C h e r n Numbers and
Rozansky-Witten
Invariants of Compact
Hyper-KahIer ManifoIds
Marc Nieper-WiBkirchen Johannes Gutenberg-Universitat Ma inz, Germany
N E W JERSEY
-
r pWorld Scientific LONDON * SINGAPORE
BElJlNG * SHANGHAI
-
HONG KONG
*
TAIPEI
*
CHENNAI
Contents
Preface
vii
Introduction
ix xiii
Notation 1. Compact hyper-Kahler manifolds and holomorphic symplectic manifolds
1.1 Basics on compact hyper-Kahler manifolds . . . . . . . . . . 1.1.1 Holonomy of Riemannian manifolds . . . . . . . . . . 1.1.2 Definition of a compact hyper-Kahler manifold . . . 1.1.3 Holomorphic symplectic manifolds . . . . . . . . . . 1.1.4 Deformations of compact complex manifolds . . . . . 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The K 3 surface . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Hilbert scheme of points on a surface . . . . . . 1.2.3 Construction of line bundles and classes in H2 on the Hilbert schemes of points on surfaces . . . . . . . . . 1.2.4 Hilbert schemes of points on K3 surfaces . . . . . . . 1.2.5 Generalised Kummer varieties . . . . . . . . . . . . . 1.2.6 Further examples . . . . . . . . . . . . . . . . . . . . . 1.3 Characteristic classes . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Symplectic sheaves . . . . . . . . . . . . . . . . . . . 1.3.2 Characteristic classes of symplectic sheaves . . . . . 1.3.3 Chern numbers of holomorphic symplectic manifolds 1.4 The Atiyah class . . . . . . . . . . . . . . . . . . . . . . . . . xix
1 1 1 3 4 7 8 8 11 13 15 16 19 20 20 21 24 26
xx
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
1.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Description in terms of Cech cohomology . . . . . . . 1.4.3 The Bianchi identity . . . . . . . . . . . . . . . . . . 1.4.4 Torsion and the Atiyah class of the tangent bundle . 1.4.5 The Atiyah class of symplectic sheaves . . . . . . . . 1.4.6 Chern-Weil theory . . . . . . . . . . . . . . . . . . . 1.5 On the second cohomology group of a hyper-Kahler manifold 1.5.1 The period map . . . . . . . . . . . . . . . . . . . . . . 1.5.2 A vanishing result for polynomials on H2 . . . . . . . 2 . Graph homology 2.1 The space of graph homology . . . . . . . . . . . . . . . . . 2.1.1 Jacobi diagrams . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Chains of Jacobi diagrams . . . . . . . . . . . . . . . 2.1.3 Glueing legs and product of Jacobi diagrams . . . . . 2.1.4 Subspaces and ideals . . . . . . . . . . . . . . . . . . 2.1.5 The graph homology spaces . . . . . . . . . . . . . . 2.2 Symmetric monoidal categories . . . . . . . . . . . . . . . . 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 k-linear categories . . . . . . . . . . . . . . . . . . . . . 2.2.3 Global sections . . . . . . . . . . . . . . . . . . . . . . 2.2.4 External tensor and symmetric algebras . . . . . . . 2.3 Metric Lie algebra objects . . . . . . . . . . . . . . . . . . . 2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Examples from the category of vector spaces . . . . . 2.3.3 Morphisms between tensor powers of metric Lie algebra objects . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 The PROP of metric Lie algebras . . . . . . . . . . . 2.3.5 The universality of the PROP of metric Lie algebras 2.4 Weight systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Constructions of weight systems . . . . . . . . . . . . 2.4.3 Modules of metric Lie algebra objects . . . . . . . . . 2.5 Operation with graphs and special graphs . . . . . . . . . . 2.5.1 Special graphs . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Closed and connected graphs . . . . . . . . . . . . . 2.5.4 Polywheels . . . . . . . . . . . . . . . . . . . . . . . . .
26 28 29 31 32
34 36 36
37 39
39 39 41 42 44 45 47 47 49 50 51
52 52 54
55 59 61 64 65 65 66 67 67 68
72 75
Contents
x i
2.5.5 The Hopf algebra structure on the space of graph homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Wheeling Theorem . . . . . . . . . . . . . . . . . . . . . 2.6.1 The wheeling element R . . . . . . . . . . . . . . . . 2.6.2 Wheeling and the Wheeling Theorem . . . . . . . . .
76 79 79 80
3 . Rozansky-Witten theory
3.1 The Rozansky-Witten weight system . . . . . . . . . . . . . 3.1.1 The derived category . . . . . . . . . . . . . . . . . . 3.1.2 A metric Lie algebra object in the derived category . 3.1.3 Rozansky-Witten weight systems . . . . . . . . . . . 3.1.4 Properties of the Rozansky-Witten weight system . . 3.1.5 An inner product on the cohomology of a holomorphic symplectic manifold . . . . . . . . . . . . . . . . . . . 3.1.6 Rozansky-Witten invariants . . . . . . . . . . . . . . 3.1.7 Complex genera and Rozansky-Witten invariants . . 3.2 Some applications . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Chebyshev polynomials . . . . . . . . . . . . . . . . . 3.2.2 An application of the Wheeling Theorem . . . . . . . 3.2.3 On the genus tdi(a)tdi(P)of an irreducible holomorphic symplectic manifold . . . . . . . . . . . . . . 3.2.4 The Ln-norm of the Riemannian curvature tensor of a compact hyper-Kahler manifold . . . . . . . . . . . 3.2.5 The Beauville-Bogomolov form . . . . . . . . . . . . 3.2.6 A Hirzebruch-Riemann-Roch formula . . . . . . . .
4 . Calculations for the example series 4.1 More on the geometry of the Hilbert schemes of points on surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The universal family . . . . . . . . . . . . . . . . . . 4.1.2 The incidence variety X[n*n+ll. . . . . . . . . . . . . 4.1.3 Calculations in various K-groups . . . . . . . . . . . 4.1.4 Chern numbers of the Hilbert schemes . . . . . . . . 4.2 Genera of Hilbert schemes of points on surfaces . . . . . . . 4.2.1 Two decomposition results . . . . . . . . . . . . . . . 4.2.2 A structural result on genera of Hilbert schemes of points on surfaces . . . . . . . . . . . . . . . . . . . . . 4.2.3 Genera of the generalised Kummer varieties . . . . .
83 83 83 85 88 90 92 94 95 99 99 100 101
105 105 107 109 109 109 110 112 113 116 116 119 121
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
xxii
4.3 Calculations of the power series A+l B41C+ and D+ . . . . 4.3.1 Bott’s residue formula . . . . . . . . . . . . . . . . . 4.3.2 How to calculate C+ and D4 . . . . . . . . . . . . . . 4.3.3 The calculation of A+ and B+ . . . . . . . . . . . . . 4.3.4 Chern numbers for the example series . . . . . . . . 4.4 Calculations of Rozansky-Witten invariants . . . . . . . . . 4.4.1 A lemma from umbra1 calculus . . . . . . . . . . . . 4.4.2 More on Rozansky-Witten invariants of closed graph homology classes . . . . . . . . . . . . . . . . . . . . . 4.4.3 A structural result on the Rozansky-Witten weights of closed connected graphs on the example series . . 4.4.4 Explicit calculation . . . . . . . . . . . . . . . . . . .
123
123 125 126 128 128 129 131 134 136
Biblzography
141
Index
143
Preface
The purpose of this book is to give a gentle introduction to the theory of Rozansky-Witten invariants of compact hyper-Kahler manifolds. The book should be easily accessible a t the graduate level, only basic knowledge of the geometry of complex manifolds is needed. Of course, a good sense of mathematical reasoning helps a lot. The title of the book is “Chern Numbers and Rozansky-Witten Invariants of Compact Hyper-Kahler Manifolds”. So what to expect from the book? Will it give a complete introduction into the topic of Chern numbers in hyper-Kahler geometry and a complete introduction into RozanskyWitten theory? By all means the answer is no. (And about 150 pages wouldn’t have sufficed!) In fact, the book will explain certain areas I find most interesting in detail. One of these areas is the link of the two parts of the title and most of the book is centred around the fact that Chern numbers are special Rozansky-Witten invariants. I am very grateful to Daniel Huybrechts who aroused my interest in hyper-Kahler geometry as my former thesis advisor. A lot of what I have written down here I learned by working under his supervision. Another big part of the theory I learned from Justin Sawon’s wonderful Ph.D. thesis that was very inspiring to me. The relations between the cobordism class of the Hilbert schemes of points on surface to the cobordism class of the surface itself heavily used in the last chapter of this book were finally introduced to me by Manfred Lehn. So the genesis of this book owes a lot to these mathematicians and also to numerous others not named explicitely. I also want to thank Liu Ling and Ye Qiang from World Scientific who guided me in the process of setting up and realising this book project and the department of mathematics a t the University of Cambridge for the hospitality. There I wrote the book receiving a DFG grant. And last but
vii
viii
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
not least I say a big thank you to my wife, Bettina Wifikirchen. She drew most of the graphics in the book with MetaPost, prepared a huge part of the index and scanned through the final document several times to track down any errors. Cambridge, England, 2003
M. A . Nieper- WiJkirchen
Chapter 1
Compact hyper-Kahler manifolds and holomorphic symplectic manifolds
1.1 Basics on compact hyper-Kahler manifolds We assume that all manifolds in this section are connected.
1.1.1
Holonomy of Riemannian manifolds
Let (X,g) be a Riemannian manifold, i.e. X is a connected smooth differentiable manifold and g E Cm(S2TVX)is an everywhere positive definite symmetric contravariant two-tensor on X . Let V be the Levi-Civita connection on X associated to g. This connection induces for every curve a : [O, 11 + X the parallel transport map
(1.1) Each transport map is an isomorphism of Euclidean vector spaces, the inverse map of P ( a ) is given by P ( 6 ) where 6 : t H a(l - t ) .
