ≥ 8(g − 1)(b − 1), χf ≥ 0 , i.e., χ(OS ) ≥ (g − 1)(b − 1), c2 (S) ≥ 4(b − 1)(g − 1), q ≤ b + g, 4 − g4 ≤ λ(f ) ≤ 12.

Here the meaning of the labeling is the following: (A) = Arakelov’s inequality, (B) = Beauville’ inequality, (X) = Xiao’s inequality (also known as slope inequality), (NN) = no name’s inequality, (ZS) = Zeuthen-Segre. A proof of those inequalities can be found in [Bea82] with the exception of the slope inequality, proved in [Xia87] (see also [CH88] in the semistable case). The equality cases of the first 4 inequalities are well described: • • • •

if equality holds in (A), f has constant moduli; equality holds in (B) ⇔ f has constant moduli and is smooth; for g ≥ 2, equality holds in (ZS) ⇔ f is smooth; q = b + g ⇔ f is birationally equivalent to the projection of a product B × F to the first factor.

In particular, we see that the slope is defined whenever the fibration is not a holomorphic bundle, since the denominator χf vanishes iff equality holds in Beauville’s inequality. An important consequence is the following Theorem 4 (Beauville). If X is a minimal surface of general type, then pg ≥ 2q − 4. Moreover, if pg = 2q − 4, then S is a product of a curve of genus 2 with a curve of genus q − 2. Note for later use (see next section) the following Corollary 1. If pg = q (i.e., if χ(OS ) = 1), then pg = q ≤ 4. Moreover, minimal surfaces of general type with pg = q = 4 are exactly the products of two genus 2 curves. Proof of theorem 4. The standard wedge product on 1−forms induces a natural map ∧ : Λ2 H 0 (ΩS1 ) → H 0 (ΩS2 )

Complex Surfaces of General Type: Some Recent Progress

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Recall that q = dim H 0 (ΩS1 ), pg = dim H 0 (ΩS2 ). Let us assume pg ≤ 2q − 4. By a dimension count, if pg ≤ 2q − 4, the projective linear subspace of P(Λ2 H 0 (ΩS1 )) corresponding to the kernel of the above map must intersect the Pl¨ ucker embedding of the Grasmannian G2 (H 0 (ΩS1 )) (which has dimension 2q − 4), and therefore there are two linearly independent 1−forms ω1 and ω2 such that the following holomorphic two form is identically zero: ω1 ∧ ω2 ≡ 0. By the theorem of Castelnuovo-De Franchis there is a fibration f : S → B 1 ) with base of genus b ≥ 2, and two holomorphic 1−forms α1 , α2 ∈ H 0 (ΩB ∗ such that f αi = ωi . Since S is of general type, also g ≥ 2. Then χf ≥ 0 ⇒ χ(OS ) ≥ (b − 1)(g − 1) = (b − 2)(g − 2) + b + g − 3 ≥ q − 3. So we have 1 − q + pg ≥ q − 3 ⇔ pg ≥ 2q − 4. If pg = 2q − 4, all inequalities are equalities and then, since q = b + g and (b − 2)(g − 2) = 0, S is a product of two curves of genus at least 2, and one of the two must have genus exactly 2.   1.4 Severi’s Inequality We recall that the Albanese variety Alb(X) of a compact K¨ahler manifold X is the cokernel of the natural map  : H1 (X, Z) → H 0 (Ω 1 (X))∨ defined by integrating 1−forms on 1-cycles. The Albanese morphism α : X → Alb(X) is defined (up to translations in Alb(X)) by fixing a point p0 ∈ X, and by p associating to each point p ∈ X the class in Alb(X) of p0 , where the integral is taken along any path between p0 and p. Recall that, if X is projective (as any surface of general type), Alb(X) is an abelian variety (of dimension q). The Albanese morphism is a powerful tool for studying irregular surfaces (q > 0) and in particular: Definition 4. A variety X is called of maximal Albanese dimension if the image of the Albanese morphism has the same dimension as X. This is the general case for surfaces, since otherwise the Albanese morphism is a fibration onto a smooth curve of genus q. We see then that for surfaces maximal Albanese dimension is equivalent to the non existence of a genus q pencil. We can now state the theorem known as Severi’s inequality

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Theorem 5 (Pardini [Par05]). If S is a smooth complex minimal surface of maximal Albanese dimension, then KS2 ≥ 4χ. This theorem was proved only very recently by R. Pardini, but it has a long story, which we briefly sketch in the following. Severi’s Conjecture The inequality takes its name from F. Severi, since he was the first to claim the result in the 30’s [Sev32]. His proof turned out to be wrong, as was pointed out in [Cat83], since it was based on the assertion that a surface with irregularity q either contains an irrational genus q fibration, or the sections of H 0 (ΩS1 ) have no common zero. Counterexamples were given in [Cat84], where there were constructed bidouble covers S → X of any algebraic surface with, among other properties, q(S) = q(X). If X has no irrational pencils, since the Albanese map of S factors through the cover, then also S has no irrational pencils. But any ramification point of the cover is a base point for H 0 (ΩS1 ). Therefore Severi’s inequality was posed in [Cat83] as Severi’s conjecture, a conjecture on surfaces of general type, since for surfaces with κ(S) ≤ 1 it is a straightforward consequence of the Enriques-Kodaira classification. It had also been posed as a conjecture by M. Reid (conj. 4 in [Rei79]) who proved the weaker KS2 ≥ 3χ. Proofs in Special Cases In the 80’s, Xiao’s work on surfaces fibred over a curve was mainly motivated by Severi’s conjecture. In [Xia87] he proved the slope inequality and Severi’s conjecture for surfaces having an irrational pencil. In the 90’s Konno [Kon96] proved the conjecture in the special case of even surfaces, i.e., surfaces whose canonical class is 2− divisible in the Picard group. Finally, at the end of the 90’s, Manetti [Man03] could prove the inequality for surfaces of general type whose canonical bundle is ample. Manetti’s Proof Manetti considers the tautological line bundle L of the P1 −bundle π : P(ΩS1 ) → S; standard computations give 3(KS2 − 4χ) = L2 · (L + π ∗ KS ). Then, using the fact that ΩS1 is generically globally generated, he can write the right hand side of the above equation as 2KS E + (L + π ∗ KS )C for an effective 1−cycle C in P(ΩS1 ), and where E is the maximal effective divisor

Complex Surfaces of General Type: Some Recent Progress

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in S such that h0 (ΩS1 (−E)) = h0 (ΩS1 ). Thus the problem is reduced to the nonnegativity of the term (L + π ∗ KS )C. This is obvious if Ω 1 (KS ) is nef, but in general it requires a very detailed and complicated analysis of the 1−cycle C. In fact, Pardini’s proof not only does not require the ampleness of the canonical divisor, but is much easier than Manetti’s. We should however mention that Manetti’s argument leads to a very detailed description of the equality case, showing that a surface of general type of maximal Albanese dimension, lying on the Severi line (K 2 = 4χ), and having ample canonical class, has irregularity q = 2 and is a double cover of a principally polarized Abelian surface, branched on a divisor D algebraically equivalent to 2Θ. Up to now there is no similar description of the limit case without the assumption that K be ample. Pardini’s Proof Pardini’s idea is to construct a sequence of genus gd fibrations fd : Yd → P1 such that lim gd = +∞ and lim λ(fd ) = KS2 /χ(OS ). d→∞

d→∞

Then, taking the limit of the left-hand side of the slope inequality, one gets the desired inequality KS2 /χ(OS ) ≥ 4. To construct these fibrations, she considers the Cartesian diagram p

S α


/S α

 Alb(S)

·d

 / Alb(S) ,

where d : Alb(S) → Alb(S) is multiplication by d. One observes that S  is connected since we have a surjection π1 (S) → π1 (Alb(S)) = H1 (S, Z). Clearly, KS2
= d2q KS2 , χ(OS
) = d2q χ(OS ). ∗ Let L be a very ample divisor on Alb(S) and set H := α∗ L, H  := α L. ∗ 2  2 2q−4 2  2q−2 H and KS
H = d KS H. Then p H ∼num d H , whence H = d Let now D1 , D2 ∈ |H  | be two general curves and define C1 := D1 + D2 ∈ |H1 + H2 |. Moreover, choose C2 ∈ |2H  | sufficiently general such that C1 and C2 intersect transversally. C1 and C2 define a rational pencil fd : Yd → P1 , where Yd is the blow up of S  at C1 ∩ C2 . The singular fibre induced by C1 guarantees that fd is not a holomorphic bundle, whence the slope λ(fd ) is well defined. For the invariants of Yd we get KY2d = KS2
− 4H  = d2q KS2 − 4d2q−4 H 2 2

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

χ(Yd ) = χ(S  ) = d2q χ(S) gd = 1 + KS
H  + 2H  = 1 + d2q−2 KS H + 2d2q−4 H 2 2

and therefore limd gd = +∞ as requested. Moreover, Kf2d = KY2d + 8(gd − 1) and χ(fd ) = χ(Yd ) + (gd − 1) and we see that both invariants are polynomials in d of degree 2q, whose leading terms are respectively KS2 and χ(OS ). In particular, lim λ(fd ) = lim Kf2 /χf = KS2 /χ(OS )

d→∞

d→∞

2 Surfaces with χ = 1 and the Bicanonical Map 2.1 The Bicanonical Map The behaviour of the m − th canonical map of S (i.e., the rational map associated to |mKS |) is an essential tool in the theory of surfaces of general type. As we mentioned in the introduction, the cases where m ≥ 3 are solved since long (cf. the survey [Cat87b]). The canonical map (m = 1) was first studied by Beauville in [Bea79], but there remain still many unresolved questions. The case m = 2 was particularly studied in the last years, and we have the impression that we are very close to a complete understanding. In order to fix the starting point, we summarize the results of several authors ([Fra91], [Rei88], [Cat81], [CC91], [CC93], [Xia85a]) in the following Theorem 6. Let S be a minimal surface of general type. Then • the bicanonical map is generically finite unless pg = 0 and K 2 = 1; • if KS2 ≥ 5 or pg ≥ 1, the bicanonical map is a morphism. Note that, if pg = 0 and K 2 = 1, then P2 = 2 and the bicanonical map is a rational (b = 0) fibration. In all known examples this is a genus 4 fibration, although at the moment it is only proven that its genus is 3 or 4 (see [CP05]). These surfaces are usually called numerical Godeaux surfaces. Numerical Godeaux surfaces with torsion (in the Picard group) of cardinality at least 3 are classified in [Rei78], a family with torsion Z/2 was constructed in [Bar84]. Up to last year only sporadic examples of surfaces with trivial torsion were known, but recently Schreyer [Sch05] has announced the construction of a family of the expected dimension (= 8) using a new approach based on homological algebra. The above theorem says that in all other cases the bicanonical map maps S to a surface, and it is a morphism (except for finitely many families). In the last years many people studied the degree of this map, in particular, trying to classify the surfaces such that the bicanonical map is not birational.

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The Standard Case It is well known that the bicanonical map of a smooth curve of general type (i.e., of genus at least 2) fails to be birational if and only if the curve has genus 2. This exception induces a “standard exception” to the birationality of the bicanonical map in dimension 2. Definition 5. A surface S of general type presents the standard case (for the non birationality of the bicanonical map) if there exists a dominant rational map onto a curve f : S

 B whose general fibre is irreducible of genus 2. In fact, if S presents the standard case, then the restriction of the bicanonical map of S to a general fibre factors through the bicanonical map of the fibre itself and therefore cannot be birational. The subschemes of the moduli space corresponding to surfaces presenting the standard case are not empty for infinitely many moduli spaces, and Persson [Per81] constructed many interesting surfaces considering double covers of ruled surfaces branched on relative sextics, thereby filling a big region of the convex region represented in figure 1. Bombieri ([Bom73]) showed that the standard case gives almost all exceptions to the birationality of the bicanonical map. More precisely, combining his results with those of Reider ([Rei88]) we know now that a minimal surface of general type with K 2 ≥ 10 either presents the standard case, or its bicanonical map is birational. In particular, the exceptions to the birationality of the bicanonical map not presenting the standard case belong to finitely many families and many authors are trying since then to classify them (see [Cil97] for a survey updated until ’96). Du Val’s Double Planes In the same paper [Bom73] Bombieri constructed a surface of type (K 2 , pg , q) = (9, 6, 0) not presenting the standard case. His example can be easily described as a hypersurface F14 of degree 14 in the weighted projective space P(1, 1, 2, 7), and from this description it follows rightaway that it bicanonical map is a double cover of the weighted projective space P(1, 1, 2) (isomorphic to a quadric cone in P3 ). This example is in fact a special case of a more general “geometric” situation studied first by du Val. Let S be a minimal regular surface with pg ≥ 2, such that the general canonical curve is irreducible, smooth and hyperelliptic. Since the restriction of the bicanonical map ϕ2K to a canonical curve factors through the canonical map of the curve itself, ϕ2K cannot be birational. Du Val [Duv52] gave a list of such surfaces obtained as double covers of rational surfaces. A generalization (see [CML00], [Bor03]) leads to the following:

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Definition 6 (du Val’s double planes). A smooth surface S is a du Val double plane if it is birational to D) a double cover of P2 branched over a smooth curve of degree 8; Dn ) a double cover of P2 branched over the union of a curve of degree 10 + n, n ≤ 6, with n distinct lines through a point p, such that the essential singularities of the branch curve are the following: – p is a singular point of multiplicity 2n + 2, – there is a singular point of type [5, 5] on each line, – possibly there are some quadruple points and some points of type [3, 3]; B) a double cover of the Hirzebruch surface F2 whose branch curve can be decomposed as C0 + G , G ∈ |7C0 + 14Γ | (where |Γ | is the ruling of F2 and C0 is the section with self intersection −2), whose only essential singularities are [3, 3] points that are tangent to a fibre. Recall that a singular point of type [d, d] is a singular point of multiplicity d having a further singular point of multiplicity d infinitely near to the first one. In other words, if we blow-up the singular point, the strict transform of the curve has one more singular point of multiplicity d lying on the exceptional divisor. Remark 1. In the definition of the du Val’s double planes of type Dn and B we only care about the essential singularities of the branch curve (as usual in the theory of double covers) since adding a simple singularity to the branch curve does not affect the properties of the resulting surface we are interested in. On the contrary, in the definition of the double planes of type D, we assume the branch curve to be smooth. In fact, if we take a double cover of P2 branched over a curve of degree 8 with a double point, the pull back of the pencil of lines through this point to the surface defines a pencil of curves of genus 2 (through a singular point of the surface), so the resulting surface presents the standard case. Note that this example shows that one can degenerate surfaces presenting a nonstandard case to surfaces presenting the standard case (just take a family of smooth plane curves of degree 8 degenerating to a singular one and consider the corresponding family of double covers). Borrelli proved that this list is “complete” in the following sense Theorem 7 (Borrelli [Bor03]). If S is a minimal surface of general type, not presenting the standard case, whose bicanonical map factors through a degree 2 rational map onto a rational or ruled surface: then S is the smooth minimal model of a du Val double plane. In particular, either q = 0 or pg = q = 1. The “Classification” The standard case and the du Val’s double planes do not give all possible surfaces of general type with nonbirational bicanonical map, but the remaining exceptions are really few.

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What is known about these is summarized in the following Theorem 8. Let S be a smooth minimal surface of general type whose bicanonical map ϕ2K is not birational. Then one of the following cases occur: i) S presents the standard case; ii) S is the smooth minimal model of a du Val double plane; iii) S is a surface of type (1, 1, 0) (for these automatically deg ϕ2K = 4 and S is a complete intersection of two sextics in P(1, 2, 2, 3, 3)); iv) S is of type (2, 1, 0) with Picard group having torsion Z/2Z (for these automatically deg ϕ2K = 4 and its double cover corresponding to the torsion class is a complete intersection of two quartics in P(1, 1, 1, 2, 2)); v) ϕ2K is 2 : 1 onto a K3 surface and pg = 1, q = 0, 2 ≤ K 2 ≤ 8; vi) S is of type (6, 3, 3) or of type (4, 2, 2) (for these both cases automatically ϕ2K has degree 2 and we have a nonstandard case) vii)S has pg = q ≤ 1. All these cases with the exception of pg = q ≤ 1 are now rather clear. The history of this theorem is rather complicated and combines the efforts of several authors. We try to reconstruct its more important steps here, giving some more details on each class. One of the first results in this direction is due to Xiao Gang [Xia90], giving, in the nonstandard case, and under the assumption that the degree d of the bicanonical map is at least 3, a list of the possible values of d and of the possible places in the Enriques classification of the bicanonical image Σ. In 1997, Ciliberto, Francia and Mendes Lopes [CFML97] gave a complete classification of the case pg ≥ 4, essentially confirming du Val’s list. Then, Ciliberto and Mendes Lopes, with contributions of the second author and Borrelli, worked in the next years to extend the classification to pg ≥ 2 (see [CCML98], [CML00], [CML02a], [CML02b], [Bor02]). The case pg = 1 and q = 0, giving cases iii), iv) and v) is classified in [Bor03]. In fact, cases iii) and iv) resulted already from the analysis of [Xia90] where it is proven that, if deg ϕ2K ≥ 3, then either S is of type (1, 1, 0) or of type (2, 1, 0), or with pg = q ≤ 2. The description given in iii) and iv) of the first two cases comes from the papers [Cat79] and [CD89], where all surfaces of respective types (1, 1, 0) and (2, 1, 0) are classified. In particular, it is shown that all surfaces of type (1, 1, 0) are as in iii). Remark 2. Surfaces of type (2, 1, 0) without torsion in homology, also sometimes called Catanese-Debarre surfaces, offer the following interesting phenomenon: there is an irreducible component of the moduli space such that 1) for the general surface the bicanonical map is birational, while there are subvarieties for which the bicanonical map can respectively be 2) of degree 2 onto a K3- quartic surface, 3) of degree 2 onto a rational quartic surface, 4) of degree 4 onto a smooth quadric surface.

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The surfaces in v) are usually called Todorov surfaces, since they were introduced in [Tod81]. The subspaces of the moduli spaces corresponding to them is described in [Mor88]. Finally, the largely open case vii) is very strongly related with the problem, of independent interest, of the classication of surfaces of general type with pg = q. We shall describe in the next subsection what is currently known on these surfaces, showing in particular that we have a very precise description of the two cases in vi). 2.2 Surfaces with pg = q These are the surfaces corresponding to the “vertical” piece of the bold line in figure 1. In particular, 1 ≤ K 2 ≤ 9. Surfaces with pg = q ≥ 4 This case is clear, by corollary 1 of Beauville’s theorem 4. If pg = q ≥ 4, then S is a product of two genus 2 curves and pg = q = 4. We recall that then K 2 = 8 and clearly the bicanonical map has degree 4, and we have a standard case. Surfaces with pg = q = 3 These surfaces have been first studied in [CCML98], and a complete classification has been recently achieved independently by Pirola [Pir02] and HaconPardini [HP02]. The result is the following Theorem 9. A minimal surface of general type with pg = q = 3 has K 2 = 6 or K 2 = 8 and, more precisely, • if K 2 = 6, S is the symmetric square of a genus 3 curve; • otherwise S = C2 × C3 /τ , where Cg denotes a curve of genus g and τ is an involution of product type acting on C2 as an elliptic involution (i.e., with elliptic quotient), and on C3 as a fixed point free involution. In particular, the moduli space of minimal surfaces of general type with pg = q = 3 is the disjoint union of M6,3,3 and M8,3,3 , which are both irreducible of respective dimension 6 and 5. We sketch the idea of the proof. By Debarre’s inequality (in the “stronger” form: q > 0 ⇒ KS2 ≥ 2pg ), pg = q = 3 implies K 2 ≥ 6. As in the proof of Beauville’s theorem, consider now the map

Complex Surfaces of General Type: Some Recent Progress

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∧ : Λ2 (H 0 (ΩS1 )) → H 0 (ΩS2 ). Since pg = q = 3, it is a linear map between two three dimensional spaces. If this is not an isomorphism, then (since every vector in Λ2 C3 is decomposable) there are two nontrivial 1−forms ω1 and ω2 with ω1 ∧ ω2 ≡ 0. This yields then (by Castelnuovo-De Franchis) a pencil f : S → B with b, g ≥ 2. Then, by Beauville’s inequality, b = g = 2 and the fibration is a holomorphic bundle (this forces KS2 = 8). Therefore f is induced by a map π1 (B) → Aut(F ) (where F is a smooth fibre), whose kernel induces an unramified cover ϕ : C → B Galois with group G. Since q = 3, the quotient of F by the group G has genus 1. By Hurwitz’s formula one easily sees that, if φ : F → F/G is branched in one point, then |G| ≤ 4, hence G is Abelian, contradicting that φ is ramified. Again Hurwitz’s formula shows that φ is branched in 2 points and G ∼ = Z/2. Otherwise ∧ is an isomorphism, and therefore S does not have any pencil f : S → B with b ≥ 2. In particular α(S) is a surface, a divisor Θ in Alb(S). Pirola noticed that Θ must be ample, else it would have an elliptic fibration and therefore an irrational pencil with base of genus b ≥ 2. This implies, by Lefschetz’s hyperplane theorem, that the induced map 1 1 ) → H 1 (ΩS1 ) is injective: since, for any class η ∈ H 1 (ΩAlb(S) ) H 1 (ΩAlb(S)  c1 (Θ) ∧ η ∧ η¯ > 0. In particular, h1 (ΩS1 ) ≥ 9 and this (since by Hodge theory 12χ − K 2 = c2 = 2pg − 4q + 2 + h1 (Ω 1 )) implies KS2 ≤ 7. The case KS2 = 6 was already settled in [CCML98], where it is first shown that the degree of the scheme of base points is K 2 − 6, and then that in the case KS2 = 6 α is an embedding. More precisely it is shown that its image is a theta divisor in a principally polarized abelian threefold and therefore S is the symmetric square of a genus 3 curve. What remains to prove is M7,3,3 = ∅, and this is done in [Pir02] by a careful study of the paracanonical system. The fact that the bicanonical map has degree 2 is an easy consequence of the adjunction formula by which KS is the pull back of Θ, and of the fact that the sections of OA (2Θ) are invariant, as well as Θ, for the symmetry of A sending x → −x. Surfaces with pg = q = 2 This case is still far from being classified. Ciliberto and Mendes Lopes [CML02a] classified all surfaces with pg = q = 2 and non-birational bicanonical map (not presenting the standard case). Their result, corresponding to the subcase (4, 2, 2) of case vi) of theorem 8, is the following Theorem 10. If S is a minimal surface of general type with pg = q = 2 and non-birational bicanonical map not presenting the standard case, then S is a double cover of a principally polarized abelian surface (A, Θ), with Θ

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irreducible. The double cover S → A is branched along a divisor B ∈ |2Θ|, having at most double points. In particular KS2 = 4. Note that, again by Debarre’s inequality, pg = q = 2 ⇒ K 2 ≥ 4, so Ciliberto and Mendes Lopes’ surfaces belong to the limit case. Their theorem solves completely the problem of the non birationality of the bicanonical map in this case, but of course a complete classification of minimal surfaces of general type with pg = q = 2 would be interesting by itself. Results in this direction have been recently obtained by F. Zucconi; to explain them we need to give the following definition. Definition 7. A surface S is said to be isogenous to a (higher) product if S admits an unramified finite covering which is biholomorphic to a product of two curves of respective genera at least 2. We have already seen surfaces isogeneous to a product in our analysis of surfaces with pg = q, namely all surfaces in M8,4,4 and all surfaces in M8,3,3 . Zucconi’s theorem is the following Theorem 11 (2.9 in [Zuc03]). There are two classes of minimal surfaces of general type with pg = q = 2 whose Albanese image is a surface and having an irrational pencil, and they are both isogenous to a higher product. More precisely, either they have a double cover which is a product of two genus 2 curves or they are a quotient of the product of two genus 3 curves by an action of Z/2Z. In both cases Zucconi describes precisely the group action as a diagonal action induced by actions on the two curves. The interested reader will find all details in Zucconi’s paper. Zucconi managed also to remove the hypothesis on the Albanese map, by use of a special class of surfaces isogenous to a higher product, the generalized hyperelliptic surfaces introduced in [Cat00]. Definition 8. Let C1 and C2 be two smooth curves, G a finite group with two injections respectively in Aut(C1 ) and Aut(C2 ). Then the quotient surface S = C1 × C2 /G by the diagonal action is said to be a generalized hyperelliptic surface if • the projection C1 → C1 /G is unramified; • C2 /G is rational. Then Zucconi proved Theorem 12. If S has pg = q = 2, and the image of the Albanese map is a curve, then S is a generalized hyperelliptic surface. What remains to be classified is the class of surfaces with pg = q = 2 having no irrational pencils. Chen and Hacon, in a preprint, constructed an example of surfaces with pg = q = 2, K 2 = 5 and Albanese morphism of degree 3.

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Surfaces with pg = q = 1 In this case Debarre’s inequality gives only K 2 ≥ 2. The Albanese morphism is a map onto an elliptic curve, in particular, all these surfaces have a fibration with base of genus b = 1. We summarize in the following statement what is known about these surfaces. Theorem 13. • M2,1,1 is unirational (by this we mean: irreducible and unirational) of dimension 7. The Albanese map of all these surfaces is a genus 2 fibration. • M3,1,1 has 4 connected components, all unirational of dimension 5. The Albanese map is a genus 3 fibration for the surfaces in one of those components, and a genus 2 fibration in all other cases. • M4,1,1 , M5,1,1 and M8,1,1 are non empty. Actually much more can be said, and we try to be more precise in the following. First the most mysterious cases. It remains unsettled the existence of surfaces of general type with pg = q = 1 and K 2 = 6, 7, 9. In a recent preprint by Rito appears the construction of a surface with pg = q = 1 and K 2 = 6 as a double cover of a Kummer surface modifying slightly (i.e., adding a singular point to the branch curve) Todorov’s construction of a surface with pg = 1, q = 0 and K 2 = 8 in [Tod81]. Its construction makes use of the computer program MAGMA (to find a branch curve with the right singularities). Second, the “partially understood” cases. Examples of surfaces with pg = q = 1 and K 2 = 4, 5 were constructed by the second author as bidouble covers in [Cat99]. In both cases the Albanese map turns out to be a genus 2 fibration, so they present the standard case. The case K 2 = 8 was studied by Polizzi [Pol06], who considered the cases of surfaces having bicanonical map of degree 2. He could prove that all these surfaces are isogenous to a product and that they form three components of the moduli space, one of dimension 5 and two of dimension 4. All these surfaces do not contain any genus 2 pencil and they are in fact du Val double planes. Finally, the cases K 2 = 2, 3 are completely classified. The first to be settled was K 2 = 2, done by the second author in [Cat81], representing all those surfaces as double covers of the symmetric square of their Albanese curve. The case K 2 = 3 was first studied in [CC91] where it was shown, among other things, that the Albanese map could be either a genus 2 or a genus 3 fibration. The case g = 3 was then classified in [CC93], showing that it gives a unirational family of dimension 5. Note that, if there is surface with pg = q = 1, K 2 ≤ 3 and nonbirational bicanonical map not presenting the standard case, it must belong to this family.

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The question whether such a surface exists is still open. Recently Polizzi [Pol05] has shown that a general surface in this component has birational bicanonical map, but this is not true for all of them, since Xiao [Xia85b] has found a subfamily of dimension 1 having a genus 2 pencil. The classification of the case K 2 = 3 was completed in [CP05] classifying all those surfaces having Albanese fibres of genus 2. The main tool for this classification is a new method for studying fibrations f : S → B of genus 2, and fibrations of genus 3 with general fibre non hyperelliptic, basically giving generators and relations of their relative canonical algebra R(f ) = ⊕Vn , seen as a sheaf of algebras over B. Let us recall the vector bundles Vn introduced in the previous section as ⊗n . Roughly speaking then, R(f ) is a bundle whose fibres are the Vn = f∗ ωS|B canonical rings of the fibres of f . We state here only the theorem for genus 2 fibrations, since it is the one used in order to complete this classification. Theorem 14. A genus 2 fibration f : S → B is determined by the following 5 data • • • • •

the base curve B; the rank 2 vector bundle V1 := f∗ ωS|B over B; an effective divisor τ on B; a class ξ ∈ Ext1OB (S 2 (V1 ), Oτ )/(AutOB (Oτ ) yielding V2 ; letting A be the subring of the relative canonical algebra generated by V2 , V3+ the (+1) eigenbundle for the hyperelliptic involution on the fibres, and defining A˜6 := Hom((V3+ )2 , A6 ) (where A6 is the image in A of S 3 (V2 )), the last datum is an element w ∈ P(H 0 (A˜6 )).