Remark 1.1 (Holonomy groupoid) W e can associate to each connected Riemannian manifold ( X , g ) a groupoid Hol(X,g) whose objects are the points of X and whose morphisms are given by all distinct transport maps as defined above. Definition 1.1 (Holonomy group) The holonomy group of X at IC E X is the subgroup Hol, of O(TX,,g,) generated by all transport maps P ( a ) with a : [O, 11 + X a curve and a(0)= a(1)= IC. Let us fix a point x E X and an orthonormal basis 81,. . . ,a, of the tangent space of X at IC. This identifies O(TX,,g,) with O ( n ) and Hol, with a subgroup G of O(n). 1
2
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Definition 1.2 (Holonomy) The conjugacy class [GI of this subgroup in O ( n ) is well-defined and independent of x and called the holonomy of
(X,9 ) . The holonomy group tells us something about the constant tensors of a manifold: Let p , q E No, and set TtX := (TX)*P 8 (TvX)*q. Fix z E X. The holonomy group Hol, acts naturally on the tensor powers TZX,.
Theorem 1.1 (Holonomy principle) Suppose S E Cm(X,TtX) is a covariantly constant tensor, i.e. V S = 0. Then S, is invariant under the action of Hol, for all x E X . Conversely, each S, E TtX, for x E X that is invariant under the action of Hol, extends uniquely to a covariantly constant tensor S E C" (X,TiX) on X . Proof. The idea of the proof is that the covariantly constant tensors are exactly those that are invariant under parallel transport. I7
Example 1.1 (Holonomy of a Kahler manifold) Let (X, g , I ) be a Kahler manifold, i.e. (X,g ) is a Riemannian manifold and I E C"(TVX @ TX) an integrable complex structure such that its Kahler form := g(I.,.) E
H~(x,R)
(1.2)
is closed. Then the holonomy of ( X , g ) lies in U ( n ) c O ( 2 n ) if 2 n is the real dimension of X .
Proof. Recall that the Kahler condition dw = 0 implies that I is covariantly constant, i.e. VI = 0. By theorem 1.1, the holonomy group Hol, for any fixed point x E X respects I,, in other words, it is conjugate to U ( n ) c O ( 2 n ) = O(TX,,g,). Here, we have again chosen an orthogonal I7 basis of (TX, ,9,). On the other hand, every Riemannian manifold with holonomy sitting inside U ( n ) is a Kahler manifold: Fix x E X . Suppose that I , is a complex structure on TX, that is respected by the action of the holonomy group Hol, at x. (Such an I, exists if and only if the holonomy sits inside U(n) if 2 n is the real dimension of X.) Let I E C"(X, T V X@ TX) be the covariantly constant tensor that extends I,.
Proposition 1.1 (U(n)-manifolds are Kahler.) The almost-complex Riemannian manifold (X, g , I ) is in fact a Kahler manifold.
Compact hyper-Kahler manafolds
3
Proof. The tensor I o I E C"(T"X 8 T X ) is also covariantly constant and equals -1 at x , and is therefore -1 everywhere, i.e. I is an almost complex structure. Recall that I defines a complex structure if and only if the Nijenhuis tensor NI E C"(hz(T"X) @I TX) given by
+
+
N I ( X ,Y ) = [ X ,Y ] I ( [ I X ,Y ] [ X ,I Y ] )- [ I X ,I Y ]
(1.3)
for all open subsets U E X and local vector fields X,Y E C"(U,TX) vanishes. As I is covariantly constant and the connection is torsion-free, i.e. V x Y - V y X = [ X ,Y ] ,we can substitute [ I X ,Y ]= V I X Y- I ( V y X ) in N J ( X , Y ) . Similarly, [ X , I Y ] = - V J Y X I ( V x Y ) and [ I X , I Y ] = I ( V 1 x Y ) - I ( V 1 y X ) . Using this and I 2 = -1 yields
+
N I ( X ,Y ) = [ X ,Y ]- V x Y + V y X
(1-4)
which vanishes as V is torsion-free. It remains to show that w := g(I.,.) is Kahler, i.e. closed. It is wellknown that this condition is equivalent to the fact that I is covariantly 0 constant, so the proof is finished.
1.1.2
Definition of a compact hyper-Kahler manifold
Now we shall define the objects of our studies.
Definition 1.3 (Compact hyper-Kahler manifold) A compact Riemannian manifold ( X ,g) of real dimension 4n, n = 0,1,2,. . ., is called a compact hyper-Kuhler manifold, if its holonomy lies inside Sp(n) c O(4n). The manifold ( X ,g) is called irreducible compact hyper-Kuhler if its holonomy equals Sp(n). As Sp(n) c U(2n), by proposition 1.1 the manifold X carries in fact a Kahler structure with Kahler metric g. In fact, even more is true: (Complex structures on hyper-Kahler manifolds) Let ( X ,g) be a compact hyper-Kahler manifold. Then there exist three complex structures I , J , K o n X such that
Proposition 1.2
(1.5)
4
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
and IJ
=
-JI = K ,
and ( X , g ,aI + bJ
JK
=
-KJ = I ,
KI
=
-IK
= J,
(1.6)
+ c K ) for ( a ,b, c) E S 2 c R3 is a Kuhler manifold.
Let us fix an IC E X . As Hol, c Sp(n) c O(4n) after choosing a suitable basis d l , . . . ,a, for TX,, there exist in fact three complex structures I,, J,,K, on TX, that are related with each other as in (1.5) and such that aI, + bJ, cK, for a , b, c E R with a2 b2 c2 = 1 is again a complex structure. By proposition 1.1 we can extend I,, J,, K , to global Kahler structures I , J , K on X . As I , J , K are covariantly constant, the relations (1.5) also extend globally. 0
Proof.
+
+ +
Definition 1.4 (Quarternionic structure) Let ( X , g ) be a Riemannian manifold. We call a family of complex structures ( I ,J , K ) its in proposition 1.2 a quarternionic structure o n ( X , g ) . Proposition 1.3 (Quaternionic structure) Let ( X , g ) be a compact Riemannian manifold that carries a quarternionic structure ( I ,J , K ) . Then ( X ,9 ) is a compact hyper-Kahler manifold.
Proof.
1.1.3
The proof goes along the same lines as the proof of example 1.1. []
Holomorphic symplectic manifolds
Definition 1.5 (Holomorphic symplectic manifold) A pair ( X ,a ) consisting of a compact Kahler manifold X of complex dimension 2n and a section a E Ho(X,Q$) is called a compact holomorphic symplectic manifold if the power an E Ho(X,R2,”)is a nowhere vanishing holomorphic 2n-form on X . (In this case, we call a non-degenerate.) The manifold ( X ,a) is called irreducible compact holomorphic symplec) 0, i.e. X is simplytic if in addition dimHo(X,0%)= 1 and T ~ ( X = connected.
As we shall not be interested in non-compact examples of holomorphic symplectic manifolds, we shall call a compact holomorphic symplectic manifold by abuse of notion just holomorphic symplectic from now on. Remark 1.2 (Trivial canonical sheaf) As g n is nowhere vanishing, the canonical sheaf wx of the manifold X is trivial, in particular q ( X ) = 0 . Later we will see that all odd Chern classes of X vanish.
5
Compact hyper-Kahler manifolds
Remark 1.3 (Non-degenerate symplectic form) The two-form c is nondegenerate, i.e. it induces an isomorphism O x + Rx given by 0 H 4 0 , .)
(1.7)
for all open subsets U of X and 0 E Qx(U).(Here, we have used the natural identification A2Rx N i.e. we view as a section in Ho(X,R2Rx).) This follows from the fact that on is nowhere vanishing.
05,
Corollary 1.1 (Mirror-symmetry of Hodge numbers) The Hodge numbers h P i Q := dimHQ(X,R5) with p , q = 0,. . . , 2 n of the holomorphic symplectic manifold (X, a ) satisfy the following "mirror-symmetry ": h P > 4 = h2n-P,9
(1.8)
Proof. By the triviality of R T , the above isomorphism of O x N R x , Serre duality and Hodge decomposition, we have the following isomorphisms of C-vector spaces: HQ(X,
QP,)
E
H2"-Q(X,
@
Apex)
N
Han-Q (X,
fig)
II
HQ(X, (1.9)
0 Kahler manifolds with vanishing first Chern class are subject t o the following Decomposition Theorem by Bogomolov:
Theorem 1.2 (Decomposition of compact Kahler manifolds with vanishing first Chern class) Let X be a compact Kahler manifold with vanishing first Chern class. Then X has a finite unramified cover as a product of Kahler manifolds of the f o r m Y
(1.10) i= 1
i= 1
where T is a complex torus, each Y, is simply-connected with holono m y Sp(dimRK/4) and each Zi is simply-connected with holonomy SU(dimR &/2) and Ho(X,R i i ) = 0 .
Proof.
See [Beauville et. a1 (1985)] or [Gross et. al. (2003)].
0
Proposition 1.4 (Compact hyper-Kahler manifolds are holomorphic symplectic.) Let ( I ,J , K ) be a quarternionic structure on a Riemannian manifold (X, 9 ) . Let W I , W J and W K E H2(X,R) be the corresponding
6
C h e r n Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
+
Kahler forms. Then D := W J iwK E H 2 ( X , C ) is a holomorphic twof o r m with respect to the complex structure I and ((X,I ) ,). is a compact holomorphic symplectic manifold. If the holonomy of ( X , g ) equals Sp(n) for 4n the real dimension of X , the compact holomorphic symplectic manifold ( ( X ,I ) ,0 ) is irreducible.
Proof. Let X , Y E C'(U)TX) open subset U of X . Then
be two local vector fields of X over an
(1.11)
a$)c H 2 ( X ,C )
which shows that c is in fact holomorphic, i.e. c E H o ( X , (with respect to the complex structure I ) . Obviously, one can write W J = !j(o 8).Therefore,
+
=1 2n
4"
(,>,k,2n-k
(1.12)
k=O
The left hand side of this equation is a nowhere vanishing top-form. Due t o holomorphic degree reasons, the right hand side can be simplified to n 41- (2")unsn, from which follows that u" is nowhere vanishing, i.e. CT is in fact symplectic. Let us now suppose that ( X , g ) is irreducible as a compact hyperKahler manifold. It remains t o show that X is simply-connected and that dim Ho(X,Q$) = 1 (with respect t o the complex structure I ) . The latter assertion follows easily from the holonomy principle, i.e. theorem 1.1: any form in H o ( X ,Q$) is covariantly constant and therefore invariant under the action of the holonomy group at every point. However, as the holonomy equals Sp(n), there exists up to a multiple only one such holomorphic two-form. Analogously, one can show that dim H o ( X , = 1 if k = 0,4,6, . . . , 2 n and dimHo(X,S2$) = 0 if k = 1,3,5,. . . . Therefore, x ( X , O x ) = n 1. Since H'(X,Ox) = 0 , the Decomposition Theorem 1.2 shows that the fundamental group of X is finite. Now, let 7r : -+ X be the universal finite unramified cover of X . The metric 7r*g induces again a structure of an irreducible compact hyper-Kahler manifold on X , so in particular x(X, 0 2 ) = x(X, OX) = n + 1. From this one concludes that the degree [] of the cover 7r is one and therefore that x l ( X ) = 0.
a$)
x
+
Compact hyper-Kahler manifolds
7
So we have proven that every compact hyper-Kahler manifold carries the structure of an irreducible holomorphic symplectic manifold whereas we can use methods of algebraic geometry to study the differential-geometric objects “compact hyper-Kahler manifolds”. The converse -every holomorphic symplectic manifold carries a hyperKahler metric - is also true. The proof makes use of the Calabi Conjecture proven by Yau.
Theorem 1.3 (Holomorphic symplectic manifolds are hyper-Kahler.) Let X be an irreducible holomorphic symplectic manifold. Let (Y E H2(X,R) lie in the Kahler cone, i.e. there exists a Kahler f o r m w whose Kahler class is a. Then there exists a unique Kahler metric g with Kahler class Q o n the underlying real manifold of X such that (X, g) is an irreducible compact hyper-Kahler manifold. Proof. First note that every hyper-Kahler metric g is Ricci-flat ( i e . the holonomy of g lies in SU(2n) when 2n is the complex dimension of X). By the Calabi-Yau theorem, there exists a unique Ricci-flat Kahler metric g with Kahler class a. Here, we need that q ( X ) = 0. This proves the uniqueness of g. It remains to show that ( X ,g) is an irreducible compact hyper-Kahler manifold: by the Decomposition Theorem 1.2 and due to the fact that X is simply-connected, the Kahler manifold (X,g) splits into a product fly=/=,Y , x Zj of simply-connected manifolds with Sp- and SUholonomy. As is unique up to a scalar and non-degenerate, y = 1 and z = 0. 0
n;=/=,
1.1.4
Deformations of compact complex manifolds
In this subsection we want to gather a couple of results on the deformation theory of compact complex manifolds that we will need later on. Let X be a compact complex manifold. A deformation of X is a smooth proper holomorphic map X -+ S where (S,O) is a pointed analytic space (or the germ of such a space) and the fibre XOover 0 E S is isomorphic to
X. Remark 1.4 (Deformation of Kahler manifolds) Let X be Kuhler. Then there exists a semi-universal deformation denoted by X -+ Def(X), i.e. (Def(X),O) is the germ of an analytic space and X -+ Def(X) is a deformation such that for every other deformation Xs 4 S over ( S ,0 ) there ex-
8
Chern Numbers and RW-Inuariants of Compact Hyper-Kahler Manifolds
ists an analytic map (which is in general not unique!) ( S ,0 ) 4 (Def(X),0 ) such that xs = x X D e f ( x ) s. The Zariski tangent space of (Def(X),O) i s naturally isomorphic to H1(X, Ox). IfHo(X,O X ) = 0, i.e. X has n o infinitesimal automorphisms, the deformation Def(X) is universal, i.e. S -+ Def(X) i s uniquely determined.
Theorem 1.4 (Tian and Todorov) Any compact Kahler manifold X with q ( X ) = 0 has unobstructed deformations, i.e. Def(X) i s smooth. Proof. 1.2
For a proof see, e.g., [Gross et. al. (2003)].
[]
Examples
In this section we want to give some examples for irreducible holomorphic symplectic manifolds. They will nearly exhaust the list of known deformation types of irreducible holomorphic symplectic manifolds. 1.2.1
The K 3 surface
Definition 1.6 (K3 surface) A connected compact complex manifold X of dimension two (i.e. a surface) with trivial canonical bundle and vanishing first Betti number bl (X) := rk H1(X, Z) is called a K3 surface.
Example 1.2 (Smooth quartic in P&) A smooth quartic surface S in P& (i.e. a smooth hypersurface defined by a homogeneous polynomial of degree four) is a K3 surface.
Proof.
By the adjunction formula, the canonical bundle of S is given by (1.13)
The Lefschetz Hyperplane Theorem for the pair 5' cf P&tells us that H1(S,Q) = H1(P&) = 0, i.e. b l ( S ) = 0. The connectedness of S follows 0 also from the Lefschetz hyperplane theorem.
Remark 1.5 (K3 surfaces) One can show that in fact x l ( S ) = 0 , i.e. S is simply-connected. I t is a deep result that every K3 surface is deformationequivalent to S and therefore also simply-connected. Furthermore by a result of Siu, every K3 surface is Kahler. Details can be found in [Beauville et. a1 (1985)].
9
Compact hyper-Kahler manajolds
Further examples of K3 surfaces are given by the so-called Kummer surfaces: Let A be a complex torus of dimension two, i.e. A = C2/F, where r c C2 is a discrete lattice of rank four. There is a natural involution map o : A 4 A given by sending [ z l ,t 2 ] [--z1, - t 2 ] . This involution map has exactly sixteen fixpoints. The fixpoint set is given by the image of in A. Let us call the blow-up of the surface A in these sixteen fixpoints A. Near each exceptional curve is covered by three smooth open affines given by UO:= Spec C[zlrtl ,z2rt1 1, U := Spec C[q,E] and 0 := Spec C[q,<] and with the following transition functions: --f
:4
-
21
= q = qJ,
22
= qJ = q.
(1.14)
The involution o on A induces an involution 6 on A. Near the exceptional curves, 6 acts by Zl++-Zl,Z2++-Z2,
q- 4 - t .
q--q,E-E,
(1.15)
The quotient K := A/{ida,6}, called the K u m m e r surface (on A), is smooth: Outside the exceptional curves, 6 acts freely, so we only have to look near the exceptional curves. The involution 6 respects the three open affines Uo, U and 0 and as one can see, the quotients Spec C[Z;',
I,
,
t,"]" = Spec C[z1 1 2 ~ 1 ~t2 12 2,
SpecC[q,J]" = SpecC[q2,J] and
SpecC[ij,i]' = SpecC[ij2,$] (1.16)
are smooth.
Example 1.3 (Kummer surface) The Kummer surface K is a K3 surface.
Proof. The process of blowing-up A doesn't change its odd Betti numbers, in particular the first. In other words, the natural map H'(A,Q) 4 H1(A,Q) is an isomorphism. By a result of Grothendieck (see [Grothendieck (1957)]), the first cohomology group with rational coefficients of the quotient K is given by those classes in H1(A, Q) being invariant under 6,i.e. by those classes in H1(A,Q) being invariant under o. However. since
H1(A, Q) = { a : I?
4
Q : a is Z-linear}
(1.17)
and a*(&)= -a for these a, there are no invariant classes. Therefore
b l ( K ) = 0.