Moreover, deg V1 = χ(OS ) − (b − 1), deg τ = KS2 − 2χ(OS ) − 10(b − 1). We want to explain here the geometry behind this theorem, which at a first glance can appear slightly technical. The vector bundles Vn yield the degree n part of the canonical ring of each fibre. So each of these vector bundles induces a rational map, the relative ncanonical map, from S to the corresponding projective bundle P(Vn ), mapping each fibre via its n-canonical map. The multiplication map of degree 1 forms give a morphism of sheaves S 2 (V1 ) → V2 which fits into an exact sequence 0 → S 2 (V1 ) → V2 → Oτ → 0 for an effective divisor τ on B supported on the image of the “bad” fibres (those which are not 2-connected, i.e., the fibres that can be decomposed as A + B with A, B effective divisors such that A · B = 1). ξ is the class of this extension. Therefore ξ yields V2 and determines the relative bicanonical map. Since the bicanonical map of a genus 2 curve is

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a double cover of a conic (branched in 6 points), this map has degree 2 onto a conic subbundle C of the P2 -bundle P(V2 ). We have C = P roj(A). In [CP05] it is proven that the relative bicanonical map is a morphism contracting at most some rational curves with self-intersection (−2), which implies that the branch curve has no “essential” singularities. In fact the 5th datum w determines the branch curve divC (w). In fact, sections of S 3 (V2 ), or of a twist of it, are “equations” of divisors in P(V2 ) which cut a cubic curve on each fibre. Taking the quotient by the subsheaf corresponding to the equations vanishing on the conic bundle C, one gets an equation for a divisor on the conic bundle, which cuts 6 points (intersection of a cubic and a conic) on a general fibre, and gives our branch curve. Let us come back to the case pg = q = 1, K 2 = 3 and g = 2. One needs to construct a suitable genus 2 fibration over an elliptic curve, with, in the sense of theorem 14, deg V1 = deg τ = 1. This is done in [CP05], by studying vector bundles on elliptic curves, and three different families are found. Let us finally mention that P(V1 ) is the symmetric square of B. In fact, our double cover is birational to a double cover of it. The behaviour of this double cover was described in [CC91], characterising these surfaces as double covers of the symmetric product of an elliptic curve with branch locus belonging to a certain algebraic system with prescribed singularities. This new method shows then, rather surprisingly, that this algebraic system is not connected. Surfaces with pg = q = 0 The class of surfaces with pg = q = 0 is one of the most complicated and intriguing classes of surfaces of general type. By the standard inequalities we have: 1 ≤ KS2 ≤ 9. We have already mentioned the case K 2 = 1, of the numerical Godeaux surfaces, the only case for which the bicanonical map is not finite, so let us restrict to KS2 ≥ 2. These surfaces are very far from being classified. From the point of view of the bicanonical system, this case was object of an intensive analysis by Mendes Lopes and Pardini in the last years. What it is known on the degree of the bicanonical map can be summarized in the following Theorem 15 ([MLP05], [MLP02]). Let S be a surface with pg = q = 0. Then • • • •

if K 2 = 9 ⇒ deg ϕ2K = 1, if K 2 = 7, 8 ⇒ deg ϕ2K = 1 or 2, if K 2 = 5, 6 ⇒ deg ϕ2K = 1, 2 or 4, if K 2 = 3, 4 ⇒ deg ϕ2K ≤ 5 and if moreover ϕ2K is a morphism, then deg ϕ2K = 1, 2 or 4,

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• if K 2 = 2 then obviously, since the image of the bicanonical map is P2 , then the bicanonical map is non birational, and obviously we have: deg ϕ2K ≤ 8, equality holding if and only if ϕ2K is a morphism. Let us recall that, by Reider’s theorem, the bicanonical map is a morphism as soon as K 2 ≥ 5. In fact the bicanonical map of all known examples of surfaces with pg = q = 0 and K 2 ≥ 2 is a morphism. So one could suspect4 that the bicanonical map is always a morphism whenever K 2 ≥ 2. Langer ([Lan00]) has proven that the bicanonical map of a minimal surface of general type with pg = q = 0 and KS2 = 4 has no fixed part. Mendes Lopes and Pardini gave also a description of some of these surfaces having non birational bicanonical map, in particular for K 2 ≥ 6. We summarize here some of their results Theorem 16 ( [MLP03], [MLP01], [MLP04a], [MLP04b]). Let S be a minimal surface of general type with pg = q = 0 whose bicanonical map is not birational. Then the image of the bicanonical map is a rational surface unless K 2 = 3, deg ϕ2K = 2, and the image is an Enriques sextic. These last surfaces form an irreducible and unirational family of dimension 6 of the moduli space. Moreover, • if K 2 = 8, S has an isotrivial genus 3 rational fibration whose general fibre is hyperelliptic with 6 double fibres; • if K 2 = 7, S has a genus 3 rational fibration whose general fibre is hyperelliptic with 5 double fibres and a fibre with reducible support, consisting of two components; • if K 2 = 6 and deg ϕ2K = 2, S has a genus 3 rational fibration whose general fibre is hyperelliptic with 4 or 5 double fibres; • if K 2 = 6 and deg ϕ2K = 4, S is a Burniat surface. Remark 3. Surfaces with pg = q = 0 and K 2 = 3, 4 were constructed by Keum ([Keu88]) and Naie ([Nai94]). For K 2 = 3 the degree of the bicanonical map can be equal to 2 and to 4, and it is an open question if it can be birational. Concerning the classification of surfaces with pg = q = 0 there has been recent progress. We would like to mention that in [BCG05b] a complete classification of surfaces with pg = q = 0 isogenous to a product (this forces K 2 = 8) is given. All the known surfaces with K 2 = 8, pg = q = 0 have the bidisk as universal covering. In the case pg = q = 0 and K 2 = 9 all the surfaces in question are, by Yau’s theorem ([Yau77] and [Yau78]), quotient of the complex unit ball in C2 by a discrete group Γ acting freely. The first effective example of such 4

Added in proof: Mendes Lopes and Pardini give in math.AG/0602633 an example of surfaces with K 2 = 2, pg = q = 0 such that |2KS | has base points.

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surfaces, called fake projective planes since they have the same Betti numbers as the projective plane, was given by Mumford ([Mum79]) using 2-adic uniformization. Other examples were given later in [IK98], while recently Keum ([Keu05]) gave an explicit geometric construction as a cyclic cover of a particular Dolgachev surface. In a recent preprint ([PY05]) G. Prasad and S.K. Yeung, using the arithmeticity of Γ , give twelve rather explicit lists of fake √ projective planes, each corresponding to an imaginary quadratic field Q( −a) and a prime p which ramifies in it. The interesting geometric features of these examples are that: i) all these groups Γ are indeed contained in SU (2, 1), hence the canonical divisor K is divisible by 3, ii) all these surfaces have a nontrivial first homology group H1 (S, Z).

3 Surfaces with pg = 4 In Enriques’ book on algebraic surfaces [Enr49] much emphasis was put on the effective construction of surfaces whose canonical map is birational, particularly for surfaces with pg = 4, where the canonical image is a surface in P3 . Later on, in particular in the last thirty years, many authors studied surfaces with pg = 4, with particular interest in the construction of surfaces with pg = 4, birational canonical map and K 2 as high as possible. If the canonical map of a minimal surface of general type S with pg = 4 is birational, then the standard inequalities give 5 ≤ K 2 ≤ 45. Nowadays we know examples, by the contribution of several authors, for every value of KS2 in the range 5 ≤ KS2 ≤ 28 (cf. e.g. [Cil81], [Cat99]). An example with KS2 = 31 has been recently obtained in [Lie03], although the example constructed has a big fixed part of the canonical system so that its canonical image has “only” degree 12. Moreover, the first two authors together with F. Grunewald have constructed a canonical surface in the projective 3space with K 2 = 45. This surface is obtained as a Galois covering of the plane with group (Z/5Z)2 , branched over a configuration of lines introduced by Hirzebruch (cf. [BCG05c]). In this case we have a rigid surface such that its canonical system has a fixed part. Obviously also in this case classification is the biggest challenge: for which values of KS2 is it possible to classify all possible minimal surfaces of general type with pg = 4? And more ambitiously: for which values of KS2 , q it is possible to completely describe the moduli space MK 2 ,4,q ? 3.1 K 2 = 4, 5 The cases K 2 = 4, 5 were already treated by Enriques ([Enr49], section 2, chapter VIII, pp.268–271), and the corresponding moduli spaces were completely understood already in the 70’s.

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We briefly recall these results. By Debarre’s inequality, all these surfaces are regular (in fact, this is true for K 2 ≤ 7). The canonical map of surfaces with K 2 = 4 and pg = 4 is a morphism of degree 2 onto an irreducible quadric in P3 , i.e., either a smooth quadric or a quadric cone. The general surface is a double cover of a smooth quadric branched over a smooth complete intersection with a sextic surface. A detailed analysis of the corresponding moduli space can be found in [Hor76], where the following is proven. Theorem 17. M4,4,0 is irreducible, unirational of dimension 42, its singular locus is irreducible of codimension 1 and corresponds exactly to the surfaces whose canonical image is a quadric cone. We have two classes of minimal surfaces of general type with K 2 = 4: let us say surfaces of type I (double covers of a smooth quadric) and II (double covers of a quadric cone). Correspondingly we have a stratification of M4,4,0 as a union of two locally closed strata, both irreducible, which we denote simply by I and II, of respective dimension 42 and 41. To draw a picture of this moduli space we need the following notation: Definition 9. Let A and B be two (locally closed) irreducible strata of a moduli space MK 2 ,pg ,q . If we write “A → B”, it means that there is a flat family with base a small disc ∆ε ⊂ C, whose central fibre is of type B and whose general fibre is of type A. In other words it means that the closure of the stratum A intersects the stratum B. With this notation a picture of M4,4,0 is the following: 42

I

41

 II

Note that at the left of each stratum stands the dimension of the corresponding irreducible stratum. The case K 2 = 5 is slightly more complicated, and completely described in [Hor75]: the canonical map is either a birational morphism to a quintic in P3 (type I), or a rational map of degree 2 onto an irreducible quadric, which can be as in the previous case either smooth (type IIa ) or a quadric cone (type IIb ). Theorem 18. M5,4,0 has two irreducible components, both unirational of dimension 40, intersecting in a 39 dimensional subvariety.

Complex Surfaces of General Type: Some Recent Progress

25

Here is the picture for M5,4,0 40

39

IIa I >> >> || | >> | >> ||  ~|| IIb

The moduli space has two irreducible unirational components of dimension 40 whose general point corresponds to surfaces with canonical image respectively a quintic or a smooth quadric. The surfaces whose canonical image is a quadric cone form a 39-dimensional subvariety of this moduli space, the intersection of the two irreducible components. 3.2 K 2 = 6 This case is much more complicated and surfaces with K 2 = 6 and pg = 4 were completely classified by Horikawa in [Hor78], obtaining a stratification of M6,4,0 in 11 strata. We will not enter here the details of this classification. A complete understanding of M6,4,0 is still missing, since it is not clear how exactly these 11 strata “glue” toghether. Theorem 19 ([Hor78], [BCP04]). M6,4,0 has 4 irreducible components, all unirational, one of dimension 39, the others of dimension 38. Moreover, the number of connected components of M6,4,0 is at most two. The main question on this moduli space remains the following: Question 1. Is M6,4,0 connected? Here is a partial picture:

39

37

IVa1 Ia { x { x {{ xx {{ xx { x |x  }{ IVa2 IVb1

36

 IVb2

38

IIIa {{ { { {{ }{{  V1 IIIb o || || | |  }|| V2

II

 Ib

This picture is partial because up to now it is not known whether all possible arrows are drawn. More precisely, M6,4,0 is connected if and only if one of the two following degenerations is possible: Ia → V1 or Ia → V2 .

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

This picture was done by Horikawa in [Hor78] with the exception of the horizontal line IIIb ← II, recently obtained in [BCP04]. We are going to explain how this arrow was obtained. We need to construct a flat family of surfaces whose central fibre is of type IIIb and whose general fibre is of type II. The difficulty lies in the obvious fact that, by reasons of dimension, the general surface of type IIIb cannot deform to a surface of type II. First however we need to explain what surfaces of type II and of type IIIb are. Surfaces of type II are defined as follows Definition 10. A minimal surface of general type with pg = 4 and K 2 = 6 is of type II if the canonical map has degree 3. It is immediate, that the canonical image is a quadric cone. In fact, one can see that their canonical models X are hypersurfaces of degree 9 in the weighted projective space P(1, 1, 2, 3), so that the canonical divisor of X is divisible by 2 as a Weil divisor. Definition 11. A minimal surface of general type with pg = 4 and K 2 = 6 is of type IIIb if it has no genus 2 pencil and the canonical system has a fixed component. Horikawa gave a very concrete description of this class (theorem 5.2 in [Hor78]) showing, among other things, that the canonical map is a double cover of a quadric cone. In both cases there is a pencil L on S, the strict transform of the ruling of the quadric cone, such that KS − 2L is effective. Computing intersection numbers one sees (cf. also [MLP00] for case II) that in both cases we have a decomposition KS = 2L+Z where Z is a fundamental cycle (Z 2 = −2, KZ = 0) and L is a genus 3 pencil with one simple base point. The main difference between the two cases is that in case IIIb all curves in the pencil are hyperelliptic, while in case II the general curve is nonhyperelliptic. The idea is then to construct a family preserving this decomposition and therefore the genus 3 pencil. Since Z is a fundamental cycle, one can consider the canonical model X of S, the surface with rational double points obtained contracting all fundamental cycles. The canonical class of X is then ˜ L ˜ being the image of L on X. 2−divisible (as a Weil divisor), since KX = 2L, ˜ the name semicanonWe consider the semicanonical ring R = R(X, L): ical being selfexplanatory, since the subring generated by the homogeneous elements of even degree is exactly the canonical ring of X (and of S). In order to compute the ring R we use the hyperplane section principle ˜ [Rei90a], first computing the quotient ring R(X, L)/(x 0 ), where x0 is a general homogeneous element of degree 1, i.e., corresponding to a general element C ˜ of the pencil L.

Complex Surfaces of General Type: Some Recent Progress

27

It is not difficult to prove that this ring equals the ring  3 3m R(C, P ) := ]P )) H 0 (C, OC ([ 2 2 m (we refer for the definition of the ring structure to [BCP04]), which is a suitable subring of the ring R(C, P ), where P is the base point of L (and a Weierstrass point for the hyperelliptic curve C). Then the strategy is the following: i) take any hyperelliptic genus 3 curve C, a Weierstraß point P on it, and compute the ring R(C, 32 P ) ii) “deform” it, adding an element of degree 1 in a flat way, to get the semicanonical ring of a surface of type IIIb iii) construct a flat family of rings (say with parameter t), whose central fibre (t = 0) is the ring constructed in ii), and whose general fibre (t = 0) is a hypersurface ring of type II Step i) is easy, steps ii) and iii) depend on the result of step i), concerning R(C, 32 P ). It turns out that R(C, 32 P ) is a Gorenstein ring of codimension 4, which can be expressed in a nice way: Proposition 1. Let C be a hyperelliptic curve of genus 3, p ∈ C a Weierstraß point. Then R(C, 32 p) ∼ = C[x, y, z, w, v, u]/I, where deg(x, y, z, w, v, u) = (1, 2, 3, 4, 5, 6) and the ideal I is generated by the 4 × 4 Pfaffians of the skewsymmetric ’extra-symmetric’ matrix ⎞ ⎛ 0 0 z v y x ⎜ 0w u z y ⎟ ⎟ ⎜ ⎜ 0 P˜9 u v ⎟ ⎟, ⎜ M =⎜ 0 w2 zw⎟ ⎟ ⎜ ⎝ 0 0⎠ −sym 0 where P˜9 is homogeneous of degree 9 in the variables x, y, z, w. The graded matrix M has the nice property to be “extrasymmetric”. Extrasymmetric matrices. Extrasymmetric matrices were introduced by Miles Reid and Duncan Dicks ([Rei90a], [Rei89]). Let A be a polynomial ring and let M be a skew ’extrasymmetric’ matrix of the form ⎛ ⎞ 0 ab c d e ⎜ 0f g h d⎟ ⎜ ⎟ ⎜ 0 i g c⎟ ⎜ ⎟ ⎜ 0 pf pb ⎟ ⎜ ⎟ ⎝ 0 pa⎠ −sym 0

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

where a, b, c, d, e, f, g, h, i, p ∈ A. Then 6 of the 15 (4 × 4) Pfaffians belong to the ideal generated by the other 9; moreover, if the entries are general enough, the ideal generated by these pfaffians has exactly 16 independent syzygies, which can all be written explicitely as functions of the entries of the matrix. This implies that, if we have a ring presented in this form, and the ring has no further syzygies, deforming the entries of the matrix (preserving the symmetries), we obtain automatically a a flat deformation of the ring. We mention that recent studies lead to generalizations of this “format”: here we need only this special “original” case. By proposition 1 R(C, 32 P ) is presented by an extrasymmetric matrix. Therefore we are reduced to add “x0 ” (step ii)) and “t” (step iii)) such that the obtained matrix is still extrasymmetric and has homogeneous entries and Pfaffians. Moreover, we have to take care that for t = 0 we obtain a fibration by hyperelliptic curves, whereas for t = 0 the general curve of the fibration has to be non hyperelliptic. A crucial ingredient is The zero of degree zero. Notice that the second entry of the first row of M is equal to 0, and it corresponds to a homogeneous element of degree 0. This allows to substitute in this entry a parameter t. For t = 0 the upshot is that from the 9 Pfaffians we can eliminate the variables w, v, u and we are left with the variables x0 , x1 , y, z and with a single equation of degree 9: that is, we have a semicanonical ring of type II, and we have succeeded! We obtain thus the following result: Theorem 20. Consider the ring C[x0 , x1 , y, z, w, v, u] with variables of respective degrees (1, 1, 2, 3, 4, 5, 6). Consider a family of skew extrasymmetric matrices, with parameter t ⎞ ⎛ 0 t z v y x1 ⎜ 0 w u P3 y ⎟ ⎟ ⎜ ⎜ 0 P9 u v ⎟ ⎟. Mt = ⎜ ⎜ 0 wP4 zP4 ⎟ ⎟ ⎜ ⎝ 0 tP4 ⎠ −sym 0 where the Pi ’s are homogeneous of degree i in the first 5 variables of the ring, and let Jt be the ideal generated by the 4 × 4 pfaffians of Mt . Then, for general choice of the polynomials Pi , C[x0 , x1 , y, z, w, v, u]/Jt is, for t = 0, the semicanonical ring of a surface of type IIIb , and for t = 0 the semicanonical ring of a surface of type II. The surfaces of type IIIb whose semicanonical ring can be presented as in the above theorem form a codimension 2 subscheme of the corresponding 38-dimensional stratum of M6,4,0 , lying in the intersection with the 38dimensional stratum II.

Complex Surfaces of General Type: Some Recent Progress

29

It is still unclear whether this is the whole intersection of the closures of these two strata. A priori one can only say that this intersection has at most dimension 37, and what we found has only dimension 36. Therefore there remains the following: Question 2. Exactly which surfaces of type IIIb lie in the closure of the stratum II? The above question is of course related to the singularity type of the local moduli space. 3.3 K 2 = 7 This is the last case for which there is a complete classification. In [Bau01] the first author gives a very precise description of surfaces with pg = 4, K 2 = 7 according to the behaviour of the canonical map, allowing to show that the moduli space M7,4,0 has three irreducible components M36 , M36 and M38 of respective dimensions 36, 36 and 38. Moreover, it is shown that the two irreducible components of dimension 36 intersect, whereas it is up to now not yet clear whether the component of dimension 38 is indeed a connected component or it intersects M36 . We encounter here a very similar situation as for K 2 = 6. There are two families, one in M36 , the other in M38 , where the first consists of surfaces admitting a non hyperelliptic genus 3 pencil, whereas the surfaces in the other family admit a hyperelliptic genus 3 pencil. In fact, the first and last author have been able to calculate the relative canonical algebra for the hyperelliptic case, which is Gorenstein of codimension 6. For this very high codimension there are yet no flexible formats known to organize the equations. Hopefully it will be possible to understand the deformations of this family.

4 Surfaces Isogeneous to a Product, Beauville Surfaces and the Absolute Galois Group Surfaces isogenous to a (higher) product were introduced and extensively studied by the second author in [Cat00], where it is proven that any surface S isogenous to a higher product has a unique minimal realization as a quotient S = (C1 × C2 )/G. Here C1 and C2 are smooth algebraic curves of genus at least 2 and G is a finite group acting freely, and with the property that no element acts trivially on one of the factors Ci . Moreover, it was shown that the topology of a surface isogenous to a product determines its deformation class up to complex conjugation. The following result contains a correction to Theorem 4.14 of [Cat00] (cf. theorem 3.3 of [Cat03]).

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Theorem 21. Let S = (C1 × C2 )/G be a surface isogenous to a product. Then any surface S  with the same topological Euler number and the same fundamental group as S is diffeomorphic to S. If moreover S  is orientedly ¯ In other diffeomorphic to S, then S  is deformation equivalent to S or to S. top dif f is either irreducible words, the corresponding moduli space MS = MS and connected or it contains two connected components which are exchanged by complex conjugation. This class of surfaces and their higher dimensional analogues provide a wide specimen of examples where one can test or disprove several conjectures and questions (cf. e.g. [Cat03], [BC04], [BCG05a], compare also the next section). ¯ Moreover, the absolute Galois group Aut(Q/Q) acts on the moduli spaces of this class of surfaces: we shall outline a direct connection with Grothendieck’s dream of “dessins d’enfants”. In the following we shall concentrate on a class of surfaces isogenous to a product, namely the rigid ones. We recall that an algebraic variety X is rigid if and only if it does not have any non trivial deformations (e.g., the projective space is rigid). There is another (stronger) notion of rigidity, which is the following Definition 12. An algebraic variety X is called strongly rigid if any other variety homotopically equivalent to X is either biholomorphic or antibiholomorphic to X. Remark 4. 1) It is nowadays wellknown that smooth compact quotients of symmetric spaces are rigid (cf. [CV60]). 2) Mostow (cf. [Mos73]) proved that indeed locally symmetric spaces of complex dimension ≥ 2 are strongly rigid, in the sense that any homotopy equivalence is induced by a unique isometry. These varieties are of general type and the moduli space of varieties of general type is defined over Z, and naturally the absolute Galois group ¯ Gal(Q/Q) acts on the set of their connected components. So, in our special ¯ case, Gal(Q/Q) acts on the isolated points which parametrize rigid varieties. In particular, rigid varieties are defined over a number field and work of Shimura gives a possible way of computing explicitly their fields of definition. By this reason these varieties were named Shimura varieties (cf. Deligne’s Bourbaki seminar [Del71]). A quite general question is Question 3. What are the fields of definition of rigid varieties? What is the ¯ Gal(Q/Q)-orbit of the point in the moduli space corresponding to a rigid variety? Much simpler examples of rigid varieties were found by the second author (cf. [Cat00]).

Complex Surfaces of General Type: Some Recent Progress

31

Beauville Surfaces Inspired by a construction of A. Beauville of a surface with K 2 = 8, pg = q = 0 (cf. [Bea78]) as a quotient of the product of two Fermat curves of degree 5 by the action of the group Z/5Z, in [Cat00] the following definition was given Definition 13. A Beauville surface is a compact complex surface S which 1) is rigid, i.e., it has no nontrivial deformation, 2) is isogenous to a higher product, i.e., it is a quotient S = (C1 × C2 )/G of a product of curves of resp. genera ≥ 2 by the free action of a finite group G. Notice that, given a surface isogenous to a product, we obtain always three more, exchanging C1 with its conjugate curve C¯1 , or C2 with C¯2 : but only if we conjugate both C1 , C2 we obtain an orientedly diffeomorphic surface. These four surfaces could however be all biholomorphic to each other. If S is a Beauville surface and X is orientedly diffeomorphic to S, then ¯ theorem 21 implies: X ∼ = S or X ∼ = S. In other words, the corresponding subset of the moduli space MS consists of one or two points (if we insist on keeping the orientation fixed, else we may get up to four points). Definition 14. C is a triangle curve if there is a finite group G acting effectively on C and satisfying the properties i) C/G ∼ = P1C , and ii) f : C → P1C ∼ = C/G has {0, 1, ∞} as branch set. Remark 5. The rigidity of a Beauville surface is equivalent to the condition that (Ci , G0 ) is a triangle curve, for i = 1, 2 (G0 ⊂ G is the subgroup of index ≤ 2 which does not exchange the two factors). Recall now the classical Theorem 22. (Riemann’s Existence Theorem) There is a natural bijection between: 1) Equivalence classes of holomorphic mappings f : C → P1C , of degree n and with Branch set Bf ⊂ B, (where C is a compact Riemann surface, and f : C → P1C , f  : C  → P1C are said to be equivalent if there is a biholomorphism g : C  → C such that f  = f ◦ g). 2) Conjugacy classes of monodromy homomorphisms µ : π1 (P1C −B) → Sn (here, Sn is the symmetric group in n letters, and µ ∼ = µ iff there is an  −1 , (∀γ). element τ ∈ Sn with µ(γ) = τ µ (γ)τ Moreover: 3) C is connected if and only if the subgroup Im(µ) acts transitively on {1, 2, . . . n}. 4) f is a polynomial if and only if ∞ ∈ B, the monodromy at ∞ is a cyclical permutation, and g(C) = 0.

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

Remark 6. 1) Assume that ∞ ∈ B, so {∞, b1 , . . . bd } = B: then π1 (P1C − B) is a free group generated by γ1 , . . . γd and µ is completely determined by the local monodromies τi := µ(γi ). Grothendieck’s enthusiasm was raised by the following result, where Belyi ([Bel79])made a very clever and very simple use of some explicit polynomials, m+r m r now called the Belyi polynomials, of the form (m+r) mm rr z (z − 1) in order to ¯ reduce the number of critical values of an algebraic function defined over Q. Theorem 23. (Belyi) An algebraic curve C can be defined over Q if and only if there exists a holomorphic map f : C → P1C with branch set only {0, 1, ∞}. The word ”dessin d’ enfant” = child’s drawing is due to the fact that the monodromy of f is determined by the ’dessin d’ enfant’ f −1 ([0, 1]), a bipartite graph (the vertices have label 0 or 1 according to their image) such that at each vertex one has a cyclical order of the edges incident in the vertex (this property holds because the graph is contained in a complex curve C thus we choose the corresponding cyclical counterclockwise order). It is clear that the triangle curves correspond to a certain class of ’dessins d’ enfants’, those which admit a group action with quotient the interval [0, 1]. Let us parenthetically observe that Gabino Gonzalez was recently able to extend Belyi’s theorem to the case of complex surfaces (in terms of Lefschetz maps with three critical values) (cf. [Gon04]). Grothendieck ([Gro97] )proposed to look at the ’dessins d’ enfants’ in order to get representations of the absolute Galois group Gal(Q, Q). We just explained that a Beauville surface is defined over Q, and that the Galois group Gal(Q, Q) operates on the discrete subset of the moduli space MS corresponding to Beauville surfaces. This action is rather strictly related to the action on the ’dessins d’ enfants’, but in this case, by theorem 21, the Galois group Gal(Q, Q) may transform a Beauville surface into another one with a non isomorphic fundamental group. Phenomena of this kind were already observed by J.P. Serre (cf. [Ser64]): here the idea is not to consider this as a pathology, but as a source of information, and to actually try to understand the representation of the Galois group Gal(Q, Q) on the class of groups which are fundamental groups of Beauville surfaces (and of their higher dimensional analogues). It looks therefore interesting to investigate these surfaces and to address the following problems: Question 4. Existence and classification of Beauville surfaces, i.e., a) which finite groups G can occur? b) classify all possible Beauville surfaces for a given finite group G. Question 5. Is the Beauville surface S biholomorphic to its complex conjugate ¯ surface S?

Complex Surfaces of General Type: Some Recent Progress

33

Is S real (i.e., does there exist a biholomorphic map σ : S → S¯ with σ = id)? 2

Another motivation to find these surfaces was also given by the following FRIEDMAN-MORGAN’S SPECULATION ( [FM88] 1987): DEF ⇐⇒ DIFF (Differentiable equivalence and deformation equivalence coincide for surfaces). A series of counterexamples were given by several authors and the simplest examples were given using non rigid surfaces isogenous to a product. Great part of the next section will be devoted to these equivalence relations. In order to reduce the description of Beauville surfaces to some group theoretic statement, we need to recall that surfaces isogenous to a higher product belong to two types: • S is of unmixed type if the action of G does not mix the two factors, i.e., it is the product action of respective actions of G on C1 , resp. C2 . • S is of mixed type, i.e., C1 is isomorphic to C2 , and the subgroup G0 of transformations in G which do not mix the factors has index precisely 2 in G. The datum of a Beauville surface can be completely described group theoretically, since it is equivalent to the datum of two triangle curves with isomorphic groups. Definition 15. Let G be a finite group. 1) A quadruple v = (a1 , c1 ; a2 , c2 ) of elements of G is an unmixed Beauville structure for G if and only if (i) the pairs a1 , c1 , and a2 , c2 both generate G, (ii) Σ(a1 , c1 ) ∩ Σ(a2 , c2 ) = {1G }, where Σ(a, c) :=

∞

{gai g −1 , gci g −1 , g(ac)i g −1 }.

g∈G i=0

We write U(G) for the set of unmixed Beauville structures on G. 2) A mixed Beauville quadruple for G is a quadruple M = (G0 ; a, c; g) consisting of a subgroup G0 of index 2 in G, of elements a, c ∈ G0 and of an element g ∈ G such that i) G0 is generated by a, c, ii) g ∈ / G0 , / Σ(a, c). iii) for every γ ∈ G0 we have gγgγ ∈ iv) Σ(a, c) ∩ Σ(gag −1 , gcg −1 ) = {1G }. We write M(G) for the set of mixed Beauville quadruples on the group G. Remark 7. We consider here finite groups G having a pair (a, c) of generators. Setting (r, s, t) := (ord(a), ord(c), ord(ac)), such a group is a quotient of the triangle group T (r, s, t) := x, y | xr = y s = (xy)t = 1.

(1)

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Ingrid C. Bauer, Fabrizio Catanese, and Roberto Pignatelli

It is now easy to explain how to get a surface from the above data, if we remember Riemann’s Existence Theorem which we recalled just above. We take as base point ∞ ∈ P1C and consider B := {−1, 0, 1}. We choose the following generators α, β of π1 (P1C − B, ∞) (γ := (α · β)−1 ):

Let now G be a finite group and v = (a1 , c1 ; a2 , c2 ) ∈ U(G). We get surjective homomorphisms π1 (P1C − B, ∞) → G,

α → ai , γ → ci

(2)

and Galois coverings λi : C(ai , ci ) → P1C ramified only in {−1, 0, 1} with ramification indices equal to the orders of ai , bi , ci and with group G (by Riemann’s existence theorem). Remark 8. 1) Condition (1), ii) ensures that the action of G on C(a1 , c1 ) × C(a2 , c2 ) is free. −1 −1 −1 2) Let be ι(a1 , c1 ; a2 , c2 ) = (a−1 1 , c1 ; a2 , c2 ). Then S(ι(v)) = S(v) (note −1 −1 in fact that α ¯ = α , γ¯ = γ ). 3) One can verify that the required conditions automatically imply: g(C(a1 , c1 )) ≥ 2 and g(C(a2 , c2 )) ≥ 2. One has: Proposition 2. Let G be a finite group and v = (a1 , c1 ; a2 , c2 ) ∈ U(G). Assume that {ord(a1 ), ord(c1 ), ord(a1 c1 )} = {ord(a2 ), ord(c2 ), ord(a2 c2 )} and that ord(ai ) < ord(ai ci ) < ord(ci ). Then S(v) ∼ = S(v) if and only if there are inner automorphisms φ1 , φ2 of G and an automorphism ψ ∈ Aut(G) −1 such that, setting ψj := ψ ◦ φj , we have ψ1 (a1 ) = a−1 1 , ψ1 (c1 ) = c1 , and −1 −1 ψ2 (a2 ) = a2 , ψ2 (c2 ) = c2 . In particular, under the above assumption, S(v) is isomorphic to S(v) if and only if S(v) has a real structure. Remark 9. Dropping the assumption on the orders of ai , ci , we can define a finite permutation group AU (G)such that for v, v  ∈ U(G) we have: S(v) ∼ = S(v  ) if and only if v is in the AU (G)-orbit of v  . Remark 10. If G is abelian, v ∈ U(G). Then S(v) always has a real structure. We have the following results (cf. [BCG05a] for some of these, others have not yet been published):

Complex Surfaces of General Type: Some Recent Progress

35

Theorem 24. 1) An abelian group G admits an unmixed Beauville structure iff G ∼ = (Z/n)2 , (n, 6) = 1. 2) The following groups admit unmixed Beauville structures: a) the alternating group An for large n, b) the symmetric group Sn for n ∈ N with n ≥ 7 c) SL(2, Fp ), PSL(2, Fp ) for p = 2, 3, 5. With the help of the computer algebra program MAGMA all finite simple nonabelian groups of order ≤ 50000 were checked and unmixed Beaville structures were found on all of them, with the exception of A5 (where it can’t be found!). This led to the following Conjecture 1. ([BCG05a]) All finite simple nonabelian groups except A5 admit an unmixed Beauville structure. This conjecture was also checked for some bigger simple groups like the Mathieu groups M12, M22 and also matrix groups of size bigger then 2. Call now (r, s, t) ∈ N3 hyperbolic if 1 1 1 + + < 1. r s t In this case the triangle group T (r, s, t) is hyperbolic. These studies led also to the following suggestive: Conjecture 2. ([BCG05a]) Let (r, s, t), (r , s , t ) be two hyperbolic types. Then almost all alternating groups An have an unmixed Beauville structure v = (a1 , c1 ; a2 , c2 ) where (a1 , c1 ) has type (r, s, t) and (a2 , c2 ) has type (r , s , t ). The above conjectures are variations of a conjecture of Higman (proved by B. Everitt (2000), [Eve00]) asserting that every hyperbolic triangle group surjects onto almost all alternating groups. Concrete explicit examples of rigid surfaces not biholomorphic to their complex conjugate were also given: Theorem 25. The following groups admit unmixed Beauville structures v such that S(v) is not biholomorpic to S(v): 1. the symmetric group Sn for n ≥ 7 2. the alternating group An for n ≥ 16 and n ≡ 0 mod 4, n ≡ 1 mod 3, n ≡ 3, 4 mod 7. And also new examples of real points of moduli spaces which do not correspond to real surfaces: Theorem 26. Let p > 5 be a prime with p ≡ 1 mod 4, p ≡ 2, 4 mod 5, p ≡ 5 mod 13 and p ≡ 4 mod 11. Set n := 3p + 1. Then there is an unmixed Beauville surface S with group An which is biholomorphic to the complex ¯ but is not real. conjugate surface S,

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For mixed Beauville surfaces the situation is more complicated, as already the following suggests. Theorem 27. 1) If a group G admits a mixed Beauville structure, then the subgroup G0 is non abelian. 2) No group of order ≤ 512 admits a mixed Beauville structure. A general construction of finite groups admitting a mixed Beauville structure was given in [BCG05a]. Let H be a non-trivial group, and let Θ : H × H → H × H be the automorphism defined by Θ(g, h) := (h, g) (g, h ∈ H). We consider the semidirect product (3) H[4] := (H × H)
Z/4Z where the generator 1 of Z/4Z acts through Θ on H × H. Since Θ2 is the identity we find H[2] := H × H × (2Z/4Z) ∼ = H × H × Z/2Z

(4)

as a subgroup of index 2 in H[4] . We have now Lemma 1. Let H be a non-trivial group and let a1 , c1 , a2 , c2 be elements of H. Assume that 1. the orders of a1 , c1 are even, 2. a21 , a1 c1 , c21 generate H, 3. a2 , c2 also generate H, 4.(ord(a1 ) · ord(c1 ) · ord(a1 c1 ), ord(a2 ) · ord(c2 ) · ord(a2 c2 )) = 1. Set G := H[4] , G0 := H[2] as above and a := (a1 , a2 , 2), c := (c1 , c2 , 2). Then (G0 ; a, c) is a mixed Beauville structure on G. Proof. It is easy to see that a, c generate G0 := H[2] . The crucial observation is that (1H , 1H , 2) ∈ / Σ(a, c).

(5)

In fact, if this were not correct, it would have to be conjugate of a power of a, c or b. Since the orders of a1 , b1 , c1 are even, we obtain a contradiction. Suppose that h = (x, y, z) ∈ Σ(a, c) satisfies ord(x) = ord(y): then our condition 4 implies that x = y = 1H and (5) shows h = 1H[4] . Let now g ∈ H[4] , g ∈ / H[2] and γ ∈ G0 = H[2] be given. Then gγ = (x, y, ±1) for appropriate x, y ∈ H. We find (gγ)2 = (xy, yx, 2) and the orders of the first two components of (gγ)2 are the same, contradicting the above remark. Therefore the third condition is satisfied.

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We come now to the fourth condition of a mixed Beauville quadruple. / H[2] be given, for instance (1H , 1H , 1). Conjugation with Let g ∈ H[4] , g ∈ g interchanges then the first two components of an element h ∈ H[4] . Our hypothesis 4 implies the result.   As an application we find the following examples Theorem 28. Let p be a prime with p ≡ 3 mod 4 and p ≡ 1 mod 5 and consider the group H := SL(2, Fp ). Then H[4] admits a mixed Beauville structure u such that S(u) is not biholomorphic to S(u). Remark 11. Note that the smallest prime satifying the above congruences is p = 11 and we get that G has order equal to 6969600. Question 6. : which is the minimal order of a group admitting a mixed Beauville structure?

5 Lefschetz Pencils and Braid Monodromies 5.1 Braids and the Mapping Class Group The elegant definition by E. Artin of the braid group (cf. [Art26], [Art65]) supplies a powerful tool, even if difficult to handle, for the study of the differential topology of algebraic varieties, in particular of algebraic surfaces. Remark 12. We observe that the subsets {w1 , . . . , wn } ⊂ C of n distinct points in C are in one to one correspondence with monic polynomials P (z) ∈ C[z] of degree n with non vanishing discriminant δ(P ). Definition 16. The group Bn := π1 (C[z]n \{P |δ(P ) = 0}), i.e., the fundamental group of the space of polynomials of degree n having n distinct roots, is called Artin’s braid group.

n Usually, one takes as base point the polynomial P (Z) = ( i=1 (z − i)) ∈ C[z]n (or the set {1, . . . , n}). To a closed (continuous) path α : [0, 1] → (C[z]n \{P |δ(P ) = 0}) one associates the subset {(z, t) ∈ C × R | α(t)(z) := αt (z) = 0} of R3 . Figure 2 below shows two realizations of the same braid. Remark 13. Obviously there is a lifting of α to Cn , the space of n-tuples of roots of polynomials of degree n and there are (continuous) functions wi (t) n such that wi (0) = i and αt (z) = i=1 (z − wi (t)). Then to each braid is associated a naturally defined permutation τ ∈ Sn given by τ (i) := wi (1).

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Fig. 2. Relation aba = bab on braids

A very powerful generalization of Artin’s braid group was given by M. Dehn (cf. [Deh38], we refer also to the book [Bir74]). Definition 17. Let M be a differentiable manifold, then the mapping class group (or Dehn group) of M is the group M ap(M) := π0 (Dif f (M)) = Dif f (M)/Dif f 0 (M), where Dif f 0 (M) is the subgroup of diffeomorphisms of M isotopic to the identity. Remark 14. If M is oriented then we often tacitly take Dif f + (M), the group of orientation preserving diffeomorphisms of M instead of Dif f (M), in the definition of the mapping class group. But it is more accurate to distinguish in this case M ap+ (M) from M ap(M). If M is a curve of genus g, then its mapping class group will be denoted by M apg . The relation between the above two definitions is the following: Theorem 29. The braid group Bn is isomorphic to the group π0 (M ap∞ (C\{1, . . . n})), where M ap∞ (C\{1, . . . n}) is the group of diffeomorphisms which are the identity outside the circle with center 0 and radius 2n. Therefore Artin’s standard generators σi of Bn (i = 1, . . . n − 1) can be represented by so-called half-twists. Definition 18. The half-twist σj is the diffeomorphism of C\{1, . . . n} isotopic to the homeomorphism given by: - rotation of 180 degrees on the circle with center j + 12 and radius 12 , - on a circle with the same center and radius 2+t 4 the map σj is the identity if t ≥ 1 and rotation of 180(1 − t) degrees, if t ≤ 1.

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Fig. 3. A geometric base of π1 (C − {1, . . . n})

Now, it is obvious that Bn acts on the free group π1 (C\{1, . . . n}), which has a geometric basis (we take as base point the complex number p := −2ni) γ1 , . . . γn as explained in figure 3. This action is called the Hurwitz action of the braid group and has the following algebraic description • σi (γi ) = γi+1 −1 • σi (γi γi+1 ) = γi γi+1 , whence σi (γi+1 ) = γi+1 γi γi+1 • σi (γj ) = γj for j = i, i + 1. Observe that the product γ1 γ2 . . . γn is left invariant under this action. Definition 19. We consider a group G and its cartesian product Gn . The map associating to each (g1 , g2 , . . . , gn ) the product g := g1 g2 . . . , gn ∈ G gives a partition of Gn , whose subsets are called factorizations of an element g ∈ G. Bn acts on Gn leaving invariant the partitions, and its orbits are called Hurwitz equivalence classes of factorizations. Definition 20. (cf. figure 4 below) Let C be a compact Riemann surface. Then a positive Dehn twist Tα with respect to a simple closed curve α on C is an isotopy class of a diffeomorphism h of C which is equal to the identity outside an annular neighbourhood of α, while inside the annulus h rotates one boundary of the annulus by 360 degrees to the right and damps the rotation down to the identity at the other boundary. If one considers a hyperelliptic Riemann surface given as a branched cover of P1C \{1, . . . n} one sees that the Artin half twist σj lifts to the Dehn twists on the loop which is the inverse image of the segment [j, j + 1]. Dehn’s fundamental result is the following Theorem 30. The mapping class group M apg is generated by Dehn twists. Explicit presentations of M apg have been given by Hatcher and Thurston ([HT80]), which have been improved by Wajnryb ([Waj83]) who obtained a simpler presentation of the mapping class group (cf. also [Waj99]).

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Fig. 4. At the left, a half twist; at the right: its lift: the Dehn twist T and its action on the segment D

5.2 Lefschetz Fibrations The method introduced by Lefschetz for the study of the topology of algebraic varieties is the topological analogue of the method of hyperplane sections and projections of the classical italian algebraic geometers. It was classically used to describe the homotopy and homology groups of algebraic varieties. In the 70’s Moisezon and Kas realized, after the work of Smale, that Lefschetz fibrations could be used to investigate the differential topology of algebraic varieties, especially of algebraic surfaces. For instance, it is an extremely difficult problem to decide whether two algebraic surfaces which are not deformation equivalent are in fact diffeomorphic, even in the case where they are simply connected. Here, the theory of Lefschetz fibrations offers a method to prove that two surfaces are diffeomorphic ([Kas80]). Definition 21. Let M be a compact differentiable (or even symplectic) manifold of dimension 4 A Lefschetz fibration is a differentiable map f : M → P1C which a) is of maximal rank except for a finite number of critical points p1 , . . . pm which have distinct critical values b1 , . . . bm ∈ P1C , b) has the property that around pi there are complex coordinates (x, y) ∈ C2 such that locally f = x2 − y 2 + const. (in the symplectic case, in the given coordinates the symplectic form ω of M has to correspond to the natural symplectic structure on C2 ). Remark 15. 1) A similar definition can be given if M is a manifold with boundary, replacing P1C by a disc D ⊂ C.

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2) An important theorem of Donaldson ([Don99]) asserts that for symplectic manifolds there exists (as for the case of projective manifolds) a Lefschetz pencil, i.e., a Lefschetz fibration f : M  → P1C on a symplectic blow up M  of M (cf. [MS98]). 3) A Lefschetz fibration with fibres genus g curves and with critical values b1 , . . . bm ∈ P1C , once a geometric basis γ1 , γ2 , . . . , γm of π1 (P1C \{b1 , . . . , bm }) is chosen, determines a factorization of the identity in the mapping class group M apg τ1 ◦ τ2 ◦ · · · ◦ τm = Id as a product of Dehn twists. We are now ready to state the theorem of Kas (cf. [Kas80]). Theorem 31. Two Lefschetz fibrations (M, f ), (M  , f  ) are equivalent (i.e., there are two diffeomorphisms u : M → M  , v : P1 → P1 such that f  ◦ u = v ◦ f ) if and only if the two corresponding factorizations of the identity in the mapping class group are equivalent (under the equivalence relation generated by Hurwitz equivalence and by simultaneous conjugation). Remark 16. 1) A similar result holds for Lefschetz fibrations over the disc and we get a factorization τ1 ◦ τ2 ◦ · · · ◦ τm = φ of the monodromy φ of the fibration over the boundary of the disc D. 2) The fibration admits a symplectic structure if and only if each Dehn twist in the factorization is positively oriented. Assume that we are given two Lefschetz fibrations over P1C : then we can define the fiber sum of these two fibrations, which depends on a diffeomorphism chosen between two respective smooth fibers (cf. [GS99]). This operation translates (in view of the above quoted theorem of Kas) into the following definition of “conjugated composition” of factorization: Definition 22. Let τ1 ◦ τ2 ◦ · · · ◦ τm = φ and τ1 ◦ τ2 ◦ · · · ◦ τr = φ be two factorizations: then their by ψ conjugated composition is the factorization τ1 ◦ τ2 ◦ . . . τm ◦ (τ1 )ψ ◦ (τ2 )ψ ◦ · · · ◦ (τr )ψ = φ(φ )ψ . Remark 17. 1) If ψ and φ commute, we obtain a factorization of φφ . 2) A particular case is φ, φ = id and it corresponds to Lefschetz fibrations over P1 .

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5.3 Braid Monodromy and Chisini’ Problem Let B ⊂ P2C be a plane algebraic curve of degree d, and let P be a generic point not on B. Then the pencil of lines Lt passing through P determines a one parameter family of d-uples of points of C ∼ = Lt \{P }, i.e., Lt ∩ B. Therefore one gets a factorization of (∆2 )d in the braid group Bn , where (∆2 ) = (σd−1 σd−2 . . . σ1 )d is the generator of the center of the braid group. The equivalence class of the factorization does not depend on the point P (if it is chosen generic) and does not depend on B, if B varies in an equisingular family of curves. Chisini was mainly interested in the case of cuspidal curves (cf. e.g. [Chi44], [Chi55]), mainly because these are the branch curves of a generic projection f : S → P2C , for any smooth projective surface S ⊂ Pr . More precisely, a generic projection f : S → P2C is a covering whose branch curve has only nodes and cusps as singularities, and moreover is such that the local monodromy around a smooth point of the branch curve is a transposition. Maps with those properties are called generic coverings: for these the local monodromies are only Z/2 = S2 (at the smooth points of the branch curve B), S3 at the cusps, and Z/2 × Z/2 at the nodes. In such a case we have a cuspidal factorization, i.e. all factors are powers of a half twist, with respective exponent 1, 2, 3. Chisini posed the following Conjecture 3. (Chisini’s conjecture.) Given two generic coverings f : S → P2C , f  : S  → P2C , one of them of degree d ≥ 5, assume that they have the same branch curve B. Is it then true that f and f  are equivalent? Observe that the condition on the degree is necessary, since counterexamples with d = 4 are furnished by the dual curve of a smooth plane cubic (as already known to Chisini, who gave a counterexample with d = 4, d = 3, while counterexamples with d = d = 4 were given in [Cat86b]). The conjecture has been proven under the hypothesis that the degree of each covering is at least 12, essentially by Kulikov (cf. [Kul99]). In fact, Kulikov proved the result under a more complicated assumption and shortly later Nemirovski [Nem01] noticed, just by using the Miyaoka-Yau inequality, that Kulikov’s assumption was implied by the simple assumption d ≥ 12. Later on generalizations of this result were obtained for singular (normal) surfaces [Kul03] or for curves with more complicated singularities [MP02]. A negative answer instead has the following problem of Chisini (due to work of B. Moishezon (cf. [Moi94]). Chisini’ s problem: (cf. [Chi55]). Given a cuspidal factorization, which is regenerable to the factorization of a smooth plane curve, is there a cuspidal curve which induces the given factorization?

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Regenerable means that there is a factorization (in the equivalence class) such that, after replacing each factor σ i (i = 2, 3) by the i corresponding factors (e.g. , σ 3 is replaced by σ◦σ◦σ) one obtains the factorization belonging to a non singular plane curve. Remark 18. 1) Moishezon proves that there exist infinitely many non equivalent cuspidal factorizations observing that π1 (P2C \B) is an invariant defined in terms of the factorization alone. On the other hand, the family of cuspidal curves of a fixed degree form an algebraic set, hence has a finite number of connected components. These two statements together give a negative answer to the above cited problem of Chisini. The examples of Moishezon have been recently reinterpreted in [ADK03], with a simpler treatment, in terms of symplectic surgeries. 2) In fact, as conjectured by Moishezon, a cuspidal factorization together with a generic monodromy with values in Sn induces a covering M → P2C , where M is a symplectic fourmanifold. Extending Donaldson’s techniques (for proving the existence of symplectic Lefschetz fibrations) Auroux and Katzarkov ([AK00]) proved that each symplectic 4-manifold is in a natural way ’asymptotically’ realized by such a generic covering. They propose to use an appropriate quotient of π1 (P2C \B) in order to produce invariants of symplectic structures, using the methods introduced by Moishezon and Teicher in a series of technically difficult papers ( see e.g. [MT92]). It seems however that, up to now, these groups π1 (P2C \B) allow only to detect homology invariants of the projected fourmanifold ([ADKY04]). 3) Suppose we have a surface S of general type and a pluricanonical embedding. Then a generic projection to P3C gives a surface with a double curve Γ  . Now, project further to P2C and we do not only get the branch curve B, but also a curve Γ , image of Γ  . Even if Chisini’s conjecture tells us that from the holomorphic point of view B determines the surface S and therefore the curve Γ , it does not follow that the fundamental group π1 (P2C \B) determines the group π1 (P2C \(B ∪ Γ )). It would be interesting to calculate this second fundamental group, even in special cases.

6 DEF, DIFF and Other Equivalence Relations As we said, one of the fundamental problems in the theory of complex algebraic surfaces is to understand the moduli spaces of surfaces of general type, and in particular their connected components, which parametrize the deformation equivalence classes of minimal surfaces of general type. Definition 23. Two minimal surfaces S and S  are said to be def-equivalent (we also write: S ∼def S  ) if and only if they are elements of the same connected component of the moduli space.

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By the classical theorem of Ehresmann, two def-equivalent algebraic surfaces are (orientedly) diffeomorphic. In the late eighties Friedman and Morgan (cf. [FM88]) conjectured that two algebraic surfaces are diffeomorphic if and only if they are def-equivalent. We will abbreviate this conjecture in the following by the acronym def = diff. The second author would like to point out here that he had made the opposite conjecture in the early eithties (cf. [Kat83]). Donaldson’s breaktrough results had made clear that diffeomorphism and homeomorphism differ drastically for algebraic surfaces (cf. [Don83]) and the success of gauge theory led Frieman and Morgan to “speculate” that the diffeomorphism type of algebraic surfaces determines the deformation class. After the first counterexamples of M. Manetti (cf. [Man01]) appeared, there were further counterexamples given by Catanese, KharlamovKulikov, Catanese-Wajnryb, Bauer-Catanese-Grunewald (cf. [Cat03], [KK02], [BCG05a], [CW04]). In the cited papers by Catanese, Kharlamov-Kulikov, Bauer-CataneseGrunewald, the counterexamples are given by pairs of surfaces, where one is the complex conjugate of the other. One could say that somehow these counterexamples are ’cheap’, and somehow in the air (cf. the definition of strong rigidity). The second author was very recently informed by R. Friedman that also he and Morgan were aware of such ’complex conjugate’ counterexamples, but for the case of elliptic surfaces. Since the beautiful examples of Manetti yield non simply connected surfaces, it made sense to weaken the conjecture def = diff in the following way. Question 7. Is the speculation def = diff true if one requires the diffeomorphism φ : S → S  to send the first Chern class c1 (KS ) ∈ H 2 (S, Z) in c1 (KS
) and moreover one requires the surfaces to be simply connected? But even this weaker question turned out to have a negative answer, as it was shown by the second author and Wajnryb ([CW04]). Remark 19. If two surfaces are def-equivalent, then there exists a diffeomrophism sending the canonical class c1 (KS ) ∈ H 2 (S, Z) in the canonical class c1 (KS
). On the other hand, by the result of Seiberg-Witten theory we know that a diffeomorphism sends the canonical class of a minimal surface S to ±c1 (KS
). Therefore, if one gives at least three surfaces, which are pairwise diffeomorphic, one finds at least two surfaces with the property that there exists a diffeomorphism between them sending the canonical class of one to the canonical class of the other. Theorem 32. ([CW04]) For each natural number h there are simply connected surfaces S1 , . . . , Sh which are pairwise diffeomorphic, but are such that two of them are never def-equivalent.

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The above surfaces S1 , . . . , Sh belong to the class of the so-called (a, b, c)surfaces, obtained as minimal compactification of some affine surface described by the following two equations: z 2 = f (x, y), w2 = g(x, y), where f and g are suitable polynomials of respective bidegrees (2a, 2b), (2c, 2b). They can be compactified simply by bihomogenizing the polynomials f, g, thus obtaining Galois covers of P1 × P1 with Galois group Z/2Z. We remark that the above compactification is smooth if the two curves {f = 0} and {g = 0} in P1 × P1 are smooth and intersect transversally. We say that these surfaces are bidouble (i.e., Galois covers with Galois group (Z/2Z)2 ) covers of P1 ×P1 of type (2a, 2b), (2c, 2b) (cf. [Cat84], [Cat99]). The above theorem is implied by the two following results: Theorem 33. Let a, b, c, k be positive even numbers such that 1) a, b, c − k ≥ 4; 2) a ≥ 2c + 1; 3) b ≥ c + 2; and either 41 ) b ≥ 2a + 2k − 1 or 42 ) a ≥ b + 2. Furthermore, let S be an (a, b, c)-surface and S  be an (a + k, b, c − k)surface. Then S is not def-equivalent to S  . Theorem 34. Let S be an (a, b, c)-surface and S  be an (a+1, b, c−1)-surface. Moreover, assume that a, b, c − 1 ≥ 2. Then S and S  are diffeomorphic. Remark 20. Observe that the surfaces in question above are simply connected (cf. [Cat84], prop. 2.7.). The proof of the two theorems above are completely different in nature. The first theorem uses techniques which have been developped in a series of papers by the second author and by Manetti ([Cat84], [Cat87a], [Cat86a], [Man94], [Man97]). They use essentially the local deformation theory a` la Kuranishi, normal degenerations of smooth surfaces and a study of quotient singularities of rational double points and of their smoothings. One very elementary, but extremely important ingredient in the proof of the first theorem is the notion of natural deformations of a bidouble cover (introduced in [Cat84], p.494), which are parametrized by a quadruple of polynomials (f, g, φ, ψ) and given by the two equations z 2 = f (x, y) + wφ(x, y), w2 = g(x, y) + zψ(x, y),

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where f and g are polynomials of respective bidegrees (2a, 2b), (2c, 2b) as before and φ and ψ have respective bidegrees (2a − c, b), (2c − a, b). Under suitable hypotheses (rigid base, branch curve of sufficiently high degree), these natural deformations indeed give all small deformations. Moreover, since a ≥ 2c + 1, it follows that ψ ≡ 0, therefore every small deformation preserves the structure of an iterated double cover. The final point is to show that this structure also passes in a suitable way to the limit, so that we do not only have an open, but also a closed subset of the moduli space. We will now comment on the newer part, the proof of theorem 34. The key ideas are here the following: 1) Both surfaces S and S  admit a holomorphic map to P1C given by the composition of the bidouble cover with the projection to the first coordinate x, and a small perturbation of this map realizes them as symplectic Lefschetz fibrations (cf. [Don99], [GS99]). 2) The respective fibrations are, by the accurate choice of the bidegrees of the curves, and especially because of the fact that the second degree is equal to 2b in both cases (thereby allowing, locally on the base, to ’rotate’ one branch curve to the other) fiber sums of the same pair of Lefschetz fibrations over the complex disc (the global effect of this local rotation is that the first curve {f = 0} loses a bidegree (2, 0), while the second {g = 0} gains a bidegree (2, 0)) 3) Once the first fiber sum is presented as composition of two factorizations and the second as the same composition of factorizations, just conjugated by the ’rotation’ Ψ , in order to prove that the two fiber sums are equivalent, it suffices, (thanks to Auroux’s lemma, [Aur02]) to show that the diffeomorphism Ψ is in the subgroup of the mapping class group generated by the Dehn twists which appear in the first factorization. 4)Figure 5 below shows the fibre C of the fibration in the case 2b = 6: it is a bidouble cover of P1 , which we can assume to be given by the equations z 2 = F (y), w2 = F (−y), where the roots of F are the integers 1, . . . , 2b. Moreover, one sees that the monodromy of the fibration at the boundary of the disc is trivial, and the map Ψ is the diffeomorphism of order 2 given by y → −y, z → w, w → z, which in our figure is given as a rotation of 180 degrees around an axis inclined in direction north-east. The figure shows a dihedral symmetry, where the automorphism of order 4 is given by y → −y, z → −w, w → z. Moreover, between the Dehn twists which appear in the factorization there are those which correspond to the inverse images of the segments between two consecutive integers (cf. figure 5). These circles can be organized on the curve C in six chains (not disjoint) and finally we have reduced ourselves to show that the isotopy class of Ψ is the same as the product of the six Coxeter elements associated to such chains.