10
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
To show that W K is trivial, we give a nowhere vanishing holomorphic two-form on K : The canonical two-form dzl Adz2 on C 2 induces a nowhere vanishing two-form w on A. Let us denote the pull-back of this two-form on by 2 . This two-form is invariant under 6 and nowhere vanishing outside the exceptional curves. Therefore, it induces a holomorphic two-form W K on K . It remains t o show that W K is nowhere vanishing near the exceptional locus of K : Using the coordinates given in (1.16), the form W K is given by
a
WK
1 = -d(z:)
Therefore,
42122
WK
1 2
1
A d(zz) = -d(q2) A dJ = --d(ij2)
is nowhere vanishing.
2
Ad[.
(1.18)
cl
Proposition 1.5 (Numerical invariants of K3 surfaces) Let X be a IT3 surface. T h e n x(X) = 24 and dimHo(X, 52%) = dimH2(X,O x ) = 1 and
dimH1(X, Rx) = 20. (1.19)
I n particular b2(X) = 22
Proof. As wx = Q$ is trivial, we have dimHo(X,a$)= 1 and therefore, by Hodge decomposition, dim H2(X,O X )= 1 as well. By Noether's formula, we have (1.20) i.e. the topological Euler characteristic is 24. As the Euler characteristic is also the alternating sum of the Betti numbers and bl(X) = b Z ( X ) = 0, we 0 conclude that dim H1 (X, Ox)= 20. The K3 surface are the basic examples for irreducible holomorphic symplectic manifolds:
Example 1.4 (Holomorphic symplectic manifolds in dimension two) The irreducible holomorphic symplectic manifolds in dimension two are exactly the K3 surfaces.
Proof. Every K3 surface X is simply-connected and has up to a scalar a unique nowhere vanishing holomorphic two-form o E Ho(X,Q;). ). Therefore, X is irreducible holomorphic symplectic. On the other hand, an irreducible holomorphic symplectic manifold X of complex dimension two has trivial canonical bundle and bl(X) = 0 as X I3 is simply-connected. Therefore, X is a K3 surface.
Compact hyper-Kahler manifolds
1.2.2
11
The Hilbert scheme of points on a surface
Let X be a complex surface and n = 0,1,2, .... A family of zerodimensional subschemes of X of length n parametrised over S is, by definition, an analytic subspace 2 of S x X which is finite of degree n over S. Let us consider the presheaf (i.e. the contravariant functor to the category of sets) H on the category of analytic spaces that maps each analytic space S to the set of flat families of zero-dimensional subschemes of X of length n parametrised over S. By a result of Douady (following Grothendieck's construction of the Hilbert scheme), this functor is representable by an analytic space Xi"].
Definition 1.7 (Douady space) We call Xin] the Douady space of zerodimensional subspaces ((, 0 ~of )length dimc Oe = n o n X or the Douady space of n points o n X .
In fact, the C-valued points of X [ " ] are exactly the zero-dimensional subspaces of length n of X . Using the fact that X is smooth, compact and of dimension two, one can show that X [ " ] is a smooth and compact manifold of dimension 2n. Furthermore, X["l is Kahler, respectively projective, if X is. We denote the analytic space of unordered n-tuples of points of X by X ( " ) := X n / B n . One can show that there exists a natural morphism p : X["l + X ( " ) that maps a C-valued point 6 to its support supp ( := E(1ength O E , ~ . z) E X ( " ) .
(1.21)
XEE
Definition 1.8 (Douady-Barlet morphism) We call p : X["I Douady-Barlet morphism.
+ X(")
the
Remark 1.6 (Hilbert schemes and Douady spaces) I n case X is algebraic, the Douady space corresponds via the " G A G A principle" to Grothendieck's Hilbert scheme of zero-dimensional subschemes of length n of X and the Douady-Barlet morphism corresponds to the Hilbert-Chow morphism in the algebraic setting. Therefore we can work with methods of algebraic geometry whenever X is projective. B y abuse of notation, we will call the Douady spaces from now on also Hilbert schemes, and the Douady-Barlet morphism Hilbert-Chow morphism. Let us say a little bit more about the structure of the Hilbert schemes at this point.
12
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Let X g be the big diagonal in X" consisting of all n-tuples of points on X for which a t least two points coincide. Set X g ) := n-(Xg) where n- : X" -+ X ( n ) is the quotient morphism. Let X,(") be the open subset of X g ) consisting of those unordered n-tuples of points on X for which exactly two points coincide.
Remark 1.7 (Hilbert-Chow morphism) The singular locus of X ( " ) is exactly X p ) . The Hilbert scheme Xin] is a resolution of these singularities via the Hilbert-Chow morphism. We shall define several open subsets of the various spaces considered: We set X p ) := X ( n )\ ( X c ' \ X i n ) ) , i.e. the open subset of those unordered n-tuples for which at most two points coincide. Furthermore, we set X P ' := p-'(Xin)) and X r := r-'(X,'"'). The following holds true:
Remark 1.8 (Beauville) Xi"] \ Xf"' is a subspace of codimension two in XIn]. The diagonal A := X g n X y is a smooth closed subspace of codimension two in X p . Let 77 : X y + X y be the blow-up of X y along the diagonal and E its exceptional divisor. The action of the symmetric group G , prolongs to the blow-up. X f / G n can be identified with Xi"]. I n fact, one has the following commutative diagram
I-
d
(1.22)
where the map on the left is the quotient map. The ramification locus of 0 is given by E . One can see this by local calculations. From this, one can easily conclude how the Hilbert schemes look like for low n:
Example 1.5 (Hilbert schemes in low dimensions) There are natural identifications:
and
91.230
Compact hyper-Kahler manafolds
1.2.3
13
Construction of line bundles and classes in H2o n the Hilbert schemes of points o n surfaces
Let X be any compact Kahler surface and n = 0 , 1 , 2 , . . . . Let C E Pic(X) be any invertible sheaf on X . The invertible sheaf n
8 pr,tC E Pic(Xn)
(1.24)
i=l
where pri : X" t X denotes the projection on the ith factor comes with a natural linearisation for the symmetric group 6,.As the isotropy subgroups of all points of X act trivially on this sheaf, it descends to an inLet us denote by L["] the pullback of C(")by vertible sheaf L(")on X("). the Hilbert-Chow morphism.
Remark 1.9 (Homomorphism between Picard groups) The map Pic(X) 4 Pic(Xln1),c ++
dn1
(1.25)
is a homomorphism of abelian groups.
The canonical sheaf of Xi"] lies in the image of this map:
Proposition 1.6 (Canonical sheaf) The canonical sheaf of XIn] is given by WX[*] = wx[nl.
(1.26)
Proof. We shall make use of remark 1.8. The canonical sheaf of X y is given b y WX;
= v*wx; @
O ( E )= v * 7 r * ( W p I X ( " ) ) @ O ( E ) .
(1.27)
On the other hand,
(1.28) so
(1.29) AS… : Pic (X[n])… Pic(Xn ) is injective, this Yields (1.30)
14
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Finally note that X["] \ proves the proposition.
Xpl is closed of codimension two in X["], so this 0
Now we'd like to do with cohomology classes what we have done for invertible sheaves. Let a E H2(X,C ) be any cohomology class. The class n
C p r f a E H ~ ( x C) ~,
(1.31)
i= 1
is invariant under the action of 6,. Therefore there exists a unique class a(n)E H2(X(n),C ) with . r r * ~ ( = ~ ) C;=,prfa, where 7r : X" t X(") is the canonical projection. Pulling back along p, this induces a class [ n ] in H2(X["],C ) .
Remark 1.10 The map
(Homomorphisms between second cohomology groups)
H2(X,C ) -+ H2(X["],C ) ,a H a["]
(1.32)
is C-linear and respects holomorphic forms.
The maps from the Picard group and the second cohomology group, respectively, of the surface to the corresponding group of the Hilbert scheme are compatible with taking the first Chern class: Proposition 1.7 (First Chern class of invertible sheaves on Hilbert schemes) For each line bundle L on X , we have
q(L["]= ) (q(L))["l.
(1.33)
Proof. This follows easily from the naturality of the Chern class and the 0 commutative diagram of remark 1.8. Proposition 1.8 (Symplectic forms on Hilbert schemes) If a is holomorphic symplectic, i.e. is a nowhere vanishing holomorphic two-form in Ho(X,fl$), a["] is as well holomorphic symplectic, i.e. (a[nl)nE Ho(X,fl$nl) is nowhere vanishing.
Proof.
By proposition 1.6, the canonical class of the Hilbert scheme is = = Ox[=], i.e. it is trivial. Therefore, is given by #0 either everywhere non-vanishing or everywhere vanishing. That (a["])" follows easily from a local calculation away from the exceptional divisor
p-l(XA"').