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Fig. 5. The curve C with a dihedral symmetry

We recall that the Coxeter elements associated to a chain are products of the type ∆ = (Tα1 )(Tα2 Tα1 ) . . . (Tαn Tαn−1 . . . Tα1 ) of Dehn twists associated to the curves of the chain. In order to finally prove that such product (let us call it Ψ  ) of Coxeter elements and Ψ are isotopic, one observes that if one removes the above cited chains of circles from the curve C, one obtains 4 connected components which are diffeomorphic to circles. By a result of Epstein it is then sufficient to verify that Ψ and Ψ  send each such curve to a pair of isotopic curves: this last step

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needs a list of lengthy (though easy) verifications, for which it is necessary to have explicit drawings. For details we refer to the original paper [CW04]. It was observed by the second author (cf. [Cat02]) that a surface of general type has a canonical symplectic structure. In fact, he proves the following Theorem 35. A minimal surface of general type S has a canonical symplectic structure, unique up to symplectomorphism, such that the class of the symplectic form is the class of the canonical sheaf OS (KS ). We give the proof under the assumption that KS be ample, since the basic idea becomes clear in this simpler case. For the general case we refer to the original article. Proof. Let m be such that mKS is very ample (by Bombieri’s result it suffices any m ≥ 4), i.e., the pluricanonical map φm := φmKS : S → PPm −1 , where Pm := h0 (S, OS (mKS )) is the m-th plurigenus of S, is an embedding. We define ωm on S as follows: ωm :=

1 ∗ 1 φ ( ∂∂log|z|2 ), m m 2πi

i.e., we divide by m the pull back of the Fubini-Study form, whence ωm yields a symplectic form on S. It remains to show that the symplectomorphism class of (S, ωm ) is indeed independent of m. For this suppose that also φn gives an embedding of S: then the same holds for mn, whence it is sufficient to see that (S, ωm ) and (S, ωnm ) are symplectomorphic. Observe that the pull back of the Fubini-Study form under the n-th Veronese map vn is n times the Fubini-Study form and vn ◦ φm is  a linear projection of φmn . Then by Moser’s theorem we are done.  Therefore it seems natural to ask the following Question 8. Are the diffeomorphic (a, b, c)-surfaces of theorem 34, endowed with their canonical symplectic structure, indeed symplectomorphic? Remark 21. 1) In [Cat02] the second author shows that Manetti’s examples are indeed symplectomorphic. 2) A possible way of showing that the answer to the question above is yes (and therefore exhibiting symplectomorphic simply connected surfaces which are not def-equivalent) goes through the analysis of the braid monodromy of the branch curve of the “perturbed” (corresponding to the Lefschetz fibration) quadruple covering, and one would like to show that the involution ι on P1 , ι(y) = −y can be written as the product of braids which show up in the factorization. Anyhow, this approach turned out to be more difficult than the corresponding analysis which has been made in the mapping class group, because

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the braid monodromy contains very many ’tangency’ factors which do not come from local contributions to the regeneration of the branch curve from the union of the curves f = 0, g = 0 counted twice. In the rest of the paragraph we will discuss another equivalence relation the so-called Q.E.D. equivalence relation, which was introduced by the second author (cf. [Cat05]), and which seems worthwhile to examine for surfaces of general type. Observe that for any number g ≥ 2 there is a smooth curve of genus g, which is an ´etale covering of a curve of genus 2. Therefore all the smooth curves of Kodaira dimension 1 are equivalent by the equivalence relation generated by deformation and by ´etale maps. (Obviously, also all curves of Kodaira dimension 0, resp. −∞ are equivalent by this equivalence relation). Remark 22. More remarkable is what happens for algebraic surfaces of Kodaira dimension 0. Enriques surfaces admit an ´etale double cover which is a K3-surface, hyperelliptic surfaces have an ´etale cover which is a torus (in fact, the product of two elliptic curves). Therefore, in order to have some analogue to the curve case, one should “link” K3-surfaces and tori by ´etale maps and deformations. Obviously, this is not possible, since tori are K(π, 1)’s and K3-surfaces are simply connected. But the solution is simple: divide the torus by the involution x → −x, and obtain the (singular!) Kummer surface. A smoothing of this Kummer surface gives a K3-surface. The price we have to pay for going from curves to surfaces is that we have to allow morphisms which are not necessarily ´etale, but only ´etale in codimension 1. Moreover, we have to allow mild singularities: ordinary double points in this case, canonical singularities in a more general setting. This remark justifies the following Definition 24. We consider for complete algebraic varieties with canonical singularities defined over a fixed algebraically closed field the equivalence relation generated by 1) birational maps; 2) flat proper algebraic deformations π : X → B, with base B a connected algebraic variety, and all fibres having canonical singularities; 3) quasi ´etale morphisms f : X → Y , i.e., surjective morphisms which are ´etale in codimension 1 on X (i.e., there is Z ⊂ X of codimension ≥ 2 such that f |(X − Z) is ´etale). We will call this equivalence relation a.q.e.d.-relation, which means algebraic quasi´etale-deformation relation and it will be denoted by X ∼a.q.e.d. X  . It is rather clear that a completely analogous equivalence relation (called then C-q.e.d.-relation) can be defined also in the setting of compact complex spaces with canonical singularities. We refer to [Cat05] for more details.

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Remark 23. Trivially, the dimension of a variety is a q.e.d. invariant. By Siu’s recent result (cf. [Siu02]) also the Kodaira dimension is an invariant of a.q.e.d.-equivalence, if we restrict ourselves to projective varieties with canonical singularities (defined over the complex numbers). For surfaces of special type, i.e., of Kodaira dimension ≤ 1 the situation is as for curves. Theorem 36. Let S and S  be smooth complex algebraic surfaces of the same Kodaira dimension ≤ 1. Then S and S  are a.q.e.d.-equivalent. The ingredients of the proof of the above theorem are the Enriques classification of surfaces, the detailed knowledge of the deformation types of elliptic surfaces and the orbifold fundamental group of a fibration. The following question seems natural. Question 9. Is it possible to determine the q.e.d. equivalence classes inside the class of varieties with fixed dimension n, and with Kodaira dimension k? For curves and special algebraic surfaces over C there is only one a.q.e.d. class, but as shown in an appendix to [Cat05] by Fritz Grunewald, already for surfaces of general type the situation is completely different. Theorem 37. There are infinitely many q.e.d.-equivalence classes of algebraic surfaces of general type. The above surfaces are constructed from quaternion algebras (along general lines suggested by Shimura and explicitly described by Kuga and Shavel, cf. [Sha78]) are rigid, but the q.e.d.-equivalence class contains countably many distinct birational classes. The main points of the construction are the following: 1) the surfaces are quotients S = H×H/Γ of the two dimensional polydisk H × H via the free action of a discrete group Γ constructed from a quaternion algebra A over a totally real quadratic field k 2) since S is rigid, it suffices to show that if Γ  is commensurable with Γ , then also Γ  acts freely on H × H . 3) One sees by general theorems that Γ  has as Q-linear span the same quaternion algebra A as Γ . 4) If Γ  does not act freely, taking the tangent representation at a fixed point, we see by 3) that A contains a cyclotomic extension whose degree divides 4. 5) Using Hasse’s theorem, one chooses A such that the set of primes where it ramifies contains, one for each possible intermediate field K  between the quadratic field k of A and one of the finitely many possible cyclotomic extensions above, a prime P such that K  ⊗ kP is not an integral domain: this however contradicts 4) hence shows the desired assertion.

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It remains open whether there are for instance varieties which are isolated in their q.e.d.-equivalence class (up to birational equivalence, of course). An interesting question is to determine, for surfaces of general type, the non standard a.q.e.d. classes (standard means: equivalent to a product of curves)

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[Par05] [Per81] [Pir02] [Pol05] [Pol06]

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[Sch05]

[Ser64] [Sev32]

[Sha78] [Siu02]

[Tod81]

[Waj83] [Waj99] [Wit94] [Xia85a] [Xia85b] [Xia87] [Xia90] [Yau77] [Yau78]

[Zuc97] [Zuc03]

Schreyer, F.-O.: An experimental approach to numerical Godeaux surfaces. in Report n. 7/2005, Workshop “Komplexe Algebraische Geometrie”, Feb. 13–19, 2005. 434–436. M.F.O., Oberwolfach (2005) Serre, P.: Exemples de vari´et´es projectives conjugu´ees non hom´ eomorphes. C. R. Acad. Sci. Paris 258, 4194–4196 (1964) Severi, F.: La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica. Comment. Math. Helv. 4, 268–326 (1932) Shavel, I.H.: A class of algebraic surfaces of general type constructed from quaternion algebras. Pacific J. Math. 76, no. 1, 221–245 (1978) Siu, Y.T.: Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Bauer, I. (ed) et al. Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday, 223–277. Springer, Berlin (2002) Todorov, A.N.: A construction of surfaces with pg = 1, q = 0 and 2 ≤ (K 2 ) ≤ 8. Counterexamples of the global Torelli theorem. Invent. Math. 63, no. 2, 287–304 (1981) Wajnryb, B.: A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45, no. 2–3, 157–174 (1983) Wajnryb, B.: An elementary approach to the mapping class group of a surface, Geom. Topol. 3, 405–466 (1999) Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1, no. 6, 769–796 (1994) Xiao, G.: Finitude de l’application bicanonique des surfaces de type g´en´eral. Bull. Soc. Math. France 113, no. 1, 23–51 (1985) Xiao, G.: Surfaces fibr´ees en courbes de genre deux. Lecture Notes in Math., 1137. Springer, Berlin (1985) Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276, no. 3, 449–466 (1987) Xiao, G.: Degree of the bicanonical map of a surface of general type. Amer. J. Math. 112, no. 5, 713–736 (1990) Yau, S.T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. U.S.A. 74, no. 5, 1798–1799 (1977) Yau, S. T.: On the Ricci curvature of a compact K¨ ahler manifold and the complex Monge-Amp`ere equation. I. Comm. Pure Appl. Math. 31, no. 3, 339–411 (1978) Zucconi, F.: A note on a theorem of Horikawa. Rev. Mat. Univ. Complut. Madrid 10, no. 2, 277–295 (1997) Zucconi, F.: Surfaces with pg = q = 2 and an irrational pencil. Canad. J. Math. 55, no. 3, 649–672 (2003)

Characteristic 0 and p Analogies, and some Motivic Cohomology Manuel Blickle, H´el`ene Esnault, Kay R¨ ulling
Universit¨ at Duisburg-Essen, Mathematik, 45117 Essen, Germany [email protected] [email protected] [email protected]

Introduction The purpose of this survey is to explain some recent results about analogies between characteristic 0 and characteristic p > 0 geometry, and to discuss an infinitesimal variant of motivic cohomology. Homotopy invariance for motivic cohomology implies, in particular, that the Picard group of the affine line over a field k is trivial, i.e. Pic(A1k ) = 0. However, if instead of considering the Picard group, we consider the group of isomorphism classes of pairs (L, t) consisting of a line bundle L on A1k , and ∼ = an isomorphism t : OSpec(OA1 /mn ) −→ LSpec(OA1 /mn ) where m is the maximal ideal at the origin, then one obtains the group of 1 units in k[[t]]/(tn ), that ×
is {isom. classes (L, t)} ∼ = 1 + tk[[t]]/(1 + tn k[[t]]) . This group is indeed a ring, namely the ring of big Witt vectors Wn−1 (k) of length n − 1 over k. On the other hand, the theory of additive higher Chow groups, or Chow groups with modulus condition 2, developed in [10] and [11], allows to realize i of k as a group of 0-cycles (Theorems 2.1 and absolute differential forms Ωk/Z 2.3). In Sect. 2 these groups of 0-cycles and their higher modulus n generalization are presented. Theorem 2.7 – the main result of Sect. 2 – asserts that these groups of 0-cycles with higher modulus n compute big Witt differential forms Wn−1 Ωki . Cutting out the p-isotypical component in characteristic p > 0 by suitable correspondences, and extending Theorem 2.7 to smooth local rings over a perfect field would then describe crystalline cohomology of a smooth proper variety over a perfect field of characteristic p > 0 as the hypercohomology of a complex of sheaves of 0-cycles. If X is a smooth complex variety, the Riemann–Hilbert correspondence establishes an equivalence of categories between holonomic DX -modules (coherent) and constructible sheaves (topological). Emerton and Kisin developed


This work has been partly supported by the DFG Schwerpunkt “Komplexe Mannigfaltigkeiten” und by the DFG Leibniz Program

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in [22] for characteristic p > 0 a correspondence with properties analogous to the complex theory. The “topological like” side consists of sheaves of finite dimensional Fp -vector spaces which are locally constant in the ´etale topology, the “D-module like” side consists of locally finitely generated unit OX,F modules (see Theorem 3.1), which are modules on which the Frobenius acts in a certain way (unit). Even if the unit condition is quite restrictive – as will be explained shortly, it essentially corresponds to “slope zero” – the theory has a vaste range of applications and many objects naturally carry such a structure. For example OX itself is such a unit OX,F module. In Sect. 3 we investigate singularities of Y (at a point x ∈ Y ⊆ X, where X is smooth) from this viewpoint and obtain striking analogies for local invariants over the complex numbers and in characteristic p > 0. Postponing the definition of these invariants, which were introduced by Lyubeznik in [43], the main result (Theorem 3.1) of Sect. 3 asserts that these invariants can be expressed in terms of ´etale cohomology Hxi (Y´et , Fp ) in characteristic p > 0 and respectively in terms of singular cohomology Hxi (Yan , C) over the complex numbers, by virtually the same expression. This extends earlier results in the isolated singular analytic case of [43] and [33]. This extension is made possible by the similarity between the two correspondences which allows to essentially treat both settings (char. p > 0 and char. 0) formally as one. In Sect. 1, we review recent results on congruences modulo q-powers for the number of Fq -rational points of algebraic varieties. They are all based on Deligne’s philosophy which predicts a deep analogy between the level of congruences for varieties defined over Fq and the Hodge level for varieties over the complex numbers. If a variety has Hodge level ≥ κ over the complex numbers, one expects that “over” Fq it will have the same number of rational points as Pn modulo q κ . Of course, the challenge is to make precise “over” as it can’t be the same variety. Divisibility of eigenvalues of the geometric Frobenius acting on -adic cohomology is one method to show the existence of congruences, slope computation in crystalline cohomology, or, in the singular case, in Berthelot’s rigid cohomology is another one. Of course, ideally one would wish to prove a motivic statement which would imply all those results at once. But it is often beyond reach. The main results are Theorem 1.5, which in particular gives a positive answer to the Lang-Manin conjecture asserting that Fano varieties over a finite field have a rational point, Theorem 1.7, asserting that the mod p reduction of a regular model of a smooth projective variety defined over a local field, the -adic cohomology of which is supported in codimension 1, carries one rational point modulo q, Theorem 1.11, asserting that two theta divisors on an abelian variety defined over a finite field carry the same number of points modulo q. Theorem 1.11 answers positively the finite consequence of a conjecture of Serre, and appears as a consequence of the slope Theorem 1.10 asserting that the slope < 1 piece of rigid cohomology is computed by Witt vector cohomology. Theorem 1.5 can be proven using either -adic cohomology or crystalline cohomology. Indeed, the geometric result behind is that Fano varieties are rationally connected and therefore

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their Chow group of 0-cycles satisfies base change. Theorem 1.7 relies on a generalization to local fields of Deligne’s integrality theorem over finite fields (Theorem 1.8).

1 Hodge Type Over the Complex Numbers and Congruences for the Number of Rational Points over a Finite Field 1.1 Deligne’s Integrality Theorem over a Finite Field Deligne developed the theory of weights for complex varieties via the weight filtration in his mixed Hodge theory [17], and, for varieties defined over finite fields, via the absolute values with respect to any complex embedding of the eigenvalues of the geometric Frobenius operator acting on -adic cohomology [18]. The philosophy of motives, as conceived by him and Grothendieck, predicts a closed analogy between those two concepts of weights. It has led to many very fundamental results, the first of which being Deligne’s proof of the Weil conjecture for Hodge level one complete intersections and for K3 surfaces (Invent. math. 15 (1972)). On the other hand, Deligne shows the fundamental integrality theorem Theorem 1.1 (Deligne [19], Corollaire 5.5.3). Let X be a scheme of finite type defined over Fq . Then the eigenvalues of the geometric Frobenius acting on compactly supported -adic cohomology Hci (X ×Fq Fq , Q ) are algebraic integers. If X is defined over the complex numbers, its Hodge filtration F j satisfies grjF Hci (X) = 0 for j < 0 (see Hodge III [17]). In Deligne’s motivic philosophy, the integrality theorem 1.1 is analogous to the Hodge filtration starting in degrees ≥ 0 in Hodge theory. This analogy has been less studied in the past than the weight analogy. The purpose of this section is to demonstrate on some examples how the analogy works between the F -filtration in Hodge theory and the integrality over a finite field for -adic cohomology. It has led in very recent years to a series of results on congruences for the number of points on varieties defined over finite fields. Theorem 1.1 implies Theorem 1.2. Let X be a scheme of finite type defined over Fq . Then the eigenvalues of the geometric Frobenius acting on -adic cohomology H i (X ×Fq Fq , Q ) are algebraic integers. Strictly speaking, Deligne shows this for X smooth via duality, but applying de Jong’s alteration theorem, one easily reduces the theorem to the smooth case as in [20], Corollary 0.3.

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1.2 Divisibility and Rational Points The Grothendieck-Lefschetz trace formula [34] |X(Fq )| =

∞ 

(−1)i Trace Frobenius|Hci (X ×Fq Fq , Q )

(1)

i=0

together with Theorem 1.1 implies that if the eigenvalues of the geometric ¯ but also, for i ≥ 1, they Frobenius are not only algebraic integers, i.e. ∈ Z, ¯ then one has are q-divisible as algebraic integers, i.e. ∈ q · Z, |X(Fq )| ≡ dim Hc0 (X ×Fq Fq ) mod q.

(2)

X proper and geometrically connected ¯ with eigenvalues of geom. Frob. ∈ q · Z =⇒ |X(Fq )| ≡ 1 mod q.

(3)

So we conclude

Purity, for which the smoothness condition is definitely necessary, together with Theorem 1.2 implies Theorem 1.3. Let X be a smooth scheme of finite type defined over Fq . Then the eigenvalues of the geometric Frobenius acting on -adic cohomology i (X ×Fq Fq , Q ) with supports along A are algebraic integers divisible HA× F Fq

q

by q κ , if A ⊂ X is a closed subscheme of codimension ≥ κ.

(See [26], Lemma 2.1 for the analogous proof in crystalline cohomology). If X is no longer smooth, the conclusion of Theorem 1.3 is no longer true (see [20], Remark 0.5). But it remains true for generic hyperplanes for example, as purity is then true (see [27], Theorem 2.1). In Deligne’s philosophy, Theorem 1.3 is analogous to the Hodge level statement Theorem 1.4. Let X be a smooth scheme of finite type defined over C. Then the graded pieces for the Hodge filtration F on de Rham cohomology with i (X) = 0 for a < κ, if A ⊂ X is a closed subscheme of support fulfill graF HA codimension ≥ κ. The proof, based on purity, is the same as for Theorem 1.3. 1.3 Chow Group of 0-Cycles, Coniveau and Divisibility for Smooth Proper Varieties So if X is smooth proper geometrically connected over Fq , Theorem 1.3 implies that (3) holds if H i (X ×Fq Fq , Q ) is supported in codimension ≥ 1 for i ≥ 1. An equivalent terminology is to say that H i (X ×Fq Fq , Q ) has coniveau ≥ 1. According to Bloch’s decomposition of the diagonal ([8], Appendix to Lecture 1), this is the case if the Chow group of 0-cycles over a field containing the field of rational functions Fq (X) is trivial. We conclude

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Theorem 1.5 ([26], Corollary 1.2, Corollary 1.3). Let X be a smooth, proper, geometrically connected variety over Fq . Assume that CH0 (X ×Fq Fq (X)) = Q. Then |X(Fq )| ≡ 1 mod q. In particular, Fano varieties have a rational point, as conjectured by Lang [42] and Manin [45]. Originally, Bloch decomposed the diagonal in order to recover Mumford’s theorem (and its variants) asserting in its simplest form that if X is a smooth projective surface over C, and if gr0F H 2 (X) = 0, then it can’t be true that CH0 (X)deg=0 ∼ = Alb(X). Furthermore, that Fano varieties, that is smooth projective geometrically irreducible varieties X so that the inverse of the du−1 is ample, are rationally connected over any algebraically alizing sheaf ωX closed field, is a consequence of Mori’s break and bend theory, and has been proven independently by Koll´ ar-Miyaoka-Mori and Campana (see [41] and references there). 1.4 Singular Varieties Defined by Equations: Ax-Katz’ Theorem and Divisibility If X is no longer smooth, not only one can’t apply Theorem 1.3 to get divisibility of eigenvalues, but also Bloch’s decomposition of the diagonal does not work. Indeed, the diagonal has only a homology class, so it does not act on cohomology, while Grothendieck-Lefschetz trace formula (1) allowing to count points needs cohomology. Yet, Deligne’s philosophy on the analogy between Hodge type over C and eigenvalue divisibility over Fq , is still at disposal. There is an instance where one can directly generalize the Leitfaden sketched for the proof of Theorem 1.5, by refining the motivic cohomology used. Rather than considering CH0 (X) of the singular variety, one embedds X ⊂ Pn in a projective space, and considers relative motivic cohomology π → Pn is an alteration so that H 2n (P × U, Y × U, n) ([12]), Sect. 1), where P − −1 Y = π (X) is a normal crossings divisor and U = Pn \ X. Then the graph of π has a cycle class in this relative motivic group and one shows ([12], Theorem 1.2) that if X is a hypersurface in Pn of degree ≤ n, then this class decomposes in a suitable sense, generalizing Bloch’s decomposability notion in H 2dim(X) (X, dim(X)) ∼ = CH0 (X) when X is smooth. This implies immediately Hodge type ≥ 1 over C as well as eigenvalue q-divisibility over Fq , and yields a motivic proof of Ax’ theorem asserting the congruence (3) for hypersurfaces of degree ≤ n in Pn over Fq . Ax’ theorem generalizes the mod p congruence due to Chevalley-Warning. Another instance for which one can make Deligne’s philosophy work concerns closed subsets X of Pn defined by equations of degrees d1 ≥ . . . ≥ dr , without any other assumption. Those equations could be chosen in a highly non-optimal way. For example, one could take r times the same equation. r ]}, and Ax and Katz in [1], [40], assign to X a level κ := max{0, [ n−d2 −...−d d1 n κ show that |X(Fq )| ≡ |P (Fq )| mod q . On the other hand, one can compute

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([24] and [25]) that over C, graF Hci (Pn \ X) = 0 for a < κ. Thus one expects eigenvalue divisibility. Indeed one has Theorem 1.6 ( [27], Theorem 1.1, [28], Theorem 2.1). Let X ⊂ Pn be a closed subset defined by equations of degrees d1 ≥ . . . ≥ dr . Then the eigenvalues of the geometric Frobenius acting on Hci ((Pn \ X) ×Fq Fq , Q ) are ¯ in q κ · Z. According to (1), Theorem 1.6 implies Ax-Katz’ theorem. However, the proof of Theorem 1.6 is not good as it uses Ax-Katz’ theorem. One would like to understand a motivic proof in the spirit of [12]. We are very far from it, as, even if X is a smooth hypersurface of low degree, we do not know how to compute its Chow groups of higher dimensional cycles. 1.5 Singular Varieties in Families Singular varieties which are degenerations of smooth ones have more structure. Fakhruddin and Rajan ([31], Corollary 1.2) generalize the motivic method of Theorem 1.5 in a relative situation: if f : X → S is a proper dominant morphism of smooth irreducible varieties over a finite field k with CH0 (X ×S k(X)) = Q, then for any s ∈ S(k), one has |f −1 (s)| ≡ 1 mod |k|. Similarly on the Hodge side one proves ( [29], Theorem 1.1) that if f : X → S is a proper morphism with S a smooth connected curve and X smooth, then if gr0F H i (f −1 (s0 )) = 0 for some s0 in the smooth locus of f and all i ≥ 1, then gr0F H i (f −1 (s)) = 0 for all s and all i ≥ 1. Those two statements, the first one in equal characteristic p > 0 with its strong motivic assumption, the second one in equal characteristic 0 with its minimal Hodge type assumption, suggest, using Deligne’s philosophy, that the mod p reduction of a smooth projective variety in characteristic zero with gr0F H i (X) = 0 for all i ≥ 1 has eigenvalue q-divisibility, and therefore by (1), its number of rational points is congruent to one modulo q. One shows Theorem 1.7 ([30], Theorem 1.1, Section 4, Proof of Theorem 1.1). Let X be a smooth projective variety over a local field K with finite residue ¯ Q ) lives in codimension ≥ 1 for all i ≥ 1. field Fq . Assume H i (X ×K K, Then the eigenvalues of the geometric Frobenius acting on H i (Y ×Fq Fq , Q ) for i ≥ 1, where Y is the mod p reduction of a projective regular model, are ¯ In particular, |Y (Fq )| ≡ 1 mod q. lying in q · Z. If the local field K has equal characteristic p > 0, one expects Theorem 1.7 to be optimal. However, if the local field K has unequal characteristic, one would wish, following Deligne’s philosophy, to replace the assumption on the coniveau of -adic cohomology by gr0F H i (X) = 0 for all i ≥ 1. Due to the comparison of ´etale and de Rham cohomology, and due to the Hodge conjecture in codimension 1, those two assumptions are equivalent for surfaces. In general, Grothendieck’s generalized Hodge conjecture in codimension 1 predicts that if gr0F H i (X) = 0 then H i (X) lives in codimension ≥ 1. Thus

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those two conditions are expected to be equivalent in general. So in unequal characteristic, Theorem 1.7 is optimal for surfaces, but in higher dimension, in absence of a proof of the generalized Hodge conjecture in codimension 1, one would wish to have another proof. On the other hand, a generalization of Bloch’s decomposition of the diagonal implies that the motivic assumption CH0 (X0 ⊗K0 Ω) = Q, where K0 ⊂ K is a subfield of finite type over the prime field over which X is defined, i.e. ¯ in unequal X = X0 ×K0 K, and Ω contains K0 (X0 ), so for example, Ω = K characteristic, implies the coniveau assumption of the theorem. So the motivic assumption implies the following direct corollary of Theorem 1.7: Let X be a smooth projective variety over a local field K with finite residue field Fq . Assume CH0 (X0 ×K0 Ω) = Q. Then the eigenvalues of the geometric Frobenius acting on H i (Y ×Fq Fq , Q ) for i ≥ 1, where Y is the mod p reduction of ¯ In particular, |Y (Fq )| ≡ 1 mod a projective regular model, are lying in q · Z. q ([30], Corollary 1.2). It is to be noted that for surfaces in characteristic zero, we are very far from knowing a positive answer to Bloch’s conjecture, asserting that gr0F H i (X) = 0 for i ≥ 1 implies the motivic condition. Thus the range of applicability of Theorem 1.7 is much larger than the one of its corollary. The proof of Theorem 1.7 relies on the specialization map, on Gabber’s purity theorem [32], Theorem 2.1.1, de Jong’s alteration [16], Theorem 6.5, and on the direct generalization of Deligne’s integrality theorem 1.1 to the local field case Theorem 1.8 ([20], Theorem 0.2, Corollary 0.3). Let X be a scheme of finite type defined over a local field K with finite residue field. Then the eigen¯ value of a lifting of the geometric Frobenius in the Galois group Gal(K/K) of i i ¯ ¯ the local field acting on Hc (X ×K K) and H (X ×K K) are algebraic integers for all i. For the proof of Theorem 1.7, one needs a form of the integrality theorem over local fields which is weaker than the one stated in Theorem 1.8, and which can be proven directly using alterations. Theorem 1.8 itself is a corollary of a more general integrality theorem for -adic sheaves. As a corollary, one has, as in Theorem 1.3 over finite field, the divisibility statement for smooth varieties Theorem 1.9 ([20], Corollary 0.4). Let X be a smooth scheme of finite type defined over a local field K with finite residue field Fq . Then the eigenvalues ¯ of a lifting of the geometric Frobenius in the Galois group Gal(K/K) of the i ¯ local field acting on HA×K K¯ (X ×K K, Q ) are algebraic integers divisible by q κ , if A ⊂ X is a closed subscheme of codimension ≥ κ. 1.6 Witt Vector Cohomology Intuitively, if instead of -adic cohomology, we consider crystalline or rigid cohomology, we expect a more direct link between Hodge type and slopes,

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and consequently congruences for the number of points if the ground field is finite. The theorem of Bloch and Illusie asserts that if X is smooth proper over a perfect field k of characteristic p > 0, W = W (k), K = Frac(W ), then there is a functorial isomorphism ([7], III, 3.5 and [38], II, 3.5) ∼ =

H i (X/K)<1 − → H i (X, W OX )K .

(4)

Here H i (X/K)<1 denotes the maximal subspace of crystalline cohomology on which Frobenius acts with slopes < 1, the subscript K denotes tensorisation with K, and the right hand side is Witt vector cohomology, as considered by Serre in [49]. On the other hand, over C, one has a functorial surjective map ([24], Proposition 1.2, [29], Proof of Theorem 1.1) surj

i H i (X, OX ) −−→ grF 0 H (X).

(5)

This gives an upper bound on the Hodge type in the sense that if the left hand side, which is usually easy to compute as this is coherent cohomology, dies, then so does the right hand side, which is a topological invariant. A weak version of (5) allows one for example to remark that if Θ ⊂ A is a theta divisor on an abelian variety A of dimension g over C (i.e. effective, ample, with h0 (A, OA (Θ)) = 1), then one has an isomorphism iso

g g −→ grF grF 0 Hc (A \ Θ) − 0 H (A).

(6)

There is a generalization of (4) and (5) Theorem 1.10 ([2], Theorem 1.1). Let X be a proper scheme over a perfect field k of characteristic p > 0, then (4) holds true, where the left hand side is replaced by Berthelot’s rigid cohomology. So Witt vector cohomology computes the slope < 1 piece of rigid cohomology. Then the finite field version of (6) asserts Theorem 1.11 ([2], Theorem 1.4). Let A be an abelian variety defined over Fq , and let Θ, Θ be two theta divisors. Then |Θ(Fq )| ≡ |Θ (Fq )| mod q. This answers positively the finite field consequence of a conjecture of Serre asserting that the difference of the motives of Θ and Θ should be divisible by the Lefschetz motive over any field. Remarks 1.12. As concluding remarks to this section, let us first observe that the motivic philosophy discussed here leads to various questions anchored directly in geometry. As an example, as already mentioned in [26], Sect. 3, Gorenstein Fano varieties X in characteristic 0 fulfill gr0F H i (X) = 0 for i ≥ 1. This suggests the existence of a good definition of rational singularities in characteristic p > 0 so that a Gorenstein Fano variety over Fq would have one rational point modulo q. This would also require a generalization of Mori’s break and bend method to varieties with this type of mild singularities.

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Next, let us remark that we did not discuss in this survey higher congruences, that is congruences modulo q κ , κ ≥ 2. Indeed, Theorem 1.3 implies that if X is projective smooth over Fq , then if H i (X ×Fq Fq , Q ) is supported in codimension ≥ κ up to the class of the j-th self-product of the polarization if i = 2j, then (1) yields |X(Fq )| ≡ |Pn (Fq )| mod q κ . However, following the Leitfaden explained in subsection 1.3, the coniveau κ condition is implied by triviality of CHi (X ×Fq Fq (X)), for all i < κ, and this is a condition we can basically never check. Finally, by Theorem 1.10, Witt vector cohomology computes the slope < 1 part of rigid cohomology. In particular, it is a topological cohomology theory. We do not know the relation between rigid and -adic cohomology if X is proper but not smooth. On the other hand, the slope = 0 part of crystalline cohomology, when X is smooth, is easier to understand as it is described by coherent cohomology. This viewpoint is developed in Sect. 3.

2 Additive Higher Chow Groups with Higher Modulus of Type (n,n) over a Field 2.1 Additive Higher Chow Groups Let X be a smooth variety over a field k. In [9] Bloch develops a theory of higher Chow groups, which are isomorphic to the motivic cohomology groups 2p−n CHp (X, n) ∼ = HM (X, Z(p)), p, n ≥ 0 (see [51]), by using the scheme ∆n = n 1 Spec k[t0 , . . . , tn ]/ ( i=0 ti − 1) or, in the cubical  definition,  the scheme (P \ n ti = 1 by ti = λ yields the {1}) . Replacing in the simplicial definition same groups as long as λ ∈ k× . The degenerate case λ = 0 is investigated in [10]. One obtains a theory of additive higher Chow groups, SHp (X, n), p ≥ 0, n ≥ 1. In analogy to the theorem of Nesterenko-Suslin and Totaro (see [47], [50]) CHn (k, n) ∼ = KnM (k), it is shown Theorem 2.1 ( [10], Theorem 5.3). Let k be field with char k = 2, then there is an isomorphism of groups n−1 SHn (k, n) ∼ . = Ωk/Z

The proof is in the spirit of the proofs in [47], [50]. We will sketch the proof for the corresponding statement with SHn (k, n) replaced by a cubical version of the higher additive Chow groups. This cubical version is defined in [11], so far only for a field and on the level of 0-cycles. The definition is as follows. n with coordinates (x, y1 , . . . , yn ) Consider the k-scheme Xn = A1 × (P1 \ {1}) n and denote the union of all faces by Yn = i=1 (yi = 0, ∞). Now denote by Z0 (k, n − 1) the free abelian group generated on all closed points of A1 \ {0} × (P1 \ {0, 1, ∞})n−1 . Let Z1 (k, n; 2) be the free abelian group generated on all irreducible curves C ⊂ Xn \ Yn satisfying the following properties

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1. (Good position) ∂ij [C] = (yi = j).[C] ∈ Z0 (k, n − 1), for i = 1, . . . , n, j = 0, ∞.  → P1 × (P1 )n be the normalization of the 2. (Modulus 2 condition) Let ν : C  compactification of C, then in Z0 (C) 2 div(ν ∗ x) ≤

n 

div(ν ∗ yi − 1).

(7)

i=1

Because of (i) one has a complex ∂=

n 

(−1)i (∂i0 − ∂i∞ ) : Z1 (k, n; 2) → Z0 (k, n − 1) → 0.

i=1

Definition 2.2 ([11], Definition 6.2). The additive higher Chow groups of type (n, n) and modulus 2 of a field k are given by the homology of the above complex, i.e. Z0 (k, n − 1) . THn (k, n; 2) = ∂Z1 (k, n; 2) It is shown in [11, Theorem 6.4] (cf. Theorem 2.1 ) Theorem 2.3. Let k be a field with char k = 2, 3, then the map

n−1 THn (k, n; 2) −→ Ωk/Z ,

 [P ] → Trk(P )/k

1 dy1 (P ) dyn−1 (P ) ··· x(P ) y1 (P ) yn−1 (P )

(8) 

is an isomorphism of groups. Furthermore, the inclusion ι : (P1 \ {1})n−1 → Xn−1 , (y1 , . . . , yn−1 ) → (1, y1 , . . . , yn−1 ) induces a commutative diagram M Kn−1 (k) dlog



n−1 Ωk/Z



/ CHn−1 (k, n − 1) ι∗

 / THn (k, n; 2)



defined on elements by {a1 , . . . , an−1 }  _ da1 a1

 ∧ ... ∧

dan−1 an−1



/ (a1 , . . . , an−1 ) _  / (1, a1 , . . . , an−1 ).

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The idea of the proof is the following. One first shows, that the map (8) is well defined, using the reciprocity law for rational differential forms on a nonsingular projective curve. Here one explicitly uses the modulus condition (7). Then one constructs an inverse map with the help of a representation of the absolute differential forms of k by generators and relations. Showing this map to be well defined is equivalent to finding 1-cycles in Z1 (k, n; 2), whose boundary yield the relations. This map is easily seen to be injective and the surjectivity follows from the fact, that the trace on the absolute differentials corresponds to the pushforward on the additive Chow groups. 2.2 Higher Modulus The cubical definition allows one to generalize the definition of the additive higher Chow groups (of type (n,n)) to higher modulus, i.e. replace the 2 in equation (7) by an integer m ≥ 2. The resulting groups are denoted by THn (k, n; m). (Notice that one has THn (k, n; 0) = CHn (A1 \ {0}, n − 1) = 0 = THn (k, n; 1).) Up to now it is not clear how to formulate this higher modulus condition in the simplicial setup. The attempt to generalize Theorem 2.3 leads to the following considerations. Let Pic(A1 , m{0}) be the relative Picard group of A1 with modulus m{0}. Then there is a natural surjective map Pic(A1 , m{0}) −→ TH1 (k, 1; m).

(9)

For this one observes that, if two 0-cycles divf and divg in the left hand side are equal, then the curve C ⊂ A1 × P1 \ {1} defined by f y = g satisfies the modulus condition ¯ m(x = 0).C¯ ≤ (y − 1).C, with C¯ ⊂ P1 × P1 the closure of C. Hence divf = divg also in the right hand side. By Theorem 2.3 this is an isomorphism for m = 2 and one can show that it is an isomorphism for all m ≥ 2. On the other hand we may identify Pic(A1 , m{0}) as a group with the additive group of the ring of big Witt vectors of length m − 1 of k via Pic(A , m{0}) ∼ = 1



1 + tk[t] 1 + tm+1 k[t]

×

∼ = Wm−1 (k).

(10)

This together with Theorem 2.3 leads to the prediction, that THn (k, n; m) is isomorphic to the group of generalized degree n − 1 Witt differential forms of length m − 1. These groups form the generalized de Rham-Witt complex of Hesselholt-Madsen generalizing the p-typical de Rham-Witt complex of BlochDeligne-Illusie (see [7], [38]). Before stating the generalization of Theorem 2.3 to the case of higher modulus, we describe the de Rham-Witt complex and some of his properties.

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Definition 2.4 (see [36], [48], cf. [38]). Let A be a ring. A Witt complex over A is a projective system of differential graded Z-algebras ((Em )m∈N , R : Em+1 → Em ) together with families of homomorphisms of graded rings (Fn : Enm+n−1 → Em )m,n∈N and homomorphisms of graded groups (Vn : Em → Enm+n−1 )m,n∈N satisfying the following relations, for all n, r ∈ N (i) (ii) (iii) (iv)

RFn = Fn Rn , Rn Vn = Vn R, F1 = V1 = id, Fn Fr = Fnr , Vn Vr = Vnr . Fn Vn = n, and if (n, r) = 1, then Fr Vn = Vn Fr on Erm+r−1 . Vn (Fn (x)y) = xVn (y), for x ∈ Enm+n−1 , y ∈ Em . Fn dVn = d, with d the differential on Em and Enm+n−1 respectively.

Furthermore, there is a homomorphism of projective systems of rings 0 (λ : Wm (A) → Em )m∈N ,

under which the Frobenius and the Verschiebung maps on the big Witt vectors correspond to Fn and Vn , n ∈ N on E 0 and satisfies (v) Fn dλ([a]) = λ([a]n−1 )dλ([a]), for a ∈ A. A morphism of Witt complexes over A is a morphism of projective systems of dga’s compatible with all the structures. This yields a category of Witt complexes over A. Theorem 2.5 (see [36], [48], cf. [38] ). The category of Witt complexes has an initial object, called the de Rham-Witt complex of A and denoted by · )m∈N . (Wm ΩA Hesselholt-Madsen prove this using the Freyd adjoint functor theorem. In case · A is a Z(p) -algebra, p = 2 a prime, Wm ΩA may be constructed following Illusie · as a quotient of ΩWm (A) . It follows · · = ΩA/Z W1 ΩA

0 Wm ΩA = Wm (A).

Remark 2.6. Let A be a Z(p) -algebra, p = 2 a prime, and denote by · )n∈N0 the p-typical de Rham-Witt complex of Bloch-Deligne-Illusie(Wn ΩA Hesselholt-Madsen, then one has  · ∼ · Wm ΩA Wn(j) ΩA , n(j) given by jpn(j) ≤ m < jpn(j)+1 . (11) = (j,p)=1

If X is a smooth variety over a perfect field, Bloch and Illusie also define · Wn ΩX , which is a complex of coherent sheaves on the scheme Wn (X). Its hypercohomology equals the crystalline cohomology. This is used for example to derive (4).

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The generalization of Theorem 2.3 to the case of higher modulus is given by the Theorem 2.7 ([48]). Let k be a field with char = 2. Then the projective sys tem ( n≥1 THn (k, n; m + 1) → n≥1 THn (k, n; m))m≥2 can be equipped with the structure of a Witt complex over k and the natural map  W∗ Ωk· → n≥0 THn+1 (k, n + 1; ∗ + 1) induced by the universality of the de Rham-Witt complex is an isomorphism. In particular Wm−1 Ωkn−1 ∼ = THn (k, n; m). In the following the Witt complex structure of  THn+1 (k, n + 1; m + 1) Tm := n≥0

is described. The multiplication of a graded commutative ring on Tm is induced by the exterior product of cycles followed by a pushforward, which is induced by the multiplication map A1 × A1 → A1 . The differential is induced by pushing forward via the diagonal A1 → A1 × P1 and then restricting to A1 × P1 \ {1}. Fr (resp. Vr ) is induced by pushing forward (resp. pulling back) via 0 = TH1 (k, 1; m + 1) A1 → A1 , a → ar . And finally the map Wm (k) → Tm is given by the composition of (10) and (9). The relations (i)-(iii) and (v) in Definition 2.4 are already satisfied on the level of cycles. That Tm is a dga with (iv), follows from the result of Nesterenko-Suslin and Totaro, since one has a surjective map  CH n−1 (k(P0 ), n − 1) → THn (k, n; m) P0 ∈A1 \{0}

induced by the inclusions {P0 }×(P1 \{1})n−1 → A1 \{0}×(P1 \{1})n−1 . Thus T is a Witt complex and this gives the natural map from Theorem 2.7. In [48] a trace map for arbitrary field extensions on the de Rham-Witt groups is constructed as well as a residue symbol for closed points of a smooth projective curve on the rational Witt differentials of this curve and a reciprocity law is proven, generalizing the corresponding notions and statements on K¨ahler differentials (and the one obtained by Witt in [53]). Using this results one can generalize the proof of Theorem 2.3 to obtain, that the map THn (k, n; m) → Wm−1 Ωkn−1   1 d[y1 (P )] d[yn−1 (P )] ··· [P ] → Trk(P )/k [x(P )] [y1 (P )] [yn−1 (P )] gives the inverse map to the one obtained by the universality of the de Rhamuller lift. Witt complex. Here [−] : k(P ) → Wm−1 (k(P )) is the Teichm¨ Finally we explain how to describe the p-typical de Rham-Witt complex over a field k via the additive higher Chow groups. Denote by

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Manuel Blickle, H´el`ene Esnault, Kay R¨ ulling i ∞  tp f = exp( − i ) ∈ 1 + tZ(p) [[t]] p i=0

the inverse of the Artin-Hasse eponential and by i fm ∈ (1 + ⊕m i=1 t Z(p) )

the truncation of f . Write m for the 0-cycle [divfm ] ∈ TH1 (k, 1; m + 1). Then it follows from the description of the additive higher Chow groups as the de Rham-Witt complex and from (11), that one has Wr Ωkn−1 ∼ = THn (k, n; pr + 1) ∗ pr , where we denote by ∗ the multiplication in the additive higher Chow groups explained above.

3 Riemann–Hilbert Type Correspondences and Applications to Local Cohomology Invariants This section starts with a very rough introduction to some aspects of the Riemann–Hilbert correspondence (over C) and, shortly thereafter, to a positive characteristic analog, developed recently by Emerton and Kisin [22]. We start at a basic level – merely motivating the correspondence with the help of a fundamental example – but progress quickly to a nontrivial construction which is central to our applications: the intermediate extension. The aim is to show how these correspondences can be put to good use in order to study singularities. Concretely we obtain a new characterization of invariants arising from local cohomology in terms of ´etale cohomology. One central interesting aspect of our approach is that the treatment is, up to the use of the respective correspondence, independent of the characteristic. The result we will discuss is a description of Lyubeznik’s local cohomology invariants in any characteristic [43] which are originally defined for a quotient A = R/I of a n-dimensional regular local ring (R, m) to be def

a (HIn−i (R))) λa,i = e(Hm

where e( ) denotes the D–module multiplicity (see Section 3.2). In [43] these are shown to be independent of the representation of A as the quotient of a regular local ring. In [4] and [5] these invariants were described in many cases in terms of ´etale cohomology; we discuss these results here as an application of the aforementioned correspondences: Theorem 3.1. Let k-be separably closed. Let A = OY,x for Y a closed k– n−i subvariety of a smooth variety X. If for i = d the modules H[Y ] (OX ) are supported at the point x then

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1. For 2 ≤ a ≤ d one has λa,d (A) − δa,d = λ0,d−a+1 (A) 2.

and all other λa,i (A) vanish.  d−a+1 dimFp H{x} (Y´et , Fp ) λa,d (A) − δa,d = d−a+1 dimC H{x} (Yan , C)

if char k = p if k = C

where δa,d is the Kroneker delta function. The apparent analogy between the situation over C and over Fp suggested by this result is somewhat misleading. As we briefly discuss at the end of this section, ´etale cohomology with Fp –coefficients is only a very small part (the slope zero part) of, say, crystalline cohomology. For crystalline cohomology on the other hand there are comparison results to singular cohomology (via de Rham theory) hence Lyubeznik’s invariants really capture a different type of information in characteristic 0 than they do in characteristic p > 0. 3.1 Riemann–Hilbert and Emerton–Kisin Correspondence Let us first fix the following notation: Throughout this section X will be a smooth scheme over a field k of dimension n. Mostly k will be either C, the field of complex numbers, or Fq , the finite field with q = pe elements, as in Section 1. The Riemann–Hilbert Correspondence Now let k = C. In its simplest incarnation the Riemann–Hilbert correspondence asserts a one to one map between     local systems of locally free coherent OX –modules ↔ with regular singular integrable connection C-vectorspaces This correspondence grew out of Hilbert’s 21st problem, motivated by work of Riemann, to find, given a monodromy action at some points, a Fuchsian (regular singular) differential equation with the prescribed monodromy action at the singular points. The correspondence is given via de Rham theory: A connection is a k– 1 linear map ∇ : M −→ ΩX ⊗OX M that satisfies the Leibniz rule ∇(rm) = r∇(m) + dr ⊗ m. One can extend ∇ in an (up to signs) obvious way to get a sequence of maps (each denoted also by ∇) ∇

∇

∇

1 2 n M −−→ ΩX ⊗ M −−→ ΩX ⊗ M −−→ . . . −→ ΩX ⊗M

and the connection is called integral if this sequence is a complex (i.e. ∇2 = 0), called the de Rham complex dR(M) associated to the connection ∇. Now, given such integrable connection its horizontal sections

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M∇ = ker ∇ = H 0 (dR(M)) is a local system. The most trivial, but at the same time most important example, is M = d 1 OX with ∇ = d the universal differential OX −−→ ΩX . This is the OX –module ∂ , . . . , ∂x∂n ) · f = 0 where associated to the system of differential equations ( ∂x 1 x1 , . . . , xn are local coordinates at some point of X. The corresponding local system (the solutions to the differential equation) is of course the constant local system C. In fact more is true: By the Poincar´e Lemma, the (analytic) de Rham complex is a resolution of the constant sheaf C and hence we can rephrase this by saying that C is quasi-isomorphic to the de Rham complex. In view of Grothendieck’s philosophy the above correspondence is flawed since the categories on either side are not closed under any reasonable functors. For example, the pushforward of a local system is generally not a local system i as the inclusion of a point {x} −−→ A1 readily illustrates (i∗ C is a skyscraper sheaf). However it is a constructible sheaf (that is one that is locally constant on each piece of a suitable stratification of X), and in fact on constructible sheaves all the functors one would like to have are defined. On the other side of the correspondence, modules with integrable connection are replaced by modules over the ring of differential operators DX . The conditions one has to impose are holonomicity (which is the crucial finiteness condition) and a further condition (namely that M is regular singualar) which will not be considered here. Without giving the precise definition, a holonomic DX –module is one of minimal possible dimension, and the category they form enjoys a strong finiteness condition: Proposition 3.2 ([15]). The category of holonomic DX –modules is abelian and closed under extensions. Every holonomic DX module has finite length. The correspondence should again be given via de Rham theory. However, it quickly becomes clear that one has to pass to the derived category. We give the following simple example as a small indication that the passage to the derived category cannot be avoided. ∂ Example 3.3. Let X = A1 = Spec C[x] such that DX = k[x, ∂x ]. Let M = 1 H{0} (OA1 ) denote the cokernel of the following injection of DX –modules

C[x] −→ C[x, x−1 ]. Then, as a C-vectorspace M has a basis consisting of de Rham complex associated to M is given

1 xi

for i > 0. The

∇

M −−→ ΩA1 1 ⊗ M where the map sends a basis element x1i −→ dx ⊗ x−i i+1 . Hence one immediately sees that ∇ is injective and that its cokernel is generated by dx ⊗ x1 , which is the skyscraper sheaf supported at 0. Hence we have an quasi-isomorphism

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i∗ C{0}

1 dR(H{0} (OA1 ))[1].

75

(12)

Note the appearance of the shift [1] which is an indication that one cannot avoid to pass to the derived category. The ultimate generalization of the basic version above is the Riemann–Hilbert correspondence as proved by Mebkhout [46], Kashiwara [39], Beilinson and Bernstein [15]: Theorem 3.4. Let X be a smooth C–variety. On the level of bounded derived categories, there is an equivalence between     constructible sheaves of holonomic DX –modules ↔ which are regular singular C-vectorspaces The correspondence is given by sending a complex M• of DX –modules to dR(M• ) = RHom(OX , M). This equivalence preserves the six standard functors f ∗ and f∗ , their duals f! and f ! and also ⊗ and Hom. Via duality on X the de Rham functor is related to the functor Sol( ) = RHom( , OX ). This functor yields therefore an anti-equivalence, and it is this anti-equivalence which can be obtained in positive characteristic. Emerton–Kisin Correspondence In positive characteristic a naive approach via de Rham theory does not work due to the failure of the Poincar´e lemma. At least one should expect that the correspondence respects the relationship “Fp corresponds to OX ” in some way or another. But due to the fact that in characteristic p there are many functions with derivative zero (namely all pth powers), the de Rham theory is ill behaved. For the same reason, D–module theory in positive characteristic is also quite different from the one in characteristic zero. So what is the correct counterpart for constructible sheaves of Fp –vectorspaces? The solution arises from the Artin–Schreier sequence: x →xp

0 −→ Fp −→ OX −−−−−→ OX −→ 0. This is an exact sequence in the ´etale topology. Hence if one views the Riemann-Hilbert correspondence as a vast generalization of de Rham theory (in terms of the Poincar´e lemma), then the correspondence of Emerton–Kisin – to be outlined shortly – is an analogous generalization of Artin–Schreier theory. Hence, the basic objects one studies in this correspondence is not DX – modules but rather quasi-coherent OX –modules M which are equipped with an action of the Frobenius F . That is, we have an OX –linear map FM : M −→ F∗ M. By adjunction, such a map is the same as a map θM : F ∗ M −→ M. Such objects have been studied in various forms for a long time, see for example [35] or [21, Exp. XXII].

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Definition 3.5. An OX,F –module (M, θ) is a quasi-coherent OX –module M together with a OX –linear map θ : F ∗ M −→ M If θ is an isomorphism (M, θ) is called unit. If is called finitely generated if M is finitely generated when viewed as a module over the non-commutative ring OX,F . The following two are the essential examples of finitely generated unit OX,F – modules. Example 3.6. For simplicity assume that X = Spec R is affine. The natural identification F ∗ OX ∼ = OX gives OX the structure of a finitely generated unit OX,F –module. Let Rf be the localization of R at a single element f ∈ R. The natural map F ∗ Rf = R ⊗F Rf −→ Rf p

sending r ⊗ ab to r abp has a natural inverse given by sending ab −→ abp−1 ⊗ 1b . Hence Rf is naturally a unit module. In fact, Rf is even finitely generated as a unit R[F ]–module, generated by the element f1 , since F ( f1 ) = f1p . The main result of [44] about finitely genrated unit OX,F –modules, which makes them a suitable analog of holonomic D–modules, is Theorem 3.7. Let X be smooth. In the abelian category of (locally) finitely generated OX,F –modules, every object has finite length. Example 3.8. Considering a Cech resolution to compute coherent cohomology with support in some subscheme Z, the easy abelian category part of the preceding theorem and the preceding examples imply that HZi (OX ) is naturally a f.g. (finitely generated) unit OX,F –module. Now the correspondence that is proven by Emerton and Kisin in [22] can be summarized as follows: Theorem 3.9. For X smooth, there is an anti-equivalence on the level of derived categories     locally finitely generated constructible sheaves of ↔ OX,F –modules Fp –vector-spaces on X´et The correspondence is given by sending a (complex of ) OX,F –modules M• to the constructible sheaf Sol(M• ) = RHom(M, OX ) and is roughly dual to the naive approach of taking the fixed points of the Frobenius (Artin-Schreier sequence) alluded to above. The correspondence preserves certain functors, namely f ! , f∗ (and ⊗) on the right hand side correspond to f ∗ , f! on the left hand side.

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Intermediate Extensions From now on we treat the two situations – characteristic zero regular singular holonomic DX –modules with the Riemann–Hilbert correspondence on one hand and positive characteristic finitely generated unit OF,X –modules with the Emerton–Kisin correspondence on the other – formally as one. The crucial property they both share is the fact that all modules have finite length. This is key to the following construction. Due to the lack of duality in positive characteristic there is no analog of the functor j! available. However there is an adequate substitute, which still exists in our context [23, 6]. The substitute we have in mind is the intermediate extension j!∗ , usually constructed as the image of the “forget supports map” from j! −→ j∗ . This usual definition does not work since j! is not available. Nevertheless it turns out that all one needs to define j!∗ is the fact that the modules have finite length: Proposition 3.10 ([6], [23]). Let j : U ⊆ X be a locally closed immersion of smooth Fp –schemes (resp. C–schemes). Let M be a finitely generated unit OU,F –module. Then there is a unique submodule N of R0 j∗ M minimal with respect to the property that f ! N = M. This submodule N is called the intermediate extension and is denoted by j!∗ M. Proof. The key point is the fact that M has finite length which ensures the existence of minimal modules with the desired property (any decreasing chain is eventually constant). Let N1 and N2 be two modules with the desired property. Since f ! Ni = M their intersection cannot be zero. On the other hand the exact sequence 0 −→ N1 ∩ N2 −→ R0 j∗ M −→ R0 j∗ M/N1 ⊕ R0 j∗ M/N2 shows that their intersection is also a finitely generated unit OX,F –module (resp. r.s. holonomic DX –module) as it is the kernel of a map of such modules.  By minimality one has N1 = N2 showing uniqueness.  Under the correspondence the intermediate extensions behave well: In the situation as above we have Sol(j!∗ M) = Image(j! Sol(M) −→ R0 j∗ Sol(M)) such that they do in fact correspond to the intermediate extensions on the constructible side, where they can be identified as the image of the “forget supports” map. For the basic computations that follow we list some key properties both correspondences enjoy: 

1. Sol(OX ) =

Fp [n] C[n]

if k = Fp if k = C

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2. There are functors f ! and f∗ which behave under the correspondence in the expected way. 3. For Y ⊆ X a subvariety, local cohomology is defined in the categories and satisfies the triangle (a highbrow way of writing the long exact sequence for cohomology with supports) RΓ[Y ] M• −→ M• −→ Rj∗ j ∗ M• −−−→ +1

where j is the open inclusion of the complement of Y into X. 4. Again, let i : Y → X be the inclusion of a closed subset, then Sol ◦RΓ[Y ] ∼ = Ri! i∗ ◦ Sol . This follows via the preceding two items and the triangle Rj! j ! L −→ L −→ +1 Ri! i−1 L −−−→ for a (complex of) constructible sheaves L. 5. The preceding items allow us to compute (see also equation (12) on page 75) Sol(RΓY (OX )) = i! i∗ Fp [n] = i! Fp |Y [n] in positive characteristic and respectively i! C|Y [n] in characteristic zero. 3.2 Lyubeznik’s Local Cohomology invariants Let A = R/I for I an ideal in a regular (local) ring (R, m) of dimension n and containing a field k. The main results of [43, 37] state that the local a cohomology module Hm (HIn−i (R)) is injective and supported at m. Therefore a (HIn−i (R))) many copies of the injective it is a finite direct sum of e = e(Hm hull ER/m of the residue field of R. Lyubeznik shows in [43] that this number def

a λa,i (A) = e(Hm (HIn−i (R)))

does not depend on the auxiliary choice of R and I. At the same time, this number e(M) is the multiplicity of the holonomic DX –module M, respectively the finitely generated unit OX,F –module M. If A is a complete intersection, these invariants are essentially trivial (all are zero except λd,d = 1 where d = dim A). In general λa,i can only be nonzero in the range 0 ≤ a, i ≤ d. These invariants were first introduced by Lyubeznik in [43] and further studied by Walther in [52]. In [33] Garcia-L´ opez and Sabbah show Theorem 3.1 in the case of an isolated complex singularity. We now indicate briefly the proof of 3.1. As shown in [4] the condition imposed on the singularities in the Theorem 3.1 easily implies (via the spectral j a+j a H[Y sequence E2a,j = H[x] ] (OX ) ⇒ H[x] (OX )) part one. Part two is the point where the correspondences enter into the picture: The idea is of course to use that a a (HYn−d (OX ))) = dim Sol(H{x} (HYn−d (OX ))) e(Hm

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by the correspondence, and then to compute the right hand side. As it is written here the right hand side is however not computable. The trick now is to replace HIn−d (R) by j!∗ HYn−d (OX )|X−{x} , which is easily checked to not affect our computation (long exact sequence for Γ{x} ). Now, the assumption on the singularity that for i = d the module HYn−i (OX ) is supported at x can simply be rephrased as HYn−d (OX )|X−{x} = RΓY −{x} (OX−{x} )[n − d]. Using that Sol commutes with j!∗ and the fact that Sol(RΓY −{x} (OX−{x} )) = i! (Fp )Y −{x} [n] one obtains that a e(Hm (HYn−d (OX ))) = dim(H −a j!∗ i! (Fp )Y −{x} [d]).

To compute the right hand side is now a feasible task that yields the desired result (feasible due to the fact that j is just the inclusion of the complement of a point, which makes it possible to effectively understand and calculate j!∗ ; see [4, 5] for details). 3.3 Comparison via Crystalline Cohomology We close this section with some remarks regarding the behaviour of these invariants under reduction to positive characteristic. There are by now classical examples that show that local cohomology does not behave well under reduction so one would expect that the invariants λa,i do not behave well either. On a superficial level, glancing at Theorem 3.1, one might however suspect a complete analogy between positive and zero characteristic. However, this is not true. The difference stems from the difference between the cohomology theories which describe λa,i . In positive characteristic, this is ´etale cohomology with coefficients in Fp , in characteristic zero however it is (topological) cohomology. Under reduction mod p the former only constitutes a very small part of the latter, namely the part of slope zero. For simplicity consider the situation of Y ⊆ Pn a smooth projective variety. Now. the local ring A of the cone of this projective embedding has an isolated singularity at its vertex and one can study the invariants λa,i (A). Theorem 3.1 shows that λa,d (A) are described (excision) by H d−a (Y´et , Fp ) and H d−a (Yan , C) in positive and zero characteristic respectively. So far everything appears in complete analogy. But let us now consider reduction mod p and let Y be the special fiber of the smooth family Y −→ Spec W (k), where W (k) is the ring of Witt vectors over k and K denotes its field of fractions. Via the comparison results between topological cohomology, de Rham cohomology and Berthelot’s crystalline cohomology (see [3]) ∼ = d−a (YK ) ∼ H d−a (YC , C) −−→ HdR = H d−a (Y /K)

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it turns out that H d−a (Y´et , Fp ) is only a very small part of the crystalline cohomology H d−a (Y /K), namely the part on which the Frobenius acts with eigenvalue zero. This is an even smaller part then the part H d−a (Y /K)<1 which was of great importance in Sect. 1.6. From this point of view it is not surprising to find the characteristic zero side to capture much more information than the positive characteristic side. Even though the positive characteristic side is therefore lacking the topological insight, the information one obtains is still very valuable. For example it is used very effectively to study L–functions in [22] and also [13], which uses a similar correspondence for that purpose.