0
Compact hyper-Kahler manifolds
1.2.4
15
Hilbert schemes of points on K3 surfaces
By proposition 1.8, we see that the Hilbert scheme Xi"] for n = 0,1,2,. . . and a compact Kahler surface X is holomorphic symplectic whenever X is holomorphic symplectic:
Example 1.6 (Hilbert schemes of points on holomorphic symplectic surfaces) If CJ E Ho(X,L?$) is a holomorphic symplectic form on X , the Hilbert scheme (X["l,&I) is holomorphic symplectic. Remark 1.11 (Holomorphic symplectic surfaces) The classification of surfaces tells us that the only holomorphic symplectic surfaces are the K3 surfaces and complex tori of dimension two. We want to know if the holomorphic symplectic manifolds given in example 1.6 are irreducible as holomorphic symplectic manifolds.
Remark 1.12 (Beauville) One can show that the fundamental group of Xin] for n 2 2 is the maximal abelian quotient of the fundamental group of X . For X a K3 surface, we therefore have .rrl(X["I) = 0 and for X a complex torus of dimension two, we have T ~ ( X [ ~=] )Z4. For a proof of this fact, see [Beauville (1983)]. This rules out the Hilbert schemes of points on an complex torus of dimension two to be irreducible hyper-Kahler manifolds. For the Hilbert schemes of points on a K3 surface however, we have:
Example 1.7 (Hilbert scheme of points on a K3 surface) Let X be a K3 surface. The Hilbert scheme XI"]is an irreducible holomorphic symplectic manifold. Proof. It remains t o show that dimHo(X["],R~r,l)= 1. This is due t o the following remark on the second cohomology group of XIn]. Recall that H'(X, C) = 0 and dimHo(X, 0%)= 1. 0
Remark 1.13 (Beauville) If X is a surface with H 1 ( X , C ) = 0 , then there exists for n 2 2 a natural isomorphism H2(X["l,C ) = i(H2(X,C ) )@ C . [El
(1.34)
compatible with Hodge stmctures. Here, i : H2(X, C ) -+ H2(X["I, C), a H a["]i s the morphism defined above and [El E H1>'(X,Z) is the class of the exceptional divisor p-l(X:)). A proof of this statement can again be found in [Beauville (1983)l.
16
Chenz Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Remark 1.14 (Beauville) The construction of the Hilbert schemes of points o n a K3 surface X gives u s therefore one example of irreducible holomorphic symplectic manifolds in each euen complex dimension. Note that XI0] is simply a point and X [ ' ] the K3 surface itself. W e have
b2(XIo1) = 0 ,
b2(X[11)= 22,
and bz(Xln])= 23
(1.35)
for n 2 2.
1.2.5
Generalised Kummer varieties
Let A be a complex torus of dimension two and n = 1 , 2 , 3 , ., . . Though A["]is not irreducible as a holomorphic symplectic manifold (recall that the fundamental group is Z4), an irreducible one sits inside. We shall give the details of what this means in this subsection. Fix an origin 0 E A. This gives rise to a summation morphism A x A + A. As the summation is commutative, we get an induced summation morphism A(,) + A. Let us denote the composition of this morphism with the Hilbert-Chow morphism p : A["]--f A(") by cr : A["]+ A. The action We denote the fibre a-l(O) c A[,] over the origin by The restriction of this action to of A on itself induces an action on A[,]. A""]]shall be denoted by u :A x
A""]]+ A[,].
(1.36)
For example if a is a zero-dimensional subspace of A given by the n distinct points a l , . . . ,a, and a E A, the image u(a,a ) is then the subspace given by the points a a l , . . . , a +a,.
+
Remark 1.15 (Decomposition of the Hilbert scheme of points on a twodimensional complex torus) The maps u and cr fit into the following cartesian diagram:
A x S[[n]]…………A[n] pr1
A
n.
(1.37) A,
where the map at the bottom is just given by multiplication with n. A s this map is a n unramified cover of degree n4, so is u. I n other words, the morphism cr is isotrivial.
17
Compact hyper-Kahler manifolds
In particular, all the fibres of CJ are isomorphic. As there are smooth fibres, one concludes that all fibres have t o be smooth. Thus we have proven: Proposition 1.9 (Beauville) The space A[[^]] is a compuct Kiihler manifold of dimension 2n - 2. Definition 1.9 (Generalised Kummer variety) The compact Kahler manifold A[[n[ is called the nth generulised Kummer variety (on A). The reason for this naming is the following: Example 1.8 (Kummer variety) The second generalised Kummer variety A"']] on A is isomorphic t o the Kummer variety constructed from A. (The first one, A[1]] consists just of a point.)
Proof. Recall A[2] = A2/& and the notions of example 1.3. There is a commutative diagram
K - A - A
1 -1
A[']
1A' -A1' 1 1
a- ( a ,-a)
c--
(a1,az)-ai
(1.38) +a2
A-A-A. The upper row is the fibre over zero of the summation morphism at the 0 bottom of the diagram. This proves K = A[n]
We define a map H2(A,C)
--f
H2(A"n11,C) respecting Hodge structures
by
, ,"rill .- ,in] H
(1.39)
IA[[nll.
Analogously, we define a map Pic(A) 4 Pic(A""I1) by
L
H L"nll
:=
A"] lA&q.
The behaviour of these two maps under following lemmata:
Y
(1.40)
is the subject of the two
Lemma 1.1 (Decomposition of locally free sheaves) For eve? invertible sheaf L on X , we have V*L["]
= pr;Ln g p r ; ~ " " ] ] .
(1.41)
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
18
Proof. As A[n] is simply-connected by the following proposition, its Picard group is discrete. Therefore, v*C["ll(alxA~[nll is independent of u E A and therefore isomorphic t o L""]].By the seesaw principle, it follows that v*C["]splits, i.e. dnI = prTC1 @ pr3L[In]1with L1 E Pic(A). It remains to calculate L1. Set L : A H A x A[[nIl,u++ (a,&*), where a0 is any zero-dimensional subspace of A concentrated in 0. We have c1 = L*v*dn1=
(p0 v
0
L)*L(")= (u H ( a , . . . ,a))*L(")=
cn.
(1.42) []
Proposition 1.10 (Beauville) The generulised K u m m e r varieties A[[n]] are simply-connected.
Proof. The exact homotopy sequence of the fibration u with typical fibre yields:
... ...
- - - TZ(A) T~(A)
T,(A""J) ~o(A"~l1)
~i(A("1) TO(A[~])
... TO(A). (1.43)
The proposition follows therefore because T ~ ( A 7ro(A["I) ), and ro(A) are trivial and because the map ~l(A""11)+ T ~ ( A )is the isomorphism of remark 1.12. 0
Lemma 1.2 (Decomposition of classes in the second cohomology) For every class a E H2(X,C ) , we have
v*&I = n . pr;a
+ pr;a""]l.
(1.44)
Proof. The proof goes along the very same lines as the proof of lemma 1.1. Here, the splitting v*a[%l= prTa1 + pr3az follows from the C ) = 0. Kunneth decomposition theorem using that H1(A[["]], Set b1 : A + A XA""]],a H ( a ,6) and LZ : AIInll + A XA""]],E H (O,<), where
+ n
(1.45)
and (1.46)
Compact hyper-Kahler manifolds
where i : proposition.
+
AInl is the natural inclusion map, thus proving the
0
Lemma 1.3 (nivial canonical bundle) The canonical bundle is trivial.
Proof. Ax
19
WA[[n]]
of
As v is unramified and the canonical bundle of A is trivial, on we have (1.47) []
From this follows:
Example 1.9 (Beauville) The generalised Kummer variety A together E H2(A, flA[[n]]) for any non-vanishing section a E Ho(A, a;) is with an irreducible holomorphic symplectic manifold of dimension 2n - 2 and second Betti number 7 (for n 2 2). is not everywhere vanishing follows again from a local Proof. That calculation away from the exceptional locus. As the canonical sheaf of is symplectic. is trivial, this suffices t o show that To conclude the rest one uses the following remark, from which it follows 0 that Ho(A[[nll,Q~[[nll) = 1 for n 2 2 in particular.
Remark 1.16 (Beauville) There exists for n 2 2 a natural isomorphism
~ ~ ( ~ "c) ~= 1 ~1 (, H ~ (c)) A , CBc . [F]
(1.48)
compatible with Hodge structures. Here, j : H2(A, C) --t H2(Al[n]l,C),a H is the morphism defined above and [F] E H1>'(A, 2) is the class of the exceptional divisor P-l(Xg))lA[[n]]. A proof of this statement can again be found an [Beauville (1983)l. 1.2.6
Further examples
With these two example series - Hilbert schemes of points on a K3 surface and generalised Kummer varieties - we have nearly exhausted the list of known irreducible holomorphic symplectic manifolds up t o deformation. Up t o the time of this writing (August 2003), only two more examples are known:
20
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Example 1.10 (O’Grady) There exists an example of a ten-dimensional irreducible holomorphic symplectic manifold with second Betti number being at least 24. This example has been constructed by O’Grady in [O’Grady ( 1999)]. Example 1.11 (O’Grady) O’Grady has also constructed a sixdimensional example of an irreducible holomorphic symplectic manifold with second Betti number eight. Details can be found in [O’Grady (2000)]. From now on, when we speak of the main examples or the two example series of irreducible holomorphic symplectic manifolds we mean the manifolds Xin], n 2 0 and n 2 1, for X a K3 surface and A a complex torus of dimension two.