References 1. Ax, J.: Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255– 261. 2. Berthelot, P., Bloch, S., Esnault, H.: On Witt vector cohomology for singular varieties, preprint 2005, 42 pages. 3. Berthelot, P., Ogus, A.: Notes on crystalline cohomology. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. vi+243 pp. 4. Blickle, M., Bondu, R.: Local cohomology multiplicities in terms of local cohomology, 2004, to appear in Ann. Inst. Fourier. 5. Blickle, M. : Lyubeznik’s invariants for cohomologically isolated singularities, preprint 2005. 6. Blickle, M.: The intersection homology D–module in finite characteristic, Math. Ann. 328 (2004), 425–450. ´ 7. Bloch, S.: Algebraic K-theory and crystalline cohomology, Publ. Math. I.H.E.S. 47 (1977), 187–268. 8. Bloch, S.: Lectures on Algebraic cycles, Duke University Mathematics Series, IV, (1980). 9. Bloch, S.: Algebraic cycles and higher K-theory, Adv. in Math. 61, No.3 (1986), 267–304. 10. Bloch, S., Esnault, H.: An additive version of higher Chow groups, Ann. Sci. ´ Ecole Norm. Sup. 4 36,No. 3 (2003), 463–477. 11. Bloch, S., Esnault, H.: The additive dilogarithm, Doc. Math., J. DMV Extra Vol. (2003), 131–155. 12. Bloch, S., Esnault, H., Levine, M.: Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces, Am. J. of Mathematics, 127 no.1 (2005), 193–207. 13. Bkle, G., Pink, R.: Cohomological Theory of crystals over function fields, uncirculated preprint, 2005. 14. Borel, A. et al: Intersection cohomology, Progress in Mathematics, vol. 50, Birkh¨ auser Boston Inc., Boston, MA, 1984, Notes on the seminar held at the University of Bern, Bern, 1983, Swiss Seminars. 15. Borel, A., Grivel, P.-P, Kaup, B., Haefliger, A., Malgrange, B., and Ehlers, E.: Algebraic D-modules, Academic Press Inc., Boston, MA, 1987. ´ 16. de Jong, A. J.: Smoothness, semi-stability and alterations, Publ. Math. I.H.E.S. 83 (1996), 51–93.

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´ 17. Deligne, P.: Th´eorie de Hodge II, Th´eorie de Hodge III, Publ. Math. I.H.E.S. 40 (1971), 5–57, 44 (1974), 5–77. 18. Deligne, P.: La conjecture de Weil. I, La conjecture de Weil. II, Publ. Math. ´ I.H.E.S. 43 (1974), 273–307, 52, (1980) 137–252. 19. Deligne, P.: Th´eor`eme d’int´egralit´e, Appendix to Katz, N.: Le niveau de la cohomologie des intersections compl`etes, Expos´e XXI in SGA 7, Lect. Notes Math. vol. 340, 363–400, Berlin Heidelberg New York Springer 1973. 20. Deligne, P., Esnault, H.: Appendix to “Deligne’s integrality theorem in unequal characteristic and rational points over finite fields”, preprint 2004, 5 pages, appears in the Annals of Mathematics. 21. Deligne, P., Katz, N.: Seminar de Geometrie Algebrique, SGA 7.II, Lect. Notes Math. vol. 340 Spinger 1973, Berlin Heidelberg New York. 22. Emerton, M., Kisin, M.: Riemann–Hilbert correspondence for unit F -crystals, Astisque 293 (2004), vi+257 pp. 23. Emerton, M., Kisin, M.: An introduction to the Riemann–Hilbert correspondence for unit F -crystals., Geometric aspects of Dwork theory. Vol. I, II, 677– 700, Walter de Gruyter GmbH & Co. KG, Berlin, 2004. 24. Esnault, H.: Hodge type of subvarieties of Pn of small degrees, Math. Ann. 288 (1990), 549 - 551. 25. Esnault, H., Nori, M., Srinivas, V.: Hodge type of projective varieties of low degree. Math. Ann. 293 (1992), 1–6. 26. Esnault, H. : Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. math. 151 (2003), 187–191. 27. 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. 28. Esnault, H., Katz. N: Cohomological divisibility and point count divisibility, Compositio Mathematica 141 01 (2005), 93–100. 29. Esnault, H.: Appendix to “Congruences for rational points on varieties over finite fields” by N. Fakhruddin and C. S. Rajan, Math. Ann. 333 (2005), 811–814. 30. Esnault, H.: Deligne’s integrality theorem in unequal characteristic and rational points over finite fields, preprint 2004, 10 pages, appears in the Annals of Mathematics. 31. Fakhruddin, N., Rajan, C. S.: Congruences for rational points on varieties over finite fields, Math. Ann. 333 (2005), 797–809. 32. Fujiwara, K.: A Proof of the Absolute Purity Conjecture (after Gabber), in Algebraic Geometry 2000, Azumino, Advanced Studies in Pure Mathematics 36 (2002), Mathematical Society of Japan, 153–183. 33. Garcia-L` opez, R., Sabbah, C.: Topological computation of local cohomology multiplicities, Collect. Math. 49 (1998), no. 2–3, 317–324, Dedicated to the memory of Fernando Serrano. 34. Grothendieck, A.: Formule de Lefschetz et rationalit´e des fonctions L, S´eminaire Bourbaki 279, 17-i`eme ann´ee (1964/1965), 1–15. 35. Hartshorne, R., Speiser, R.: Local cohomological dimension in characteristic p, Annals of Mathematics 105 (1977), 45–79. 36. Hesselholt, L., Madsen, I.: On the K-theory of nilpotent endomorphisms, Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc., Providence, RI (2001), 127–140. 37. Huneke, C. L., Sharp, Ro.Y.: Bass numbers of local cohomology modules, Trans. Amer. Math. Soc. 339 (1993), no. 2, 765–779.

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´ 38. Illusie, L.: Complexe de de Rham-Witt et cohomologie cristalline, Ann. Ec. Norm. Sup. 4 12 (1979), 501–661. 39. Kashiwara, M.: Faisceaux constructibles et syst`emes holonˆ omes d’´equations aux d´eriv´ees partielles lin´eaires ` a points singuliers r´eguliers, S´eminaire Goulaouic´ Schwartz, 1979–1980, Ecole Polytech., Palaiseau, 1980, pp. Exp. No. 19, 7. 40. Katz, N.: On a theorem of Ax, Amer. J. Math. 93 (1971), 485–499. 41. Koll´ ar, J.: Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32 (1996), Springer-Verlag, Berlin, 1996. 42. Lang, S.: Cyclotomic points, very anti-canonical varieties, and quasi-algebraic closure, preprint 2000. 43. Lyubeznik, G.: Finiteness properties of local cohomology modules (an application of D-modules to commutative algebra), Invent. math. 113 (1993), no. 1, 41–55. 44. Lyubeznik, G.: F -modules: an application to local cohomology and D-modules in characteristic p > 0, Journal f¨ ur reine und angewandte Mathematik 491 (1997), 65–130. 45. Manin, Yu.: Notes on the arithmetic of Fano threefolds, Compos. math. 85 (1993), 37–55. 46. Mebkhout, Z.: Cohomologie locale des espaces analytiques complexes, Ph.D. thesis, Universit´e Paris VII, 1979. 47. Nesterenko, Y. P., Suslin, A.: Homology of the general linear group over a local ring, and Milnor’s K-theory, Math. USSR-Izv. 34, No.1 (1990), 121–145. 48. R¨ ulling, K.: The generalized de Rham-Witt complex over a field is a complex of zero-cycles, preprint 2005. 49. Serre, J.-P.: Sur la topologie des vari´et´es alg´ebriques en caract´eristique p, in Symposium internacional de topolog´ıa algebraica (International symposium on algebraic topology), 24–53, Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City, 1958. 50. Totaro, B.: Milnor K-theory is the simplest part of algebraic K-theory, K-theory 6, No.2 (1992), 177–189. 51. Voevodsky, V.: Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 7 (2002), 7, 351–355. 52. Walther, U.: On the Lyubeznik numbers of a local ring, Proc. Amer. Math. Soc. 129 (2001), No. 6, 1631–1634 (electronic). 53. Witt, E.: Zyklische K¨ orper und Algebren der Charakteristik p vom Grad pn . Struktur diskret bewerteter perfekter K¨ orper mit vollkommenem Restklassenk¨ orper der Charakteristik p, J. Reine Angew. Math. 176 (1936), 126–140.

Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves Lesya Bodnarchuk1 , Igor Burban2 , Yuriy Drozd3 , and Gert-Martin Greuel4 1 2 3 4

Technische Universit¨ at Kaiserslautern, Erwin-Schr¨ odinger-Straße, 67663 Kaiserslautern, Germany, [email protected] Institut f¨ ur Mathematik, Johannes Gutenberg-Universit¨ at Mainz, 55099 Mainz, [email protected] Kyiv Taras Shevchenko University, Department of Mechanics and Mathematics, Volodimirska 64, 01033 Kyiv, Ukraine, [email protected] Technische Universit¨ at Kaiserslautern, Erwin-Schr¨ odinger-Straße, 67663 Kaiserslautern, Germany, [email protected]

Summary. In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms , both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

1 Overview The aim of this paper is to give a survey on the classification of vector bundles and torsion free sheaves on singular projective curves of arithmetic genus one. We include new proofs of some classical results on coherent sheaves on smooth elliptic curves, which use the technique of derived categories and FourierMukai transforms and are simpler than the original ones. Some results about singular curves are new or at least presented in a new framework. This research project had several sources of motivation and inspiration. Our study of vector bundles on degenerations of elliptic curves was originally motivated by the McKay correspondence for minimally elliptic surface singularities [Kah89]. Here we use as the main technical tool methods from the representation theory of associative algebras, in particular, a key tool in our approach to classification problems is played by the technique of “representations of bunches of chains ” or “Gelfand problems” [Bon92]. At last, but not least we want to mention that our research was strongly influenced by ideas and methods coming from the homological mirror symmetry [Kon95, PZ98, FMW99].

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Many different questions concerning properties of the category of vector bundles and coherent sheaves on degenerations of elliptic curves are encoded in the following general set-up: Problem 1. Let E −→ T be a flat family of projective curves of arithmetic genus one such that the fiber Et is smooth for generic t and singular for t = 0. What happens with the derived category Db (CohEt ), when t → 0?

In order to start working on this question one has to consider the absolute case first, where the base T is Spec(k). In particular, one has to describe indecomposable vector bundles and indecomposable objects of the derived category of coherent sheaves on degenerations of elliptic curves and develop a technique to calculate homomorphism and extension spaces between indecomposable torsion free sheaves as well as various operations on them, like tensor products and dualizing. For the first time we face this sort of problems when dealing with the McKay correspondence for minimally elliptic singularities. Namely, let S = Spec(R) be the spectrum of a complete (or analytical) two-dimensional  −→ S its minimal resolution, and E minimally elliptic singularity, π : X the exceptional divisor. Due to a construction of Kahn [Kah89], the functor M → resE (π ∗ (M )∨∨ ) establishes a bijection between the reflexive R–modules (maximal Cohen-Macaulay modules) and the generically globally generated indecomposable vector bundles on E with vanishing first cohomology.5 A typical example of a minimally elliptic singularity is a Tpqr – singularity, given by the equation k
x, y, z/(xp + y q + z r − λxyz), where p1 + 1q + 1r ≤ 1 and λ = 0. If p1 + 1q + 1r = 1, then this singularity is simple elliptic and the exceptional divisor E is a smooth elliptic curve. Thus, in this case a description 5

resE denotes the restriction to E

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of indecomposable maximal Cohen-Macaulay modules follows from Atiyah’s classification of vector bundles on elliptic curves [Ati57]. The main result of Atiyah’s paper essentially says: Theorem 1 (Atiyah). An indecomposable vector bundle E on an elliptic curve E is uniquely determined by its rank r, degree d and determinant det(E) ∈ Picd (E) ∼ = E. In Section 2.3 we give a new proof of this result. However, if p1 + 1q + 1r < 1, then S is a so-called cusp singularity and in this case E is a cycle of n projective lines En , where E1 denotes a rational curve with one node.

A complete classification of indecomposable vector bundles and torsion free sheaves on these curves in the case of an arbitrary base field k was obtained by Drozd and Greuel [DG01]. For algebraically closed fields there is the following description, which we prove in Section 3.2. Theorem 2. Let En denote a cycle of n projective lines and Ik be a chain of k projective lines, E an indecomposable torsion free sheaf on En . 1. If E is locally free, then there exists an ´etale covering π : Enr −→ En of degree r, a line bundle L ∈ Pic(Enr ) and a natural number m ∈ N such that E∼ = π∗ (L ⊗ Fm ), where Fm is an indecomposable vector bundle on Enr , recursively defined by the sequences 0 −→ Fm−1 −→ Fm −→ O −→ 0,

m ≥ 2,

F1 = O.

2. If E is not locally free then there exists a finite map p : Ik −→ En and a line bundle L ∈ Pic(Ik ) (where k, p and L are determined by E) such that E∼ = p∗ (L). This classification is completely analogous to Oda’s description of vector bundles on smooth elliptic curves [Oda71] and provides quite simple rules for the computation of the decomposition of the tensor product of any two vector bundles into a direct sum of indecomposable ones. It allows to describe the

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dual sheaf of an indecomposable torsion free sheaf as well as the dimensions of homomorphism and extension spaces between indecomposable vector bundles (and in particular, their cohomology), see [Bur03,BDG01]. We carry out these computations in Section 3.2. However, the way we prove this theorem, essentially uses ideas from representation theory and the technique of matrix problems [Bon92]. Using a similar approach, Theorem 2 was generalized by Burban and Drozd [BD04] to classify indecomposable complexes of the bounded (from the right) derived category of coherent sheaves D− (Coh(E)) on a cycle of projective lines E = En , see also [BD05] for the case of associative algebras. The situation turns out to be quite different for other singular projective curves of arithmetic genus one. For example, in the case of a cuspidal rational curve zy 2 = x3 even a classification of indecomposable semi-stable vector bundles of a given slope is a representation-wild problem.6 However, if we restrict our attention only on stable vector bundles, then this problem turns out to be tame again.7 Moreover, the combinatorics of the answer is essentially the same as for smooth and nodal Weierstraß curves8 : Theorem 3 (see [BD03, BK3]). Let E be a cuspidal cubic curve over an algebraically closed field k then a stable vector bundle E is completely determined by its rank r, its degree d, that should be coprime, and its determinant det(E) ∈ Picd (E) ∼ = k. The technique of matrix problems is a very convenient tool for the study of vector bundles on a given singular projective rational curve of arithmetic genus one. However, to investigate the behavior of the category of coherent sheaves on genus one curves in families one needs other methods. One possible approach is provided by the technique of derived categories and Fourier-Mukai transforms [Muk81,ST01], see Section 3.4. The key idea of this method is that we can transform a sky-scraper sheaf into a torsion free sheaf by applying an auto-equivalence of the derived category. In a relative setting of elliptic fibrations with a section one can use relative Fourier-Mukai transforms to construct examples of relatively semi-stable torsion free sheaves, see for example [BK2]. Theorem 4 (see [BK1]). Let E be an irreducible projective curve of arithmetic genus one over an algebraically closed field k. Then 6

7 8

An exact k–linear category A over an algebraically closed field k is called wild if it contains as a full subcategory the category of finite-dimensional representations of any associative algebra. For a formal definition of tameness we refer to [DG01], where a wild-tame dichotomy for vector bundles and torsion free sheaves on reduced curves was proven. In this paper we call a plane cubic curve Weierstraß curve. If k is algebraically closed and char(k) = 2, 3 then it can be written in the form zy 2 = 4x3 − g2 xz 2 − g3 z 3 , where g2 , g3 ∈ k. It is singular if and only if g23 = 27g32 and unless g2 = g3 = 0 the singularity is a node, whereas in the case g2 = g3 = 0 the singularity is a cusp.

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1. The group of exact auto-equivalences of the derived category Db (Coh(E)) transforms stable sheaves into stable ones and semi-stable sheaves into semi-stable ones. 2. For any rational number ν the abelian category Cohν (E) of semi-stable coherent sheaves of slope ν is equivalent to the category Coh∞ (E) of coherent torsion sheaves and this equivalence is induced by an exact autoequivalence of Db (Coh(E)). 3. For any coherent sheaf F on E such that End(F ) = k there exists a point x ∈ E and Φ ∈ Aut(Db (Coh(E))) such that F ∼ = Φ(k(x)). This theorem shows a fundamental difference between a nodal and a cuspidal Weierstraß curve. Namely, let E be a singular Weierstraß curve and s its singular point. Then the category of finite-dimensional modules over the  complete local ring O E,s has different representation types in the nodal and cuspidal cases. For a nodal curve, the category of finite dimensional representations of k
x, y/xy is tame due to a result of Gelfand and Ponomarev [GP68]. In the second case, the category of finite length modules over the ring k
x, y/(y 2 − x3 ) is representation wild, see for example [Dro72]. The correspondence between sky-scraper sheaves and semi-stable vector bundles on irreducible Weierstraß curves was first discovered by Friedman, Morgan and Witten [FMW99] (see also [Teo00]) and afterwards widely used in the physical literature under the name “spectral cover construction”. Theorem 5 (see [FMW99]). Let E be an irreducible Weierstraß curve, p0 ∈ E a smooth point and E a semi-stable torsion free sheaf of degree zero. Then the sequence ev

0 −→ H 0 (E(p0 )) ⊗ O −→ E(p0 ) −→ coker(ev) −→ 0 is exact. Moreover, the functor Φ : E → coker(ev) establish an equivalence between the category Coh0 (E) of semi-stable torsion-free sheaves of degree zero and the category of coherent torsion sheaves Coh∞ (E). This correspondence between torsion sheaves and semi-stable coherent sheaves can be generalized to a relative setting of an elliptic fibration E −→ T. In [FMW99] it was used to construct vector bundles on E which are semi-stable of degree zero on each fiber, see also [BK2]. As was shown in [BK1], the functor Φ is the trace of a certain exact auto-equivalence of the derived category Db (Coh(E)). Using this equivalence of categories and a concrete description of k
x, y/xy–modules in terms of their projective resolutions, one can get a description of semi-stable torsion free sheaves of degree zero on a nodal Weierstraß curve in terms of ´etale coverings [FM, BK1]. In Section 3.4 we give a short overview of some related results without going into details. Acknowledgement. Work on this article has been supported by the DFGSchwerpunkt ”Globale Methoden in der komplexen Geometrie”. The second-

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named author would like to thank Bernd Kreußler for numerous discussions of the results of this paper.

2 Vector Bundles on Smooth Projective Curves In this section we review some classical results about vector bundles on smooth curves. However, we provide non-classical proofs which, as we think, are simpler and fit well in our approach to coherent sheaves over singular curves. The behavior of the category of vector bundles on a smooth projective curve X is controlled by its genus g(X). If g(X) = 0 then X is a projective line P1 and any locally free sheaf on it splits into a direct sum of line bundles OP1 (n), n ∈ Z. This result, usually attributed to Grothendieck [Gro57], was already known in an equivalent form to Birkhoff [Bir13]. We found it quite instructive to include Birkhoff’s algorithmic proof in our survey. A classification of vector bundles in the case of smooth elliptic curves, i.e. for g(X) = 1 was obtained by Atiyah [Ati57]. He has shown that the category of vector bundles on an elliptic curve X is tame and an indecomposable vector bundle E is uniquely determined by its rank r, its degree d and a point of the curve x ∈ X. A modern, and in our opinion more conceptual way to prove Atiah’s result uses the language of derived categories and is due to Lenzing and Meltzer [LM93]. In the case of an algebraically closed field of characteristics zero an alternative description of indecomposable vector bundles via ´etale coverings was found by Oda [Oda71]. This classification was a cornerstone in the proof of Polishchuk and Zaslow [PZ98] of Kontsevich’s homological mirror symmetry conjecture for elliptic curves, see also [Kre01]. The case of curves of genus bigger than one is not considered in this survey. In this situation even the category of semi-stable vector bundles of slope one is representation wild and the main attention is drawn to the study of various moduli problems and properties of stable vector bundles, see for example [LeP97]. Throughout this section we do not require any assumptions about the base field k. 2.1 Vector Bundles on the Projective Line We are going to prove the following classical theorem. Theorem 6 (Birkhoff-Grothendieck). Any vector bundle E on the projective line P1 splits into a direct sum of line bundles:  OP1 (n)rn . E∼ = n∈Z

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A proof of this result based on Serre duality and vanishing theorems can be found in a book of Le Potier [LeP97]. However, it is quite interesting to give another, completely elementary proof, based on a lemma proven by Birkhoff in 1913. A projective line P1 is a union of two affine lines A1i (i = 0, 1). If (x0 : x1 ) are homogeneous coordinates in P1 then A1i = {(x0 : x1 )|xi = 0}. The affine coordinate on A10 is z = x1 /x0 and on A11 it is z −1 = x0 /x1 . Thus we can identify A10 with Spec(k[z]) and A11 with Spec(k[z −1 ]), their intersection is then Spec(k[z, z −1 ]). Certainly, any projective module over k[z] is free, i.e. all vector bundles over an affine line are trivial. Therefore to define a vector bundle over P1 one only has to prescribe its rank r and a gluing matrix M ∈ GL(r, k[z, z −1 ]). Changing bases in free modules over k[z] and k[z −1 ] corresponds to the transformations M → T −1 M S, where S and T are invertible matrices of the same size, over k[z] and k[z −1 ] respectively. Proposition 1 (Birkhoff [Bir13]). For any matrix M ∈ GL(r, k[z, z −1 ]) there are matrices S ∈ GL(r, k[z]) and T ∈ GL(r, k[z −1 ]) such that T −1 M S is a diagonal matrix diag(z d1 , . . . , z dr ). Proof. One can diagonalize the matrix M in three steps. Step 1. Reduce the matrix M = (aij ) to a lower triangular form with diagonal entries aii = z mi , where mi ∈ Z and m1 ≤ m2 ≤ · · · ≤ mr . Indeed, since k[z] is a discrete valuation ring, using invertible transformations of columns over k[z] we can reduce the first row (a11 , a12 , . . . , a1r ) of M to the form (a1 , 0, . . . , 0), where a1 is the greatest common divisor of a11 , a12 , . . . , a1r . Let M  be the (r − 1) × (r − 1) matrix formed by the entries aij , i, j ≥ 2. Since det(M ) = a1 · det(M  ) and det(M ) is a unit in k[z, z −1 ], it implies that a1 = z m1 and det(M  ) is a unit in k[z, z −1 ] too. Then we proceed with the matrix M  inductively. Note, that the diagonal entries can be always reordered to satisfy m1 ≤ m2 ≤ · · · ≤ mr . Step 2. Consider the case of a lower-triangular (2 × 2)-matrix   zm 0 M= p(z, z −1 ) z n with m ≤ n. We show by induction on the difference n − m that M can be diagonalized performing invertible transformations of rows over k[z −1 ] and invertible transformations of columns over k[z]. If m = n then we can simply kill the entry p = p(z, z −1 ). Assume now that m < n and p = 0. Without loss of generality we may suppose that p ∈ z m+1 , . . . , z n−1 . Therefore there exist two mutually prime polynomials  n−d  az a and b in k[z] such that ap + bz n = z d and m < d < n. Then b p/z d belongs to GL(2, k[z]) and

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zm 0 p zn



a z n−d b p/z d



 =

z m a z n+m−d zd 0

 .

In order to conclude the induction step it remains to note that |n + m − 2d| < |n − m|. Step 3. Let M be a lower-diagonal matrix with the diagonal elements r  z m1 , . . . , z mr with m1 ≤ m2 ≤ · · · ≤ mr . We show by induction on |mi − mj | that M can be diagonalized. This statement is obvious for 0. Assume that

r 

r 

i,j=1

|mi −mj | =

i,j=1

|mi − mj | = N > 0. Introduce an ordering on the set

i,j=1

{(i, j)|1 ≤ j ≤ i ≤ r}: (2, 1) < (2, 3) < · · · < (r − 1, r) < (3, 1) < · · · < (r − 2, r) < · · · < (r, 1). Let (i0 , j0 ) be the smallest pair such that ai0 j0 = 0. Then we can apply the algorithm from the Step 2 to the (2 × 2) matrix formed by the entries r  |mi − mj |. This (j0 , j0 ), (j0 , i0 ), (i0 , j0 ) and (i0 , i0 ) to diminish the sum i,j=1

completes the proof of Birkhoff’s lemma. Now it remains to note that 1 × 1 matrix (td ) defines the line bundle OP1 (−d). This implies the statement about the splitting of a vector bundle on a projective line into a direct sum of line bundles. 2.2 Projective Curves of Arithmetic Genus Bigger Then One Are Vector Bundle Wild In this subsection we are going to prove the following Theorem 7 (see [DG01]). Let X be an irreducible projective curve of arithmetic genus g(X) > 1 over an algebraically closed field k. Then the abelian category of semi-stable vector bundles of slope9 one is representation wild. In order to show the wildness of a category A one frequently uses the following lemma: Lemma 1. Let A be an abelian category, M, N ∈ Ob(A) with HomA (M, N ) = 0 and ξ, ξ  ∈ Ext1A (N, M ) two extensions α

β

α


β


ξ : 0 −→ M −→ K −→ N −→ 0, ξ  : 0 −→ M −→ K  −→ N −→ 0. Then K ∼ = K  if and only if there exist two isomorphisms f : M −→ M and g : N −→ N such that f ξ = ξ  g. 9

The slope of a coherent sheaf F is µ(F ) =

deg(F) . rk(F)

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Proof. The statement is clear in one direction: if f ξ = ξ  g in Ext1A (N, M ), then K∼ = K  by the 5-Lemma. Now suppose K ∼ = K  and let h : K −→ K  be an isomorphism. Then im(hα) is a subobject of im(α ). Indeed, otherwise the map β  hα : M −→ N would be non-zero, a contradiction. Therefore we get the following commutative diagram: 0

/M f

α

/K h

β

/N

/0

g


  
/ 0. / M α / K β / N 0 −1 In the same way we proceed with h . Hence f is an isomorphism, what proves the lemma.

Let us come back to the proof of the theorem. Suppose now that X is an irreducible projective curve of arithmetic genus g > 1, O := OX . Then for any two points x = y from X we have Hom(O(x), O(y)) ∼ = H 0 (X, O(y − x)) = 0 1 and the Riemann-Roch theorem implies that Ext (O(x), O(y)) ∼ = H 1 (X, O(y− g−1 ∼ x)) = k . Fix 5 different points x1 , . . . , x5 of the curve X, choose nonzero elements ξij ∈ Ext1 (O(xj ), O(xi )) for i = j and consider vector bundles F (A, B), where A, B ∈ M at(n × n, k) and F (A, B) is given as an extension 0 −→ (O(x1 ) ⊕ O(x2 ))n −→ F(A, B) −→ (O(x3 ) ⊕ O(x4 ) ⊕ O(x5 ))n −→ 0       B

A

corresponding to the element ξ(A, B) of Ext1 (A, B) presented by the matrix   ξ13 I ξ14 I ξ15 I , ξ23 I ξ24 A ξ25 B where I denotes the unit n × n matrix. If (A , B  ) is another pair of matrices, and F (A, B) → F (A , B  ) any morphism, then the previous lemma implies that there are morphisms φ : A → A and ψ : B → B such that ψξ(A, B) = ξ(A , B  )φ. Now one can easily deduce that Φ = diag(S, S, S) and Ψ = diag(S, S) for some matrix S ∈ M at(n × n, k) such that SA = A S and SB = B  S. If we consider a pair of matrices (A, B) as a representation of the free algebra kx, y in 2 generators, the correspondence (A, B) → F (A, B) becomes a full, faithful and exact functor kx, y − mod −→ VB(X). In particular, it maps non-isomorphic modules to non-isomorphic vector bundles and indecomposable modules to indecomposable vector bundles. Using the terminology of representation theory of algebras, we say in this situation that the curve X is vector bundle wild. For a precise definition of wildness we refer to [DG01]. Recall that the algebra kx, y here can be replaced by any finitely generated algebra Λ = ka1 , . . . , an . Indeed, any Λ–module M such that dimk (M ) = m is given by a set of matrices A1 , . . . , An of size m × m. One

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gets a full, faithful and exact functor Λ − mod −→ kx, y − mod mapping the module M to the kx, y–module of dimension m · n defined by the pair of matrices ⎛ ⎞ ⎞ ⎛ A1 I . . . 0 λ1 I 0 . . . 0 ⎜ 0 A2 . . . 0 ⎟ ⎜ 0 λ2 I . . . 0 ⎟ ⎜ ⎟ ⎟ ⎜ Y =⎜ . . . X=⎜ . ⎟ . . . ⎟, . .. . . .. ⎠ ⎝ .. .. . . .. ⎠ ⎝ .. 0 0 . . . λn I 0 0 . . . An where λ1 , . . . , λn are different elements of the field k. Thus a classification of vector bundles over X would imply a classifications of all representations of all finitely generated algebras, a goal that perhaps nobody considers as achievable (whence the name “wild”). 2.3 Vector Bundles on Elliptic Curves In this subsection we shall discuss a classification of indecomposable coherent sheaves over smooth elliptic curves. Modulo some facts about derived categories we give a self-contained proof of Atiyah’s classification of indecomposable vector bundles which is probably simpler than the original one. Definition 1. An elliptic curve E over a field k is a smooth projective curve of genus one having a k–rational point p0 . The category Coh(E) of coherent sheaves on an elliptic curve E has the following properties, sometimes called “the dimension one Calabi-Yau property”: • It is abelian, k–linear, Hom-finite, noetherian and of global dimension one. • Serre Duality: for any two coherent sheaves F and G on E there is an isomorphism Hom(F , G) ∼ = Ext1 (G, F )∗ , functorial in both arguments. It is interesting to note that these properties almost characterize the category of coherent sheaves on an elliptic curve: Theorem 8 (Reiten – van den Bergh [RV02]). Let k be an algebraically closed field and A an indecomposable abelian Calabi-Yau category of dimension one. Then A is equivalent either to the category of finite-dimensional k
t– modules or to the category of coherent sheaves on an elliptic curve E. This theorem characterizes Calabi-Yau abelian categories of global dimension one. We shall need one more formula to proceed with a classification of indecomposable coherent sheaves.

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Theorem 9 (Riemann–Roch formula). For any two coherent sheaves F and G on an elliptic curve E there is an integral bilinear Euler form    deg(G) deg(F )  1 .  F, G := dimk Hom(F , G) − dimk Ext (F , G) =  rk(G) rk(F )  In particular,  ,  is anti-symmetric: F, G = −G, F . Now we are ready to start with the classification of indecomposable coherent sheaves. Theorem 10 (Atiyah). Let E be an elliptic curve over a field k. Then 1. Any indecomposable coherent sheaf F on E is semi-stable. 2. If F is semi-stable and indecomposable then all its Jordan-H¨ older factors are isomorphic. 3. A coherent sheaf F is stable if and only if End(F ) = K, where k ⊂ K is some finite field extension. Proof. It is well-known that any coherent sheaf F ∈ Coh(E) has a HarderNarasimhan filtration 0 ⊂ Fn ⊂ . . . ⊂ F1 ⊂ F0 = F whose factors Aν := Fν /Fν+1 are semi-stable with decreasing slopes µ(An ) > µ(An−1 ) > . . . > µ(A0 ). Using the definition of semi-stability, this implies Hom(Aν+i , Aν ) = 0 for all ν ≥ 0 and i > 0. Therefore, Ext1 (A0 , F1 ) ∼ = Hom(F1 , A0 )∗ = 0, and the exact sequence 0 → F1 → F → A0 → 0 must split. In particular, if F is indecomposable, we have F1 = 0 and F ∼ = A0 and F is semi-stable. The full sub-category of Coh(E) whose objects are the semi-stable sheaves of a fixed slope is an abelian category in which any object has a Jordan-H¨ older filtration with stable factors. If F and G are non-isomorphic stable sheaves which have the same slope then Ext1 (F , G) = 0. Based on this fact we deduce that an indecomposable semi-stable sheaf has all its Jordan-H¨older factors isomorphic to each other. It is well-known that any non-zero automorphism of a stable coherent sheaf F is invertible, i.e. End(F ) is a field K. Since E is projective, the field extension k ⊂ K is finite. On a smooth elliptic curve, the converse is true as well, which equips us with a useful homological characterization of stability. To see this, suppose that all endomorphism of F are invertible but F is not stable. This implies the existence of an epimorphism F → G with G stable and µ(F ) ≥ µ(G). Serre duality implies dimk Ext1 (G, F ) = dimk Hom(F , G) > 0, hence, G, F  = dimk Hom(G, F ) −
dimk Ext1 (G,F ) < dimk Hom(G, F ). By Riemann-Roch formula G, F  = µ(F ) − µ(G) rk(F ) rk(G) > 0, thus Hom(G, F ) = 0. But this produces a non-zero composition F → G → F which is not an isomorphism, in contradiction to the assumption that End(F ) is a field.

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Remark 1. Usually one speaks about stability of vector bundles on projective varieties in the case of an algebraically closed field of characteristics zero. However, due to a result of Rudakov [Rud97] one can introduce a stability notion for fairly general abelian categories. The following classical fact was, probably first, proven by Dold [Dol60]: Proposition 2. Let A be an abelian category of global dimension one and F an  object of the derived category Db (A). Then there is an isomorphism ∼ H i (F )[−i], i.e. any object of Db (A) splits into a direct sum of its F = i∈Z

homologies. This proposition in particular means that the derived category Db (A) of a hereditary abelian category A and the abelian category A itself have the same representation type. However, it turns out that the derived category has a richer structure and more symmetry then the corresponding abelian category. First of all note that the group Aut(Db (Coh(E))) acts on the K–group K(E) of Coh(E) preserving the Euler form  , . Hence, it leaves invariant the radical of the Euler form rad ,  = {F ∈ K(E)|F,  = 0} and induces an acZ tion on K(E)/rad , . Since by Riemann-Roch theorem Z : K(E)/rad ,  −→ Z2 is an isomorphism, where Z(F ) := (rk(F ), deg(F )) ∈ Z2 , we get a group homomorphism Aut(Db (Coh(E))) −→ SL(2, Z). We call the pair Z(F ) ∈ Z2 the charge of F . Theorem 11 (Mukai, [Muk81]). Let E be an elliptic curve. Then the group homomorphism Aut(Db (Coh(E))) −→ SL(2, Z) is surjective. Proof. By the definition of an elliptic curve there is a k–rational point p0 on E
inducing an exact equivalence O(p0 )⊗−. Let P = OE×E ∆−(p0 ×E)−(E×p0 ) then the Fourier-Mukai transform ΦP : Db (Coh(E)) → Db (Coh(E)),

ΦP (F ) = Rπ2∗ (P ⊗ π1∗ F )

is an exact auto-equivalence of the derived category, see [Muk81]. The actions of O(p0 ) ⊗ − and ΦP on
K(E)/rad ,  in the basis {[O], [k(p0 )]} are given by  0 1 the matrices ( 11 01 ) and −1 0 which are known to generate SL(2, Z). This shows the claim. The technique of derived categories makes it easy to give a classification of indecomposable coherent sheaves on an elliptic curve. Theorem 12. Let F be an indecomposable coherent sheaf on an elliptic curve E. Then there exits a torsion sheaf T and an exact auto-equivalence Φ ∈ Aut(Db (Coh(E))) such that F ∼ = Φ(T ). Proof. Let F be an indecomposable coherent sheaf on E with the charge Z(F ) = (r, d), r > 0. Let h = g.c.d.(r, d) be the greatest common divisor, then

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there exists a matrix F ∈ SL(2, Z) such that F ( dr ) = ( h0 ). We can lift the matrix F to an auto-equivalence Φ ∈ Aut(Db (Coh(E)), then Z(Φ(F )) = ( h0 ). Since Aut(Db (Coh(E))) maps indecomposable objects of the derived category to indecomposable ones and since the only indecomposable objects in the derived category are shifts of indecomposable coherent sheaves, we can conclude that Φ(F ) is isomorphic to a shift of some indecomposable sheaf of rank zero, what proves the theorem. Let ME (r, d) denote the set of indecomposable vector bundles on E of rank r and degree d. Theorem 13 (Atiyah). Let E be an elliptic curve. Then for any integer h > 0 there exists a unique indecomposable vector bundle Fh ∈ ME (h, 0) such that H 0 (Fh ) = 0. The vector bundles Fh are called unipotent. Moreover, the following properties hold: 1. H 0 (Fh ) = H 1 (Fh ) = k for all h ≥ 1. ∼ Symh−1 (F2 ). Moreover 2. If char(k) = 0 then Fh = F e ⊗ Ff ∼ =

f −1 

Fe+f −2i−1

i=0

Sketch of the proof. Since Fh is indecomposable of degree zero, it has a unique Jordan-H¨ older factor L ∈ Pic0 (E). From the assumption Hom(O, Fh ) = 0 we conclude that L ∼ = O, so each bundle Fh can be obtained by recursive self-extensions of the structure sheaf. Since by Theorem 12 the category of semi-stable vector bundles of degree zero is equivalent to the category of torsion sheaves, we conclude that the category of semi-stable sheaves with the Jordan-H¨ older factor O is equivalent to the category of finite-dimensional k
t-modules. The exact sequence 0 −→ Fh−1 −→ Fh −→ O −→ 0. corresponds via Fourier-Mukai transform ΦP to 0 −→ k
t/th−1 −→ k
t/th −→ k −→ 0. In the same way we conclude that Hom(O, Fh ) = Homk
t (k, k
t/th ) = k. Moreover, one can show that ΦP (k
t/te ⊗k k
t/tf ) ∼ = Fe ⊗ Ff , hence we have the same rules for the decomposition of the tensor product of unipotent vector bundles and of nilpotent Jordan cells, see [Ati57,Oda71,Muk81,PH05]. Remark 2. Atiyah’s original proof from 1957 was written at the time when the formalism of derived and triangulated categories was not developed yet. However, his construction of a bijection between ME (r, d) and ME (h, 0) corresponds exactly to the action of the group of exact auto-equivalences of the derived category of coherent sheaves on the set of indecomposable objects. This was probably for the first time observed by Lenzing and Meltzer in [LM93]. For further elaborations, see [Pol03, PH05, BK3].

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Actually, Atiyah’s description of indecomposable vector bundles on an elliptic curve E is more precise. Theorem 14 (Atiyah). Let E be an elliptic curve over an algebraically closed field k. For any pair of coprime integers (r, d) with r > 0 pick up some E(r, d) ∈ ME (r, d). Then 1. 2. 3. 4.

ME (r, d) = {E(r, d) ⊗ L| L ∈ Pic0 (E)}. E(r, d) ⊗ L ∼ = O. = E(r, d) if and only if Lr ∼ The map det : ME (r, d) −→ ME (1, d) is a bijection. If char(k) = 0, then Fh ⊗ − : ME (r, d) −→ ME (rh, dh) is a bijection.

Remark 3. If k is not algebraically closed and E is a smooth projective curve of genus one over k, without k–rational points, then we miss the generator O(p0 ) in the group of exact auto-equivalences Db (Coh(X)) and the method used for elliptic curves can not be immediately applied. This problem was solved in a paper of Pumpl¨ un [Pum04]. We may sum up the discussed properties of indecomposable coherent sheaves on elliptic curves: Proposition 3. Let E be an elliptic curve over a field k. Then 1. Any indecomposable coherent sheaf F on E is semi-stable with a unique stable Jordan-H¨ older factor. 2. An indecomposable vector bundle is determined by its charge (r, d) ∈ Z2 and a closed point x of the curve E. 3. Let Cohν (E) be the category of semi-stable sheaves of slope ν. Then for any µ, ν ∈ Q ∪ {∞} the abelian categories Cohν (E) and Cohµ (E) are equivalent and this equivalence is induced by an auto-equivalence of Db (Coh(E)). 4. In particular, each category Cohµ (E) is equivalent to the category of coherent torsion sheaves. 5. If F ∈ Cohν (E), G ∈ Cohµ (E) and ν < µ then Ext1 (F , G) = 0 and dimk Hom(F , G) = deg(G)rk(F ) − deg(F )rk(G). The case ν > µ is dual by Serre duality. This gives a pretty complete description of the category of coherent sheaves on an elliptic curve and of its derived category. However, in applications one needs another description of indecomposable vector bundles, see [Pol02,PZ98]. The following form of Atiyah’s classification is due to Oda [Oda71]. It was used by Polishchuk and Zaslow in their proof of the homological mirror symmetry conjecture for elliptic curves, see [PZ98, Kre01]. Theorem 15 (Oda). Let k = C and E = Eτ = C/1, τ  be an elliptic curve, E ∈ ME (rh, dh) an indecomposable vector bundle, where g.c.d.(r, d) = 1. Then there exists a unique line bundle L of degree d on Erτ such that

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E∼ = π∗ (L) ⊗ Fh ∼ = π∗ (L ⊗ Fh ), where π : Erτ −→ Eτ is an ´etale covering of degree r. Proof. Let L be a line bundle on Erτ of degree d. Since the morphism π is ´etale, π∗ (L) is a vector bundle on Eτ of rank r. The Todd class of an elliptic curve is trivial, hence by Grothendieck-Riemann-Roch theorem we obtain deg(π∗ (L)) = deg(L) = d. Now let us show that π∗ (L) is indecomposable. To do this it suffices to prove that End(π∗ L) = C. Consider the fiber product diagram  E ⏐ ⏐ p2

p1

−−−−→ Erτ ⏐ ⏐π1 π

Erτ −−−2−→ Eτ  is a union of r copies of the elliptic curve Erτ : One can easily check that E r ! (i) = Eirτ , where each p1 : Eirτ −→ Erτ , i = 1, . . . , r can be chosen to be E i=1 (i)

the identity map and p2 (z) = z + ri τ . Since all morphisms πi , pi , i = 1, 2 are affine and flat, the functors πi∗ , πi∗ , pi∗ , p∗i are exact. Moreover, p!i = p∗i , since the canonical sheaf of an elliptic curve is trivial.10 Using the base change isomorphism and Grothendieck duality we have

  HomEτ π1∗ L, π2∗ L ∼ = HomErτ π2∗ π1∗ L, L

  ∼ = HomE p2∗ p∗ L, L ∼ = Hom p∗ L, p∗ L . rτ

1

E

1

2


(i)∗ (i)∗  It remains to note that HomE p1 L, p2 L = 0 for i = 0. If E is an indecomposable vector bundle on Eτ of rank rh and degree dh, then by Theorem 14 there exists M ∈ Pic0 (Eτ ) such that E ∼ = π∗ (L)⊗M⊗Fh . By the projection formula E ∼ = π∗ (L ⊗ π ∗ M) ⊗ Fh . Moreover, passing to an ´etale covering kills the ambiguity in the choice of M. It remains to show that π ∗ (Fh ) ∼ = Fh . To do this it suffices see that ∗ π (F2 ) ∼ = F2 , since Fh ∼ = Symh−1 (F2 ) and the inverse image commutes with all tensor operations. The only property we have to check is that π ∗ (F2 ) does not split. It is equivalent to say that the map π ∗ : H 1 (OEτ ) −→ H 1 (OErτ ) is non-zero. This follows from the commutativity of the diagram:

10

Recall that if f : X → Y is a finite morphism of Gorenstein projective schemes then HomX (F , f ! G) ∼ = HomE (f∗ F , G) for any coherent sheaf F on X and a coherent sheaf G on Y.

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ZO 2 ( 10 0r ) Z2

H1 (Eτ , Z) O

∼ =

/ H 1 (Eτ , Z)  

∼ =

 / H 1 (Erτ , Z)  

π∗

π∗

H1 (Erτ , Z)

/ H 1 (Eτ , 0) 

π∗

/ H 1 (Erτ , 0).

3 Vector Bundles and Torsion Free Sheaves on Singular Curves of Arithmetic Genus One In this paper we discuss two approaches for the study of the category of coherent sheaves on a singular projective curve of arithmetic genus one. The first uses the technique of derived categories and Fourier-Mukai transforms . Its key point is that any semi-stable torsion free sheaf on an irreducible Weierstraß curve can be obtained from a torsion sheaf by applying an autoequivalence of the derived category. This technique can be generalized to the case of elliptic fibrations: we can transform a family of torsion sheaves to a family of sheaves, which are semi-stable on each fiber. However, the approach via Fourier-Mukai transforms allows to describe only semi-stable sheaves. In order to get a description of all indecomposable torsion free sheaves, another technique turns out to be useful. Namely, we relate vector bundles on a singular rational curve X and on its normalizap  can map  −→ X. The inverse image functor p∗ : VB(X) −→ VB(X) tion X non-isomorphic bundles into isomorphic ones. The full information about the fibers of this map is encoded in a certain matrix problem. In the case of an algebraically closed field this approach leads to a very concrete description of indecomposable vector bundles on cycles of projective lines via ´etale coverings (no assumption on char(k) is needed). Combining both methods, we get a quite complete description of the category of torsion free sheaves on a nodal Weierstraß curve. 3.1 Vector Bundles on Singular Curves via Matrix Problems Let X be a reduced projective curve over a field k. Introduce the following notation:  −→ X the normalization of X; • p:X  = p∗ O ; • O = OX and O X   the conductor of O in O; • J = AnnO (O/O)   • A = O/J and A = O/J . Note that A and A are skyscraper sheaves supported at the singular locus of X. Since the morphism p is affine, p∗ identifies the category of coherent  and the category Coh  of coherent modules on the ringed sheaves Coh(X) O  Let S be the subscheme of X defined by the conductor J ,  space (X, O). S its

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 and I = I its ideal sheaf on X.  Then p∗ scheme-theoretic pull-back on X S also induces an equivalence between the category of OX /I–modules and the  category of A–modules.  A, A}  on the topological space X, denote For a sheaf of algebras Λ ∈ {O, O, by TFΛ the category of torsion free coherent Λ–modules and by VBΛ its full subcategory of locally free sheaves. The usual way to deal with vector bundles on a singular curve is to lift them to the normalization, and then work on a smooth curve, see for example [Ses82]. Passing to the normalization we loose information about the isomorphism classes of objects of VBO since nonisomorphic vector bundles can have isomorphic inverse images. In order to describe the fibers of the map VBO −→ VBO and to be able to deal with arbitrary torsion free sheaves we introduce the following definition: Definition 2. The category of triples TX is defined as follows:  1. Its objects are triples (F, M, ˜i), where F is a locally free O–module, M    ˜ is a coherent A–module and i : M ⊗O A −→ F ⊗O A is an epimorphism  of A–modules, which induces a monomorphism of A–modules i : M −→ ˜i

 M ⊗O A −→ F ⊗O A. (F,f ) 2. A morphism (F1 , M1 , ˜i1 ) −→ (F2 , M1 , ˜i2 ) is given by a pair (F, f ), where

f F  and M1 −→ M2 is a morphism F1 −→ F2 is a morphism of O-modules of A-modules, such that the following diagram

M1 ⊗O A

˜i1

f¯

/ F1 ⊗  A O F¯

  ˜i2 / F2 ⊗  A M2 ⊗O A O is commutative in CohA , where F¯ = F ⊗ id and f¯ = ϕ ⊗ id. The main reason to introduce the formalism of triples is the following theorem: Ψ

Theorem 16. The functor TFO −→ TX mapping a torsion free sheaf F to the   M = F ⊗O A and ˜i : F ⊗O ⊗O O), triple (F, M, ˜i), where F = F ⊗O O/tor(F  is an equivalence of categories. Moreover, the category of vector A −→ F ⊗O A, bundles VBO is equivalent to the full subcategory of TX consisting of those triples (F, M, ˜i), for which M is a free A–module and ˜i is an isomorphism. Ψ


Sketch of the proof. We construct the quasi-inverse functor TX −→ TFX as follows. Let (F, M, ˜i) be some triple. Consider the pull-back diagram / J F /M /0 /F 0 

0

id

/ J F

i

 / F

 / F ⊗  A O

/0

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel

in the category of O–modules. Since the pull-back is functorial, we get a Ψ
functor TX −→ Coh(X). Since the map i is injective, F −→ F is injective as well, so F is torsion free. It remains to show that the functors Ψ and Ψ  are quasi-inverse to each other. We refer to [DG01] for the details of the proof. Remark 4. There is a geometric way to interpret the above construction of the p  −→ X its normalization, s : category of triples. Let X be a singular curve, X S −→ X the inclusion of the closed subscheme defined by the conductor ideal  its pull-back on the normalization. Consider the Cartesian and s˜ :  S −→ X diagram s˜ /   X S p˜

p

 s /  S X. Theorem 16 says that a torsion free sheaf F on a singular curve X can be ∗ reconstructed from its “normalization” p∗ (F )/tor(p s∗ F on
F ), its pull-back  S and the “gluing map” p˜∗ s∗ F −→ s˜∗ p∗ F −→ s˜∗ p∗ F /tor(p∗ F ) . Now let us see how this construction can be used to classify torsion free sheaves on degenerations of elliptic curves. Let char(k) = 2 and E be a nodal Weierstraß curve, given by the equation zy 2 − x3 − zx2 = 0, s = (0 : 0 : 1) its p  −→ singular point, P1 = E E the normalization map. Choose coordinates on 1 P in such a way that the preimages of s are 0 = (0 : 1) and ∞ = (1 : 0). The previous theorem says that a torsion free sheaf F on the curve E is uniquely determined by the corresponding triple Ψ (F ) = (F, M, ˜i). Here F  is a locally free O–module, or as we have seen, a locally free OP1 -module.  Using the notation O(n) = p∗ (OP1 (n)), due to the theorem of Birkhoff  rn Grothendieck, F ∼ O(n) . = n∈Z

 Since A = O/J = k(s) and A = O/J = (k × k)(s), the sheaf M can be ˜ identified with its stalk at s and the map i : M⊗A A −→ F ⊗O A can be viewed as a pair (i(0), i(∞)) of linear maps of k–vector spaces. In order to write ˜i  in terms of matrices we identify O(n) ⊗O A with p∗ (OP1 (n) ⊗OP1 OP1 /I). The choice of coordinates on P1 fixes two canonical sections z0 and z1 of H 0 (OP1 (1)) and we use the trivializations OP1 (n) ⊗ I −→ k(0) × k(∞) given by ζ ⊗ 1 → (ζ/z1n (0), ζ/z0n (∞)). Note, that this isomorphism only depends on the choice of coordinates of P1 . In such a way we supply the    A–module F ⊗ A = F(0) ⊕ F(∞) with a basis and get isomorphisms   rn ∼ ∼   F (0) = k(0) and F (∞) = k(∞)rn . With respect to all choices the n∈Z

n∈Z

morphism ˜i is given by two matrices i(0) and i(∞), divided into horizontal blocks:

Vector Bundles on Degenerations of Elliptic Curves .. .

101

.. .

n−1

n−1

}rn

n

n

n+1

n+1

.. .

.. .

i(0)

}rn

i(∞)

From the definition of the category of triples it follows that the matrices i(0) and i(∞) have to be of full row rank and the transposed matrix (i(0)|i(∞))t has to be monomorphic. Vector bundles on E correspond to invertible square matrices i(0) and i(∞).   rn O(n) and M = kN (s), two different pairs Of course, for a fixed F = n∈Z

of matrices (i(0)|i(∞)) and (i (0)|i (∞)) can define isomorphic torsion free sheaves on E. However, since the functor Ψ : TFO −→ TX preserves isomorphism classes of indecomposable objects, two triples (F, M, ˜i) and (F, M, i ) define isomorphic torsion free sheaves if and only if there are automorphisms ¯˜i = i f¯. F : F −→ F and f : M −→ M such  thatrF n  O(n) can be written in a matrix form: An endomorphism F of F = n∈Z

F = (Fkl ), where Fkl is a rl × rk –matrix with coefficients in the vector space ∼   Hom(O(k), O(l)) = k[z0 , z1 ]l−k . In particular, the matrix F is lower triangular and the diagonal rn × rn blocks Fnn are just matrices over k. The morphism F is an isomorphism if and only if all Fnn are invertible. Let r = rank(F). With respect to the chosen trivialization of OP1 (n) at 0 and ∞ the map F¯ : kr (0)⊕kr (∞) −→ kr (0)⊕kr (∞) is given by the pair of matrices (F (0), F (∞)) and we have the following transformation rules for the pair (i(0), i(∞)):

 i(0), i(∞) → F (0)−1 i(0)S, F (∞)−1 i(∞)S , where F is an automorphism of

  rn O(n) and S an automorphism of kN . n∈Z

Note, that the matrices Fkl (0) and Fkl (∞), k, l ∈ Z, k > l can be arbitrary and Fnn (0) = Fnn (∞) can be arbitrary invertible for n ∈ Z. As a result we get the following matrix problem. Matrix problem for a nodal Weierstraß curve. We have two matrices i(0) and i(∞) of the same size and both of full row rank. Each of them is divided into horizontal blocks labeled by integers (they are called sometimes weights). Blocks of i(0) and i(∞), labeled by the same integer, have the same size. We are allowed to perform only the following transformations: 1. We can simultaneously do any elementary transformations of columns of i(0) and i(∞).

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2. We can simultaneously do any invertible elementary transformations of rows inside of any two conjugated horizontal blocks. 3. We can in each of the matrices i(0) and i(∞) independently add a scalar multiple of any row with lower weight to any row with higher weight. The main idea is that we can transform the matrix i into a canonical form which is quite analogous to the Jordan normal form. These types of matrix problems are well-known in representation theory. First they appeared in a work of Nazarova and Roiter [NR69] about the classification of k
x, y/(xy)–modules. They are called, sometimes, “Gelfand problems” or “representations of bunches of chains”. Example 1. Let E be a nodal Weierstraß curve. • The following triple (F, M, ˜i) defines an indecomposable vector bundle of  ⊕ O(n),  rank 2 on E: the normalization F = O n = 0, M = k2 (s) and matrices: i(0) =

1 0

0

0 1

n

and i(∞) =

0 1

0

λ 0

n

λ ∈ k∗ .


 • The triple O(−1), k2 , ˜i = 1 0 0 1 describes the unique torsion free sheaf that is not locally free of degree zero, which compactifies the Jacobian Pic0 (E). A Gelfand matrix problem is determined by a certain partially ordered set together with an equivalence relation on it. Such a poset with an equivalence relation is called a bunch of chains. Before giving a general definition, we give an example describing the matrix problem which corresponds to a nodal Weierstraß curve. There are two infinite sets E0 = {E0 (k)|k ∈ Z} and E∞ = {E∞ (k)|k ∈ Z} with the ordering · · · < E∗ (−1) < E∗ (0) < E∗ (1) < . . . , ∗ ∈ {0, ∞} and two one-point sets {F0 } and {F∞ }. On the union E F = (E0 ∪ E∞ ) (F0 ∪ F∞ ) we introduce an equivalence relation: E0 (k) ∼ E∞ (k), where k ∈ Z and F0 ∼ F∞ . F0 F∞ ppppp ! !