Characteristic classes
1.3
In this section we want to present some result on characteristic classes and the cohomology of holomorphic symplectic manifolds in general that are easily t o grasp and/or will be used in the following theory.
1.3.1
Symplectic sheaves
Let X be a complex manifold.
Definition 1.10 (Symplectic sheaf) A pair (&, g) consisting of a locally free Ox-module E and a section E Ho(X,A2EV)= Hom(A2E,0x) is called a symplectic sheaf if the natural map V, : & -+&” given by
(1.49) is an isomorphism of Ox-modules. In what follows, let ( E , g) be a symplectic sheaf on X.
Definition 1.11 (Symplectomorphisms) The subsheaf of spanned by all sections A E w ( & ) ( U )with
AS, s’) + g(s, As‘) = 0 and U an open subset of X and s, s‘ E & ( U )is written End(&,a).
End(€) (1.50)
Compact hyper-Kahler manifolds
21
Remark 1.17 (Symplectomorphisms) Let us identify & and IV via the isomorphism V,. This isomorphism identifies the sheaf End(&) = &v 8 & of endomorphisms of & with the sheaf &@' of Ox-modules. Under this identification, the subsheaf End(&,a ) of End(&) becomes the subsheaf S2& of Em2, Proof. We shall give a proof of the last statement: Let U be an open subset in X and A E M ( & , a ) ( U ) .By identification of & with IV we can view A as a section E (&@')(U). Let y : &@' -+ &@2 be the isomorphism that exchanges the two tensor factors. We then calculate
A
(1.51)
i.e. A is G2-invariant, whichproves thelaststatement.
1.3.2
[]
Characteristic classes of syrnplectic sheaves
Let X be a complex manifold and (&, a) a symplectic sheaf on X . As a symplectic sheaf is isomorphic to its dual, one can easily see for its Chern classes that cz(&)= C i ( & V )
=
(-l)iq(€)
(1.52)
for i = 0 , 1 , 2 , . . . , i.e. the odd Chern classes of & vanish up to two-torsion. In fact, the odd Chern classes vanish completely.
(Vanishing of odd Chern classes) For i = 1,3,5,. . . an odd integer, the Chern class ci(&) vanishes.
Proposition 1.11
Proof. We prove the proposition by induction on the rank &. For rk & = 0 the claim is obvious. By the splitting principle, there exists a manifold -% and a holomorphic map 7r : X + X such that the induced map on cohomology is injective and the pullback T * & has a locally free subsheaf of rank one. To prove that certain Chern classes of & vanish, it is enough to prove that the corresponding
22
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
Chern classes of T*& vanish. Therefore, without loss of generality, we can assume that & already has a locally free subsheaf C of rank one. Let CL be the a-orthogonal subsheaf to C of 1.Since u is symplectic, is C' locally free of rank n - 1 and C is a subbundle of C'. We have the following short exact sequences of bundles on X: 0-
C-&
-&/C
-0
and
Since a induces a symplectic structure on CL/C, by induction all odd Chern classes of this bundle of rank rk& - 2 vanish. Furthermore note that u induces an isomorphism between C and (&/C')*, so all odd Chern classes of C @ &/L' vanish. Now, the two exact sequences give us c(&)= c(C @ &/LL). c(.C'/C). Therefore, we can conclude that all odd Chern classes of & vanish. 0
For real vector Ijundles, one can define Pontrjagin classes as Chern classes of the complexified bundles: Definition 1.12 (Pontrjagin classes) Let A4 be a C"-manifold and Y a locally free C"(M)-module of finite rank, i.e. a real vector bundle of finite rank. The ith Pontrjagin class p i ( V ) of V for i = 0,1,2,. . . is defined to be Pi(v) := ( - l ) i ~ 2 i ( V@R C ) E H4i(M,Z).
(1.53)
Let 3 be any locally free sheaf of finite rank of Ox-modules. The Chern classes of F are linked with the Pontrjagin classes of 3 viewed as a sheaf of C" (X)-modules: Proposition 1.12 (Pontrjagin classes) For i = 0 , 1 , 2 , . . . we have
(1.54)
23
Compact hyper-Kahler manifolds
Proof.
By the properties of the Chern classes,
(1.55)
0 Corollary 1.2 (The Pontrjagin classes are determined by the Chern classes.) The collection of the Chern classes of F determines the Pontrjagin classes o f 3 . The first relations between the Pontrjagin classes and the Chern classes of 3 are
(1.56)
For the symplectic sheaf E these relations are invertible modulo torsion, i.e. if we view the classes as classes in rational cohomology. Proposition 1.13 (The Chern classes are determined by the Pontrjagin classes.) The collection of rational Pontrjagin classes of E, viewed as a sheaf of CM(X)-modules determine the rational Chern classes of E .
Proof. Since the odd Chern classes of E vanish by proposition 1.11, we just have to express the even rational Chern classes in terms of the rational Pontrjagin classes. By proposition 1.12, (-l)i
C2i(E)
= ,-Pi(€)
+
Pi(CZ(€),
1 . .
,C
2 4 E ) )
(1.57)
over H*( X ,Q) for i = 1,2,3, . . . where P is a polynomial in i - 1variables over Q . Inductively, we can therefore express the c z i ( € ) as polynomials in terms of the p k ( & ) , at least rationally. 0
Remark 1.18 (Chern and Pontrjagin numbers) Chern numbers of symplectic sheaves, i. e. integrals ouer Chern classes, are therefore differentiable (and in fact, topological) invariants.
24
Chern Numbers and RW-Invariants of Compact
1.3.3
Hyper-Kahler Manifolds
Chern numbers of h o l o m o r p h i c symplectic m a n i f o l d s
We shall focus now on the first part of the title of this book, i.e. giving some general results on Chern numbers of holomorpbic symplectic results. Let X be any compact complex manifold of dimension n.
Definition 1.13 (Characteristic numbers) A characteristic number of X is any integral Jx a for all a E H*( X ,Rfu) lying in the subring generated by the Chern classes C ~ ( X ) , C ~ ( X ) , C. .~.(. X ) , Remark 1.19 (Chern numbers) All characteristic numbers of X can be given as universal linear combinations of characteristic numbers of the form c x ( X ) , where X = 1x12x23A3.. is any partition of n and cx = A1 A 2 5 3 . . . . These characteristic numbers are called Chern numbers. c1 c2 cg By results of the previous subsection, the Chern numbers of a holomorphic symplectic manifold X of dimension n can be expressed in terms of Chern numbers of the form c 2 x ( X ) , where X is a partition of n / 2 .
s
sx
Example 1.12 (Chern numbers of two-dimensional holomorphic symplectic manifolds) There is only one (non-trivial) Chern number of a holomorphic symplectic manifold X of dimension two, namely cZ(X).
,s
The Hirzebruch-Riemann-Roch theorem gives us universal relations between the Hodge numbers hpyq(X) := dimHQ(X,R;) and some characteristic numbers of X :
Definition 1.14 (Hirzebruch Xy-genus) Let x P ( X ) be the holomorphic Euler characteristic of the bundle R;, in other words
(1.58)
We call
(1.59) the Hirzebruch Xy-genus of X
Remark 1.20 (Hirzebruch-Riemann-Roch theorem) By the HirzebmchRiemann-Roch theorem, the Xy-genus is expressible in terms of character-
25
Compact hyper-Kahler manifolds
istic numbers by
c n
X ? m=
p=o
YP
J
ch(Rp,) . %X).
(1.SO)
x
Inverting these relations allows u s to express some characteristic numbers in terms of Hodge numbers.
Example 1.13 (Sawon) In dimensions two, four and six, the relations provided by the Hirzebruch-Riemann-Roch theorem are enough to express all characteristic numbers of a holomorphic symplectic manifold X of dimension n in terms of its Hodge numbers: CZ(X) = 12X0(X)
(n = 21,
c ~ ( X= ) 248xo(X)- 2x1(X),
( n = 4)
cq(X) = 24xo(X)- 6x1(X),
(1.61)
~ 8x2(X), ~ cg(X) = 7272x0(X)- 1 8 4 -
(n = 6)
c ~ ( X ) C ~ (=X1368xo(X) ) -2 0 8 ~ 8x2(X), ~
+
cg(X) = 36xo(X)- 16x1(X) 4x2(X).
For a better readability, we have suppressed the integral signs on the left hand sides. This allows us t o calculate some Chern numbers (all for dimensions less or equal six) of the two example series as their Xy-genus is known. However, this method is not sufficient t o calculate all Chern numbers. In the last chapter, we demonstrate how to calculate all Chern numbers. From a result of Salamon, a certain divisibility of the Euler number e(X) = cn(X) of a holomorphic symplectic manifolds follows:
,s
Proposition 1.14 (Salamon, Gritzenko and Hirzebruch) Let X be a 4n-
dimensional differentiable manifold that can be equipped with the structure of a compact hyper-Kahler manifold. Then n . e ( X ) is divisible by 24. Proof. We can assume that X is equipped with the structure of a holomorphic symplectic manifold of complex dimension 2n. For a compact complex manifold M of complex dimension m, Salamon (see [Salamon (1996)]) has shown that 2m
ci(M)c,-l(M) JM
1
= C(-l)’(6j2 - 5 d ( 3 d + l))bj(M), j=O
(1.62)
26
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
provided that hP>Q(M) = h"-p?q(M). Specialising to our case, this reduces to 4n
c(-l)i6i26i
= n(6n
+l)e(X),
(1.63)
i=O
where we have used that e ( X ) = sition.
zfzob i ( X ) . Now use the following propo0
Proposition 1.15 (Wakakuwa) The Betti numbers of a compact hyperKahler manifold X of real dimension 4 n obey
b2i(X) 2 1 and
bzi+l
=0
(mod 4)
(1.64)
for i = 0,1,. . .,2n. Proof.
The original proof can be found in [Wakakuwa (1958)].
0
1.4 The Atiyah class The Atiyah class is the main ingredient in the algebraic construction of the Rozansky-Witten invariants to be defined later. This section is therefore reserved for a thorough discussion of this holomorphic analogue of the curvature tensor in differential geometry. 1.4.1 Definition Let X be a complex manifold and E a locally free sheaf of Ox-modules on X . A holomorphic connection V of E is a morphism E + fix @& of sheaves of C-modules such that V fulfils the Leibniz rule
V(fs) = f Vs
+ df 8 s
(1.65)
for every open subset U in X and local sections f E ( 3 x ( U ) and s E €(U). We write VQfor V(6, .) : E ( U ) + E ( U ) if 6 E Q x ( U ) is a local vector field.
Example 1.14 (The space of holomorphic connections is affine.) Let V be a holomorphic connection. For every a E H o ( X ,fix @ u ( E ) ) ,the sum
V+cU is again a holomorphic connection.
(1.66)
Compact hyper-Kahler manifolds
27
Definition 1.15 (First-order jet bundle) The sheaf
J(E)
:=&@ax@&
(1.67)
of C-modules endowed with the structure of an Ox-module given by
f . (S,Q: @ s')
= (f S,f Q: 8 S'
+ df @ s')
(1.68)
for every open subset U in X and local sections f E Ox(U) and s, s' E E(V) is called the (first-order) jet bundle of E. The importance of the jet bundle is that it allows us to view connections as morphisms of Ox-bundles:
- - --
Lemma 1.4 (Holomorphic connections as splits) The sequence 0
Ox@€
J(E)
&
0
(1.69)
where the second and third arrow are induced by the canonical inclusion, respectively projection of C-modules is a short exact sequence of Ox-modules. A n y split u : E -i J ( E ) can be written as a map of C-modules as (ids, V), where U is a holomorphic connection of E . O n the other hand, every holomorphic connection defines a split in this way.
Proof. One verifies easily that the short exact sequence of sheaves of C-modules is in fact a sequence of Ox-modules. By construction of the Ox-module structure of J ( E ) , the second component V of a section is a [] holomor phic connection. As E is a locally free, the groups Extl(E,Rx @ E ) and H'(X,Ox @
End(€)) coincide.
The short exact sequence (1.69) defines an element in
this group. Definition 1.16 (Atiyah class) The extension class of (1.69) lying in H1(X,Rx )€(dnE@, is called the Atiyah class Q:& of E. In other words, the Atiyah class of E is the obstruction for the existence of a global holomorphic connection on E.
The Atiyah class behaves well under pullback via holomorphic maps:
Lemma 1.5 (Naturality of the Atiyah class) Let Y be another complex manifold and f : Y ---f X a holomorphic map. Then (1.70)
28
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Proof.
The natural map
H1(X, O x
@End(&))
4
H1(Y, f *Ox@End( f * E ) )
(1.71)
maps the Atiyah class of & to the extension class CY of the top row of the commuting diagram 0
0
- f*Rx
f*(J(&))
1
1
- Qy
J-(f*E)
- - 1=f*&
f*&
0
(1.72)
0.
The natural map
H1(Y, f * O x @,(f*&))
4
H’(Y,Oy
(1.73) @m(f*&))
in turn maps G t o the extension class of the lower sequence, which is the Atiyah class of f * E . 0
1.4.2
Description in terms of cech cohomology
Let 0
--- E’
&
E”
0
(1.74)
be a short exact sequence of Ox-modules, and let us assume that E“ is locally free. Let ( U Z ) , Ebe ~ a covering of X by open subsets such that local splitting maps 0, : €”(U,) -+ E(U,) exist. The differences 0,j(I,n(I, O,luzn(I, for all z , E ~ I obviously factor uniquely through E’, i.e. they are given by maps : &’’(U, n U,) + &’(U, n U,).
(1.75)
It is a well-known fact that the extension class Q of the short exact sequence (1.74) lying in Extl(E”,&’) = H1(X,&’IV@ &) = H1(X,E”VB E ’ ) is given by the Cech cocycle (Q%j),,,EI
E C1((U,),E”V c 3 E’).
(1.76)
Applying this t o the short exact sequence (1.69), we see that the Atiyah class QE of the Ox-module & is given by the Cech-cocycle
(V,l(~,nu,- V j Iu,nu,)2,3cI E C1((uz), Ox @End(&))
(1.77)
where ( U , ) , ~ Iis a covering of X by open subsets and V, are holomorphic connections of E 1 (I,.
Compact hyper-Kahler manifolds
29
We shall use this description to show that the Atiyah class for invertible sheaves is basically the first Chern class:
Example 1.15 (Atiyah class of invertible sheaves) Let C be an invertible sheaf on X. Then i (1.78) q ( L )= - C q 2n in H1(X,Rx) = H (
x,Rx @ End(L)).
Proof. Let (Ui)iElbe a cover of X by open subsets such that there exist nowhere vanishing sections si E C(Ui). The first Chern class of C is given by the Cech cocycle &d(log(z))i,jEl E C1((Ui),Rx). On the other hand, local holomorphic connections Vi for L defined on Ui are given by (1.79)
+
where E C ( V ) is a local section of C over the open subset V of U . A Cech cocycle for the Atiyah class is therefore given by ((Vi -Vj)(l))i,jEr = 0 (d log E C1((Ui),R x ) which proves the claim.
2)
The B i a n c h i i d e n t i t y
1.4.3
Let the morphism (1.80) be given by
( a @ A ) @ ( P @ ’H ) (Q.P)@[A,BI
(1.81)
for all open subsets U of X and local sections a,,B E Rx(U) and A , B E M ( € ) ( U ) .Furthermore, we define a morphism
s : R x @ M ( E ) 63 R x 63 End(Ox)
--t
S2RX @ End(€)
(1.82)
given by ~
B 6A3 ~ 6 (7 3 80)
-+
( p . 7 )B A
(1.83)
for all open subsets U of X and local sections a, p, y E R x ( U ) , 6 E O X ( U ) and A E M ( E ) ( U ) . These two morphisms of Ox-modules induce morphisms L , and S, respectively, in cohomology.
30
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Proposition 1.16 (Bianchi identity) The Atiyah class a& of a locally free sheaf E of Ox-modules satisfies the following identity:
L*(a€u a&)f S*(a&u ~
(1.84)
Q x=) 0.
Proof. We will use the description of the Atiyah classes for E and the tangent sheaf OX in terms of Cech cohomology to proof the proposition. Let ( U i ) i € ~be a covering of X by open sets and for each i E I , Vf a holomorphic connection of € 1 and ~ ~ 07 a holomorphic connection of 8 x lux. The class a&u a& E H2(X,(Rx @ End(€))@") is given by the Cech cocycle
((Vf
- 0:) @
(0;
(1.85)
-Vt))i,j,k~ E ~C2(X,(Rx
Similarly, a€ Uae E H2(X,Rx @ m ( E )@ Rx @ M ( O x ) ) is given by the Cech cocycle
(1.86) By definition of S and L, a cocycle (ijk)i,j,k, I representing L (
) +
given by
(1.87)
for all and
For
set
(1.88) One verifies that ~ i j ( 6s)~is, Ox-linear in s and quadratic in 8. Therefore, this defines a Cech-cochain (Yij)i,jd
E C1(X, S2RX @ End(€)).
Finally, note that G(yij) = ( a i j k ) where 6 is the Cech differential, i.e. is cohomologous to zero.
(1.89) (aijk)
0
Compact hyper-Kahler manifolds
1.4.4
31
Torsion and the Atiyah class of the tangent bundle
Let U be an open subset of X and V a holomorphic connection of the tangent sheaf 0 x 1 of ~ U . For every open subset V of U and 8,O’ E 0 x ( V ) , let us set
(1.90) We have
Tv(Q,O’)= -Tv(e’,O)
(1.91)
and for f E Ox(V),
Tv(fe,8’)= fVee’- fVe,e-(e’. f)e- f
[e,eq-[f,eqe= fTv(6,e’). (1.92)
Therefore, (1.90) defines an element Tv in Ho(U,A2s2x @ ex).
Definition 1.17 (Holomorphic torsion) The section Tv is called the torsion of the holomorphic connection V. A holomorphic connection V is said to be torsion-free if Tv = 0. Remark 1.21 (Torsion only defined for the tangent sheaf) Note that there is no way to define the torsion for holomorphic connections of arbitrary locally free Ox -modules. Lemma 1.6 (Existence of torsion-free connections) If there exists a holomorphic connection of 0 x 1 there ~ ~ also exists a torsion-free holomorphic connection of 8 x 1 ~ . Proof. Let V be a holomorphic connection of 0 x 1 ~ . By example 1.14, we have that L
(1.93)
is again a holomorphic connection. Its torsion is given by
Tvtf(e,er)= Vet)’i.e. it is torsion-free.