E0 (k) E0 (k+1)

◦  ◦  ◦ 

ppppp ppppp ppppp

◦  E∞ (k) ◦  E∞ (k+1) ◦ 

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103

This picture contains complete information about the corresponding matrix problem. The circles denote the elements of E, the diamonds denote elements F0 and F∞ , dotted lines connect equivalent elements and vertical arrows describe the partial order in E0 and E∞ . The sets E0 ∪F0 and E∞ ∪F∞ correspond to matrices i(0) and i(∞) respectively, elements E0 (k) and E∞ (k) label their horizontal stripes, k ∈ Z. We also say that a row from the horizontal block E∗ (k) has weight k, where ∗ is either 0 or ∞. The equivalence relation E0 (k) ∼ E∞ (k) means that horizontal blocks of weight k have the same number of rows and F0 ∼ F∞ tells that i(0) and i(∞) have the same number of columns. Moreover, elementary transformations inside of two conjugated blocks have to be done simultaneously. The total order on E0 and E∞ means that we can add any scalar multiple of any row of smaller weight to any row with a bigger weight. Such transformations can be performed in the matrices E0 and E∞ independently. Definition 3. Let k be an arbitrary field. A cycle of n projective lines is a projective curve En over k with n irreducible components, each of them is isomorphic to P1 . We additionally assume that all components intersect n and the completion of the transversally with intersection matrix of type A local rings of any singular point of En is isomorphic to k
u, v/uv. In a similar way, a chain of k projective lines Ik is a configuration of projective k lines with intersection matrix of type Ak−1 . Remark 5. Let k = R and E be the cubic curve zy 2 = x3 − zx2 . Then s = (0 : 2 2  0 : 1) is the singular point of E and O E,s = R
u, v/(u + v ). Then E is not a cycle of projective lines in the sense of Definition 3 and the combinatorics of the indecomposable vector bundles on E will be considered elsewhere.  its normalization. Let E be either a chain or a cycle of projective lines and E The matrix problem we get is given by the following partially ordered set. Consider the set of pairs {(L, a)}, where L is an irreducible component  and a ∈ L a preimage of a singular point. To each such pair we attach of E a totally ordered set E(L,a) = {E(L,a) (i)|i ∈ Z}, where · · · < E(L,a) (−1) < E(L,a) (0) < E(L,a) (1) < . . . and a one-point set F(L,a) . On the union E



F=



(E(L,a) ∪ F(L,a) ),

(L,a)

we introduce an equivalence relation: 1. F(L
,a
) ∼ F(L

,a

) , where a and a are preimages of the same singular point a ∈ E. 2. E(L,a
) (k) ∼ F(L,a

) (k) for k ∈ Z and a , a ∈ L. Such a partially ordered set with an equivalence relation is called a bunch of chains [Bon92]. A representation of such a bunch of chains is given by a set

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of matrices M (L, a), for each element (L, a). Every matrix M (L, a) is divided into horizontal blocks labelled by the elements of E(L,a) . Of course, all but finitely many labels corresponds to empty blocks. The principle of conjugation of blocks is the same as for a rational curve with one node. The category of representations of a bunch of chains is additive and has two types of indecomposable representations: bands and strings [Bon92]. Hence, the technique of representations of bunches of chains allows to describe indecomposable torsion free sheaves on chains and cycles of projective lines. Let E = En be a cycle of n projective lines, {a1 , a2 , . . . , an } the set of n p  = ! Li , where each  −→ E the normalization of E, E singular points of E, E i=1

Li is isomorphic to a projective line and {ai , ai } = p−1 (ai ). Assume that ai , ai+1 ∈ Li , where an+1 = a1 . Fix coordinates on each projective line Li in such a way, that ai = (0 : 1) and ai+1 = (1 : 0). Definition 4. A band B(d, m, p(t)) is an indecomposable vector bundle of rank rmk. It is determined by the following parameters: 1. d = (d1 , d2 , . . . , dn , dn+1 , dn+2 , . . . , d2n , . . . , drn−n+1 , drn−n+2 , . . . , drn ) ∈  This sequence Zrn is a sequence of degrees on the normalized curve E. s . . . e, where e = should be non-periodic, i.e. not of the form e = ee   s times

e1 , e2 , . . . , eqn is another sequence and q = rs . 2. p(t) = tk + a1 tk−1 + · · · + ak ∈ k[t] is an irreducible polynomial of degree k, p(t) = t. 3. m ∈ Z+ is a positive integer. In particular, one can recover from the sequence d the pull-back of B(d, m, p(t))  it is on the l-th irreducible component of E: p∗l (B(d, m, p(t))) ∼ =

r 

OLl (dl+in )mk .

i=1

A string S(d, f ) is a torsion free sheaf which depends only on two discrete parameters f ∈ {1, 2, . . . , n} and d = (d1 , d2 , . . . , dt ), t > 1. Now we are going to explain the way of construction of gluing matrices of triples corresponding to bands B(d, m, p(t)) and strings S(e, f ). Algorithm 1. Bands. Let d = (d1 , d2 , . . . , drn ) ∈ Zrn be a non-periodic sequence, m ∈ Z+ and p(t) ∈ k[t] an irreducible polynomial of degree k. We have 2n matrices M (Li , ai ) and M (Li , ai+1 ), i = 1, . . . , n occurring in the triple, corresponding to B(d, m, p(t)). Each of them has size mrk × mrk. Divide these matrices into mk × mk square blocks. Consider the sequences d(i) = di di+n . . . di+(r−1)n and label the horizontal strips of M (Li , ai ) and

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105

M (Li , ai+1 ) by integers occurring in each d(i). If some integer d appears l times in d(i) then the horizontal strip corresponding to the label d consists of l substrips having mk rows each. Recall now an algorithm of writing the components of the matrix ˜i in a normal form [Bon92]: 1

1

1

1

1. Start with the sequence (L1 , a1 ) −→ (L1 , a2 ) −→ (L2 , a2 ) −→ (L2 , a3 ) −→ 1 2 2 r r 1 · · · −→ (Ln , a1 ) −→ (L1 , a1 ) −→ · · · −→ (Ln , an ) −→ (Ln , a1 ) −→ . It is convenient to imagine this sequence as a cyclic word broken at the place (L1 , a1 ). 2. Unroll the sequence d. This means that we write over each (Li , a) the corresponding term of the subsequence d(i) together with the number of its previous occurrences in d(i) including the current one: (L1 , a1 )(d1 ,1) −→ (L2 , a2 )(d2 ,1) −→ (L2 , a3 )(d2 ,1) −→ 1

1

1

· · · −→ (Ln , a1 )drn ,∗ −→ (Ln , a1 )(drn ,1) −→ . 1

r

r

3. Now we can fill the entries of the matrices M (L, a). Consider each arrow l

(L, a)(d,i) −→ . Then insert the matrix Imk in the block ((d, i), l) of the matrix M (L, a), which is defined as the intersection of the i-th substrip of the horizontal strip labeled by d and the l-th vertical strip. 4. Put at the ((drn , 1), r)-th place of M (Ln , a1 ) the Frobenius block Jm (p(t)). Strings. Let e = (e1 , e2 , . . . , es ) ∈ Zs and f ∈ {1, 2, . . . , n}. The algorithm to write the matrices for the torsion free sheaf S(d, f ) is essentially the same as for bands. The parameter f denotes the number of the component Lf of  which the sequence (Lf , a ) −→ (Lf , a ) −→ . . . −→ (Lf +s , a ) −→ E f f +1 f +s (Lf +s , af +s+1 ) starts with. Then we unroll e using the same algorithm as for bands. The only difference is that we insert instead of Imk the unit (1 × 1)– matrix. Note, that some of the matrices M (Li , ai ) and M (Li , ai ) can be nonsquare, but they are automatically of full row rank. Example 2. Let E = E2 be a cycle of two projective lines, d = (0, 1, 1, 3, 1, −2) and p(t) ∈ k
t an irreducible polynomial of degree k. Then B(d, m, p(t)) is a vector bundle of degree 3m with the normalization 

mk OL1 ⊕ OL1 (1)2mk ⊕ OL2 (−2)mk ⊕ OL2 (1)mk ⊕ OL2 (3)mk and gluing matrices

Imk

M (L1 , a1 )

=

0 Imk Imk

1 Imk

= M (L1 , a2 )

Imk Imk

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel −2

Jm

M (L2 , a1 )

=

Imk

= M (L2 , a2 ),

1 Imk

Imk Imk

3

Imk

where Jm is the Frobenius block corresponding to the k[t]-module k[t]/p(t)m . The corresponding unrolled sequence looks as follows: 1 1 1 2 (L1 , a1 )(0,1) −→ (L1 , a2 )(0,1) −→ (L2 , a2 )(1,1) −→ (L2 , a1 )(1,1) −→ 2 2 2 3 (L1 , a1 )(1,1) −→ (L1 , a2 )(1,1) −→ (L2 , a2 )(3,1) −→ (L2 , a1 )(3,1) −→ 3 3 3 1 (L1 , a1 )(1,2) −→ (L1 , a2 )(1,2) −→ (L2 , a2 )(−2,1) −→ (L2 , a1 )(−2,1) −→ . Let f = 2 and d = (−1, 0, 1, −1, 1). Then the corresponding torsion free sheaf S(d, f ) has normalization

 F = OL1 (−1) ⊕ OL1 ⊕ OL2 (−1) ⊕ OL2 (1)2 and gluing matrices

M (L1 , a1 ) =

M (L2 , a1 )

=

1

1

−1

1

0

1

−1

1

1

1

1

= M (L1 , a2 )

1 = M (L2 , a2 )

1 1

The corresponding unrolled sequence is 1 1 1 1 (L2 , a2 )(−1,1) −→ (L2 , a1 )(−1,1) −→ (L1 , a1 )(0,1) −→ (L1 , a2 )(0,1) −→ 2 2 2 2 (L2 , a2 )(1,1) −→ (L2 , a1 )(1,1) −→ (L1 , a1 )(−1,1) −→ (L1 , a2 )(−1,1) −→ 3 3 (L2 , a2 )(1,2) −→ (L2 , a1 )(1,2) −→ . Summing everything up, we get the following theorem. Theorem 17 ( [DG01]). Let E = En be a cycle of n projective lines over a filed k. Then • any indecomposable vector bundle on E is isomorphic to some B(d, m, p(t)), where d = (d1 , d2 , . . . , drn ) ∈ Zrn is a non-periodic sequence, m ∈ Z+ and p(t) = tk + a1 tk−1 + · · · + ak ∈ k[t] is an irreducible polynomial, p(t) = t. • Any torsion free but not locally free coherent sheaf is isomorphic to some S(d, t), where t ∈ {1, 2, . . . , n} and d ∈ Zrn . The only isomorphisms between indecomposable vector bundles are generated by

Vector Bundles on Degenerations of Elliptic Curves

• B(d, m, p(t)) ∼ = B(d◦ , m, q(t)), d◦ = drn , drn−1 , . . . , d1 and q(t) = • B(d, m, p(t)) ∼ = B(d
, m, p(t)), with

107

tk ak p(1/t)

d
= dn+1 , dn+2 , . . . , d2n , d2n+1 , . . . , d1 , d2 , . . . , dn . The only isomorphisms between strings are S(e, f ) ∼ = S(e◦ , f ◦ ), where e◦ is ◦ the opposite sequence e = es , es−1 , . . . , e1 . If s = nk + s with 0 ≤ s < n, then f ◦ = s + f taken modulo n. Remark 6. If E = E1 is a Weierstraß nodal curve, then there is no choice for the parameter f in the definition of a string and one simply uses the notation S(d). If the field k is algebraically closed, we write B(d, m, λ) instead of B(d, m, t − λ). As a direct corollary of the combinatorics of strings and bands we obtain the following theorem Theorem 18 ( [DG01]). Let X = In be a chain of n projective lines, then any vector bundle E on X splits into a direct sum of lines bundles. Moreover, Pic(In ) ∼ = Zn and a line bundle is determined by its restrictions on each irreducible component. A description of torsion free sheaves is similar: any indecomposable torsion free sheaf is isomorphic to the direct image of a line bundle on a subchain of projective lines. 3.2 Properties of Torsion Free Sheaves on Cycles of Projective Lines Throughout this subsection, let k be an algebraically closed field and E = En a cycle of n projective lines over k. As we have seen in the previous subsection, indecomposable vector bundles on E are bands B(d, m, λ) and indecomposable torsion free but not locally sheaves are strings S(d, f ). They were described in terms of a certain problem of linear algebra. However, in the case of an algebraically closed field there is a geometric way to present the classification of indecomposable torsion free sheaves on E without appealing to the formalism of bunches of chains. This description, the proof of which we give here for the first time, is completely parallel to Oda’s one for vector bundles on elliptic curves [Oda71]. We start with a lemma describing unipotent vector bundles on E. Lemma 2. For any integer m ≥ 1 there exists a unique indecomposable vector bundle Fm on En appearing in the exact sequence 0 −→ Fm−1 −→ Fm −→ O −→ 0,

F1 = O.

In our notation we have Fm ∼ = B(0, m, 1), where 0 = (0, 0, . . . , 0).    n times

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel

Sketch of the proof. Since the dualizing sheaf ωE ∼ = O is trivial, we have Ext1 (O, O) = k and there is a unique non-split extension 0 −→ O −→ F2 −→ O −→ 0. Then using the same arguments as in [Ati57], we can inductively construct indecomposable vector bundles Fm , m ≥ 1 such that H 0 (Fm ) ∼ = H 1 (Fm ) = k together with exact sequences 0 −→ Fm−1 −→ Fm −→ O −→ 0,

F1 = O.

On the other hand, B(0, m, 1) is the unique indecomposable vector bundle on E of rank m and normalization OEm with a non-zero section. Hence Fm ∼ = B(0, m, 1). The proof of the following proposition is straightforward: Proposition 4. Let Ψ : VB(E) −→ TE be the functor establishing an equivalence between the category of vector bundles on E and the category of triples. Then Ψ preserves tensor products: Ψ (E ⊗ F) ∼ = Ψ (E) ⊗ Ψ (F ), where  M, ˜i) ⊗ (F, N ,  (E, j) = (E ⊗O F, M ⊗A N , ˜i ⊗  j). In particular, • We have an isomorphism B((d), m, λ) ∼ = B((d), 1, λ) ⊗ Fm . • There is the following rule for a decomposition of the tensor product of two unipotent vector bundles:  Fhi , F f ⊗ Fg ∼ = i

where integers hi are the same as in the decomposition k[t]/tf ⊗k k[t]/tg ∼ =  k[t]/thi in the category of k[t]–modules. i∈Z

• In particular, if k is of characteristics zero, we have F f ⊗ Fg ∼ =

g 

Ff −g−1+2j .

j=1

Now we formulate a geometric description of indecomposable torsion free sheaves on a cycle of projective lines in the case of an algebraically closed field. Theorem 19. Let E = En be a cycle of n projective lines and Ik be a chain of k projective lines, E an indecomposable torsion free sheaf on En . 1. If E is locally free, then there is an ´etale covering πr : Enr −→ En , a line bundle L ∈ Pic(Enr ) and a natural number m ∈ N such that E∼ = πr∗ (L ⊗ Fm ).

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109

Moreover, if char(k) = 0, then integers r and m are uniquely determined and the line bundle L is unique up to the action of Aut(Enr /Er ). Other way around, for given r, m ∈ Z+ and L ∈ Pic(Enr ) the vector bundle πr∗ (L ⊗ Fm ) is indecomposable if and only if L does not belong to the image of the map πt∗ : Pic(Enr/t ) → Pic(Enr ) for any proper divisor t of r. 2. If E is not locally free then there exists a map pk : Ik −→ En and a line bundle L ∈ Pic(Ik ) such that E ∼ = pk∗ (L).

Other way around, the torsion free sheaf pk∗ (L) is indecomposable for any line bundle L ∈ Pic(Ik ). Proof. Let E be a cycle of projective lines and πE : E −→ E an ´etale covering of degree r, F a torsion free sheaf on E . In the notation of Remark 4 we have the commutative diagram πS



S

S ~? p
~~ ~ i
~~ ~~ π

  i
E ~? ~ ~ ~~p
~  ~  E

/ S

 S

/S
@




p˜

i

˜i

πE

πE

 /E @




p


/E 

in which all squares are pull-back diagrams. In order to prove the theorem we have to compute the triple describing the torsion free sheaf π∗ (F ). Note that each map I −→ E from a chain of projective lines to a cycle of projective lines factors through an ´etale covering E −→ E. So, in order to prove the second

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel

part of the theorem about the characterization of strings we may consider an ´etale covering of E as well. Note the following simple fact about pull-back diagrams: Lemma 3. Let Y

g


f


 X

/Y f

g

 /X

be a pull-back diagram, where all maps f, g, f  , g  are affine. Then for any coherent sheaf F on Y it holds g ∗ f∗ F ∼ = f∗ g ∗ F .
∗  ∗ The morphism p∗ πE∗ (F )/tor(p∗ πE∗  (F )) −→ πE∗  p (F )/tor(p (F )) is an isomorphism. Indeed, we have a surjection ∼ =

∗ ∗ ∗ p∗ πE∗ (F ) −→ πE∗  p F −→ πE∗  (p (F )/tor(p (F )))

which induces a surjective map
∗  ∗ p∗ πE∗ (F )/tor(p∗ πE∗  (F )) −→ πE∗  p (F )/tor(p (F )) of torsion free sheaves. Since both sheaves have the same rank on each ir we conclude that this map is also injective and reducible component of E, therefore an isomorphism. We need one more simple statement about ´etale coverings. Lemma 4. Let π : Y −→ X be an ´etale map of reduced schemes and F a coherent sheaf on Y. Then there is a canonical isomorphism π∗ (F /tor(F )) −→ π∗ (F )/tor(π∗ (F )). Proof. The canonical map F −→ F/tor(F ) induces the morphism π∗ (F ) −→ π∗ (F /tor(F )). Since π is ´etale, the sheaf π∗ (F /tor(F )) is torsion free and the induced morphism π∗ (F /tor(F )) −→ π∗ (F )/tor(π∗ (F )) is an isomorphism since it is an isomorphism on the stalks. ∗ Let ε : p˜∗ i∗ (F ) −→ i (p∗ (F )/tor(p∗ (F ))) be the gluing map describing the torsion free sheaf F in the corresponding triple. From the commutativity of the diagram

Vector Bundles on Degenerations of Elliptic Curves

111


 / ˜i∗ p∗ πE∗ (F )/tor(p∗ πE∗ (F ))

p˜∗ i∗ πE∗ (F )


 ˜i∗ π p∗ (F )/tor(π p∗ (F )) E∗ E∗ 
 ˜i∗ π (p∗ (F )/tor(p∗ (F )) E∗

 p˜∗ πS∗ i∗n (F )  ∗ πS∗ p i∗ (F )

πS∗ ( ε)

 / π i ∗
p∗ (F )/tor(p∗ (F ))  S∗

we conclude that the direct image sheaf πE∗ F is described by the gluing maε), which are exactly the matrices constructed in the Algorithm 1. trices πS∗ ( This completes the proof. Remark 7. For a given cycle of projective lines E over an arbitrary field k there is always an ´etale covering π : E −→ E of a given degree r. For example, let E = E1 be a rational curve with one node, Xi = P1 (i = 1, 2), fi : Spec(k × k) −→ Xi be two closed embeddings with the image 0 and ∞ and gi : Xi −→ E two normalization maps mapping the points 0 and ∞ on Xi to the singular point of E. Then the push-out of X1 and X2 over Spec(k × k) (in the category of all schemes and affine maps) is a cycle of two projective lines E2 and the induced map g : E2 −→ E1 is an ´etale covering of degree two. The general case can be considered in a similar way. Note that this is quite different to the case of elliptic curves, where the existence of an ´etale covering of a given degree strongly depends on the arithmetics of the curve. Similarly to the proof of Theorem 19 we have the following proposition. Proposition 5. Let πr : Enr −→ En be an ´etale covering of degree r, n nr  nr = ! L be the normalizations of En and Enr . Let  n = ! Li and E E j i=1

j=1

{a1 , a2 , . . . , an } be the set of singular points of En and {b1 , b2 , . . . , bn , bn+1 , . . . , bnr } the singular points of Enr and πr−1 (ai ) = {bi , bi+n , . . . , bi+(r−1) }.  Al , ˜i), where Assume E is a vector bundle of rank l, given by the triple (E,  E ∼ = E1 ⊕ E2 ⊕ · · · ⊕ En and ˜i is given by matrices M (L1 , a1 ), M (L1 , a2 ), . . . , M (Ln , a1 ). Then the pull-back πr∗ (E) corresponds to the triple (E , Al , ˜i ), ∼ where E |L
= Ei , 0 ≤ j ≤ r − 1 and i is given by matrices M (L , b ) = i+nj

M (L, a ) and M (L , b ) = M (L, a ) if πr (L ) = L and πr (b) = a. From this proposition follows the following corollary:

Corollary 1. Let Let E = En be a cycle of n projective lines and πr : Erd −→ En an ´etale covering of degree r. Then

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel

1. πr∗ B(d, 1, λ) ∼ = B(dr , 1, λr ). 2. If char(k) = 0, then πr∗ (Fm ) ∼ = Fm . In particular, we have an isomorphism B(d, m, λ) ∼ = B(d, 1, λ) ⊗ Fm . Proof. The proof of the first part is straightforward. To prove the second, m , Am , ˜i), where ˜i note that πr∗ (Fm ) is given by a triple, isomorphic to (O   is given by matrices M (L1 , a1 ) = Im , M (L1 , a2 ) = Im , . . . , M (Ln , an ) = Im , M (Ln , a1 ) = Jm (1)r , where Jm (1) is the Jordan (m × m)–block with the eigenvalue 1. If char(k) = 0 then Jm (1)r ∼ Jm (1) and we get the claim. Note, that in the case char(k) = p we have Jp (1)p = Ip that implies πp∗ (Fp ) ∼ = Op . To complete the proof of the second claim note, that B(d, m, λ) ∼ = πr∗ (L(d, λ) ⊗ Fm ) ∼ = πr∗ (L(d, λ) ⊗ πr∗ Fm ) ∼ = πr∗ (L(d, λ)) ⊗ Fm ∼ = B(d, 1, λ) ⊗ Fm .

Remark 8. As we have already seen, the technique of ´etale coverings requires special care in the case of positive characteristics. For example, let E = E1 be a Weierstraß nodal curve and π2 : E2 −→ E1 an ´etale covering of degree 2. 2 , k2 (s), ˜i), where Then the vector bundle π2∗ (O) corresponds to the triple (O ˜i is given by matrices i(0) =

1 0

and i(∞) =

0 1

0 1 1 0

Then for char(k) = 2 we have 0

1

1

0

∼

1 0 0 −1

and π2∗ O ∼ = O ⊕ B(0, 1, −1). However, for char(k) = 2 0

1

1

0

∼

1 1 0 1

and π2∗ O ∼ = F2 . From Theorem 19 one can derive formulas for the cohomology groups of indecomposable torsion free sheaves, a formula for the dual of an indecomposable torsion free sheaf and rules for the computation of the direct sum decomposition of two indecomposable vector bundles. This is what we are going to describe now.

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113

Lemma 5 ( [BDG01,BK1]). If d = (d1 , . . . , drn ) ∈ Zrn , e = (e1 , e2 , . . . , ek ), λ ∈ k∗ , m ≥ 1, 1 ≤ f ≤ n, we have: (i) B(d, m, λ)∨ ∼ = B(−d, m, λ−1 )

 (−1, 0, . . . , 0, −1) (ii)S(e, f ) ∼ = S(κ − e, f ) with κ = −2 ∨

if k ≥ 2 if k = 1.

Proof. If f : X → E is a finite morphism, F a coherent sheaf on X and G a locally free sheaf on E then there is a natural isomorphism of f∗ OX –modules f∗ HomX (F , f ! G) ∼ = HomE (f∗ F , G). Recall that f ! ωE is a dualizing sheaf on X if ωE is one on E. In our situation ωE ∼ = OE and we obtain an isomorphism f∗ HomX (F , ωX ) ∼ = HomE (f∗ F , OE ) ∼ = (f∗ F )∨ . To show (i), we consider X = En and f = πn . The claim follows now from ∨ ∼ ωEn ∼ = OEn , Fm = Fm and L(d, λ)∨ ∼ = L(−d, λ−1 ) on En . For the proof of (ii) we let X = Ik and f = pk . Now ωIk ∼ = L(κ) and the result follows from L(d)∨ ∼ = L(−d) on Ik . Using the description of indecomposable vector bundles via ´etale coverings it is not difficult to compute their cohomology. Lemma 6 ( [DGK03]). There is the following formula for the cohomology of indecomposable vector bundles dimk H 0 (B(d, m, λ)) = m

rn


 (di + 1)+ − θ(d) + δ(d, λ)

i=1

and dimk H 1 (B(d, m, λ)) = rm − dimk H 0 (B(d, m, λ), where δ(d, λ) = 1 if d = (0, . . . , 0), λ = 1 and 0 otherwise; k+ = k if k > 0 and zero otherwise. The number θ(d) is defined as follows: call a subsequence p = (dk+1 , . . . , dk+l ), where 0 ≤ k < rn and 1 ≤ l ≤ rn a positive part of d if all dk+j ≥ 0 and either l = rn or both dk < 0 and dk+l+1 < 0. For such a positive part put θ(p) = l if either l = rs or p = (0, . . . , 0) and θ(p) = l + 1 otherwise. Then θ(d) = θ(p), where we take a sum over all positive subparts of d. In order to compute the tensor product of two indecomposable vector bundles we shall need the following lemma. Lemma 7. Let E be a cycle of projective lines, πi : Ei −→ E two ´etale coverings i = 1, 2 and Ei a vector bundle on Ei . Let E be the fiber product of E1 and E2 over E:

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Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel p1

E −−−−→ E1 ⏐ ⏐ ⏐p2 ⏐π1 π

E2 −−−2−→ E. Denote by π  : E −→ E the composition π1 p1 , then
 π1∗ (E1 ) ⊗ π2∗ (E2 ) ∼ ∗ p∗1 (E1 ) ⊗ p∗2 (E2 ) . =π
 Proof.
By the base change and
projection formula π ∗
p∗1 (E1 ) ⊗ p∗2 (E2 )   π2∗ p2∗ p∗1 (E1 ) ⊗ p∗2 (E2 ) ∼ = π2∗ π2∗ π1∗ (E1 ) ⊗ E2 = π2∗ p2∗ p∗1 (E1 ) ⊗ E2 ∼ π1∗ (E1 ) ⊗ π2∗ (E2 ).

∼ = ∼ =

The following proposition describes the fiber product of two ´etale coverings of a given cycle of projective lines. Proposition 6 ( [Bur03]). Let En be a cycle of n projective lines and πi : Edi n −→ En two ´etale covering of degree di , i = 1, 2. Choose a labeling of the irreducible components of each cycle Ed1 n and Ed2 n by consecutive nonnegative integers 0, 1, 2, . . . , such that πi maps the zero component into a zero component, i = 1, 2. Let d = g.c.d.(d1 , d2 ) be the greatest common divisor and D = [d1 , d2 ] the smallest common multiple of d1 and d2 . Then the fiber product d  = ! E(i) and p1 : E(i) −→ Ed n is the ´etale covering determined by the is E 1 Dn Dn i=1

(i)

assumption that it maps the i-th component of EDn to the 0-th component of (i) Ed1 n . The second morphism p2 : EDn −→ Ed2 n is the ´etale covering, mapping the zero component to the zero component. These properties allow to describe a decomposition of the tensor product of any two indecomposable vector bundles into a direct sum of indecomposable ones. In particular, in the case of a nodal Weierstraß curve we get the following concrete algorithm, obtained for the first time in [Yud01]. Theorem 20 ( [Yud01, Bur03]). Let E be a Weierstraß nodal curve over an algebraically closed field k of characteristics zero, B(d, 1, λ) and B(e, 1, µ) two vector bundles on E of rank k and l respectively, d = d1 d2 . . . dk and e = e1 e2 . . . el . Let D be the smallest common multiple and d the greatest common divisor of k and l. Consider d sequences f 1 = d1 + e1 , d2 + e2 , . . . , dk + el , f 2 = d1 + e2 , d2 + e3 , . . . , dk + e1 , .. . f d = d1 + ed , d2 + ed+1 , . . . , dk + ed−1 , of length D. Then the following decomposition holds:

Vector Bundles on Degenerations of Elliptic Curves

B(d, 1, λ) ⊗ B(e, 1, µ) ∼ =

d 

l

115

k

B(fi , 1, λ d µ d ).

i=1

If some fi is periodic, then we use the isomorphism B(g l , 1, λ) =

l 

√ l B(g, 1, ξ i λ),

i=1

where g l = gg . . . g and ξ a primitive l–th root of 1.    l

Even possessing a complete classification of indecomposable torsion free sheaves on a Weierstraß nodal curve, an exact description of stable vector bundles is a non-trivial problem. It can be shown by many methods that for a pair of coprime integers (r, d) ∈ Z2 , r > 0 the moduli space of stable vector bundles of rank r and degree d is k∗ , see for example [BK3]. However, for applications it is important to have a description of stable vector bundles via ´etale coverings. In order to get such a classification note the following useful fact. Lemma 8 ( [Bur03]). Let E be an irreducible Weierstraß curve. Then a coherent sheaf F on E is stable if and only if it is simple i.e. End(F ) = k. In general, for irreducible curves stability implies simplicity, but in the case of irreducible curves of arithmetic genus one both conditions are equivalent. Then one can prove the following theorem: Theorem 21 ( [Bur03]). Let E be a nodal Weierstraß curve and E a stable vector bundle on E of rank r and degree d, 0 < d < r. Then g.c.d.(r, d) = 1, E∼ = B(d, 1, λ) and d can be obtained by the following algorithm. 1. Let y = min(d, r−d), x = max(d, r−d). If x = y, then d = (0, 1). Assume now x > y. Consider the triple (x, y, x + y) and write x + y = (k + 1)y + s, where 0 < s < y and k ≥ 1. If s > y − s then replace (x, y, x + y) by (s, y − s, y) and say say that (x, y, x + y) is obtained from (s, y − s, y) by the blow-up of type (A, k). If s < y − s then replace (x, y, x + y) by (y − s, s, y) and say that (x, y, x + y) is obtained from (y − s, s, y) by the blow-up of type (B, k). 2. Repeat this algorithm until we get the triple (p, 1, p + 1). Consider the (C1 ,k1 )

sequence of reductions (x, y, x + y) = (x0 , y0 , x0 + y0 ) −→ (x1 , y1 , x1 + (Cn ,kn )

y1 ) −→ · · · −→ (xn , yn , xn + yn ) = (p, 1, p + 1), where Ci ∈ {A, B} and ki ≥ 1 for 1 ≤ i ≤ n. Now we can recover the vector d :

116

Lesya Bodnarchuk, Igor Burban, Yuriy Drozd, and Gert-Martin Greuel

1. Start with sequence α, α, . . . , α, β, which corresponds to the triple (xn , yn ,    p times

xn + yn ) = (p, 1, p + 1). If Cn = A then replace each letter α by the block α, α, . . . , α and each β by the block α, α, . . . , α. Between these new       kn + 1

kn

blocks insert the letter β. If Cn = B then replace each letter α by the block α, α, . . . , α and each letter β by the block α, α, . . . , α. Between these new       kn

kn + 1

blocks insert the letter β again. We have got a new sequence of letters α and β of total length xn−1 + yn−1 with xn−1 letters α and yn−1 letters β. 2. Proceed inductively until we get a sequence of length r with max(d, r − d) letters α and min(d, r − d) letters β. 3. If d > r − d then replace each letter α by 1 and each letter β by 0. In case d ≤ r − d replace α by 0 and β by 1. The resulted sequence is the vector d we are looking for. Example 3. Let rank r = 19 and degree 11. The sequence of reductions is (B,1)

(A,1)

(11, 8, 19) −→ (5, 3, 8) −→ (2, 1, 3). Using the algorithm we get the sequence of blowing-ups (A,1)

α, α, β −→ α, α, β, α, α, β, α, β (B,1)

α, α, β, α, α, β, α, β −→ α, β, α, β, α, α, β, α, β, α, β, α, α, β, α, β, α, α, β and hence d = (1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0). This result was generalized by Mozgovoy [Moz] to get a recursive description of semi-stable torsion free sheaves of arbitrary slope. Note the following important difference between smooth and singular curves of arithmetic genus one. In the smooth case any indecomposable coherent sheaf is either locally free or torsion free and is automatically semi-stable. This is no longer true for singular curves, in particular, in that case there are indecomposable coherent sheaves which are neither torsion nor torsion free. Example 4. Let E be a nodal Weierstraß curve, s its singular point