1 1 vole- [O,e’]- -Tv(e,O‘) + -Tv(8’,8) 2 2
= 0, (1.94)
[]
We will use this lemma to prove the following symmetry result for the Atiyah class of the tangent sheaf:
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
32
Proposition 1.17 (Symmetry of the Atiyah class) The Atiyah class aeX of the tangent sheaf of X lies in the direct summand H1 (X, S2Rx @ ax) of
H1(X,Rx@ W ( @ x ) ) . Proof. Let (Vi)iElbe a covering of X by open subsets such that holomorphic connections V i of OX[^, exist. By lemma 1.6 we can assume that the V i are torsion-free. By the description of the Atiyah class in terms of Cech cohomology, we know that
(1.95) is a Cech cocycle in C l ( ( U i ) RZ2 , @ O X ) representing O x (Vi n V j ) ,we have
sex. For 8,e’ E
+
aij(8,e’)- aij(e’,e) = vi,&‘ - V i p e - Vj,&’ Vj,ete = T~~(8,el) - T~~(0, e l ) = 0 , i.e.
( a i j )lies
1.4.5
in C1((Ui), S2Rx @‘OX), which proves the proposition.
(1.96) El
The Atiyah class of symplectic sheaves
Let X be a complex manifold and ( E , a ) a symplectic sheaf on X .
Definition 1.18 (Holomorphic symplectic connection) Let U be an open subset of X. A holomorphic connection V of Elu is called symplectic if d(a(s, s’)) = ~ ( V Ss’) ,
+
O(S,
VS’)
(1.97)
for all open subsets V of U and sections s, s’ E E(V).
Lemma 1.7 (Existence of symplectic connections) If there exists a holomorphic connection of E[u, there also exists a symplectic holomorphic connection of € 1 ~ . Proof.
For V an open subset of X and s, s’ E E(V) let us define
F ( s , s’) := - (d(a(s,s’)) + ~ ( V Ss’) , - O ( S , VS’)). 1
2
We have F ( s ,fs’) = f F ( s ,s’) for all f E d(s)E ( O x @ E ) ( V )by setting O(V(S),
(1.98)
Ox(V),i.e. it is possible to define
s’) = F ( s , s’).
(1.99)
Compact hyper-Kahler manafolds
33
One easily verifies that this defines another holomorphic symplectic connection 0 of E[u. This connection is in fact symplectic: d(a(s, s’)) - a ( V s , s‘) - a(s, Vs’) = d(a(s, s‘)) - F ( s , s‘)
+ F(s’, S) = 0,
which concludes the proof.
(1.100)
0
Proposition 1.18 (Symmetry of the Atiyah class of symplectic sheaves) The Atiyah class a&of the symplectic sheaf (E, a ) lies in the direct summand H1(X, fix 8 End(&,a ) ) ofH1(X, fix @End(&)). Proof. The proof goes along the same lines as the proof of proposition 1.18. Let (Ui)icl be a covering of X by open subsets such that holomorphic connections Vi of €lu%exist. By lemma (1.7) we can assume that the Vi are symplectic. Then (1.101)
:= (0% - Vj)i,jGl
(Crij)
is a Cech cocycle in C1((Ui), R x 60 End(€)) representing For 8 E Ox(& n U j ) and s, s’ E E(Ui n U j ) ,we have C(&j
(8, ). ,s’)
+ C(s ,
= “(Vi,ss, s‘)
“ij
+
QE.
(8, s’ ) ) O(S,
Vip~’) a ( V j , o ~s’) , - O ( S , Vj,es’) = d(a(s, s’)) - d(a(s, s’)) = 0,
i.e.
(aij)lies
in C ’ ( ( U ~ )~x , 8
m(€, a)).
(1.102)
0
Corollary 1.3 (Symmetry of the Atiyah class of symplectic sheaves) Let (OX,a ) be a symplectic sheaf. Identifying OX with R x via V, induces an identification ofH*(X, fix @ End(Ox)) with H*(X, Under this identification, the Atiyah class as, lies in the direct summand H’(X, &Ox) ofH’(X, 0%’).
@z3).
Proof. By proposition 1.17, the Atiyah class ae, lies in the direct summand H’(X, S28x @ OX). At the same time, it also lies in the summand H1(X,Ox @ SzOx) by proposition 1.17 and remark 1.17. This proves the corollary. 0 Let us see how the Bianchi identity, proposition 1.16, looks for symplectic tangent sheaves:
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
34
Let (OX, c) be a symplectic sheaf. Let the morphism P : Az(S38x) S48x be given by
+
(eB3 8 egr3)- (el@38 eB3) H
a(e,el) . (e 8 e 8 el
@el
+
el
8 el 8 e 8 e
+ e 8 e‘ 8 e 8 el
+ el B e 8 e’ 8 e + el B e B e 8 el + e 8 el 8 el 8 e)
(1.103)
for all open subsets U of X and local sections 8,O’ E Rx(U). It induces a morphism P, in cohomology. Proposition 1.19 (Bianchi identity for holomorphic symplectic manifolds) Let (@x,u)be a symplectic tangent sheaf. Its Atiyah class satisfies the following identity:
P*(aexu sex)
(1.104)
= 0.
Proof. After identifying RX with OX via V, and restricting to S38x O,: the map S L : (S3Ox)@’ + S20x 8 0$2looks like
+
[email protected]/@3
.(e, e / ) . ((e . e l ) 8 (e 8 el + el 8 e) +
el2
-+
8 ( e B 2 ) ) . (1.105)
Let T : S2Bx 8 Og2 H (ex)4be the canonical lift given by (01 . 4 2 ) €4 (03 8 0 4 ) )
H
(01 8 02
+ 62 €4 01) €4 (63 8 64))
(1.106)
for all 81,. . . ,Q4 E Ox(U). The restriction of T o ( S + L ) to AzS3Ox factors as P over the inclusion S4Ox -+ Finally, we use the fact that agt lies in the direct summand H2(X,A ~ S ~ Q X of )H2(X, (&ex)@’) and that ( S , L * ) ( a g t )= 0 to conclude the proof.
+
1.4.6
Chern- W e d theory
For any complex manifold X, any locally free sheaf E on X and any nonnegative integer n = 0,1,2,. . ., let the morphism Tr : R p @End(€)@” + 0%be given by
(a18 ... €4 a,) 8 A1 8 ... €4 A,
H
(a1A
... A an)8 tr(A1 o ... o A,) (1.107)
for all open subsets U of X, local sections A1,. . . ,A, E M ( E ) ( U ) . Consider the class
al,
. . . ,a,
E Rx(U) and
(1.108)
Compact hyper-Kahler manifolds
35
Proposition 1.20 (“Chern-Weil theory”) The class c?l(E) coincides with the Chern character ch(E) of E . Proof. We have to show that c?l fulfils the defining properties of the [] Chern character. These are summarised in the following lemmata. Lemma 1.8 (“Whitney sum formula”) Let O-E‘---+&-&ff
-0
(1.109)
be a short exact sequence of Ox-modules. Then c?l(E) = c?l(E’)
+ c?l(€”).
Proof. We have the following short exact sequence of the short exact sequences defining the Atiyah classes of E l , E and Eft, respectively:
0
0
0
-
1 1 1 1
ax @ € I
Ox@&
0
0
0
- 1 -1 - 1 -1 J(E’)
El
0
J(&)
&
0
1 1
1
0
0
(1.110)
In this situation, there exist extension classes 4 E Extl(E,Slx @ E l ) and $ E Extl(Etf,R x @€) such that 4 H a,p and $ H a p by the natural maps. Further more, if we denote by 01 the image of 4 under the natural map Extl(E, R x @ C )-+ Ext’(€, R x @ E ) and by a 2 the image of $ under the natural map Extl(E”, R x @ E ) --+ Extl(€, @ E ) , we have QE = a1 a 2 . This can be seen, for example in the description of the extension groups in terms of Cech cohomology. By the naturality of the trace, it follows from this that Tr(az) = Tr(aF,) Tr(az,,),which proves the proposition. 0
+
+
Lemma 1.9 (“Naturality”) Let Y be another compZex manifold and f : Y + X a holomorphic map. Then
f* : H*(X,a>)+ H*(Y,a$), cX(&)H c?l(f*&).
(1.111)
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
36
Proof.
This follows immediately from lemma 1.5.
0
Lemma 1.10 (“Norm”) For an invertible sheafC o n X , we have ~ X ( L )= exp(c1 (c>>.
Proof.
1.5
1.5.1
(1.112)
0
This follows again immediately from example 1.15.
On the second cohomology group of a hyper-Kahler manifold The period m a p
Let X be an irreducible holomorphic symplectic manifold of dimension 2n. Let us consider the polynomial gx : H2(X,C)-+ C given by
gx(a) :=
s,
a2*.
(1.113)
The polynomial gx is homogeneous and therefore it defines a hypersurface (gx)in P ( H 2 ( X , C ) ) . Let Q x be the open (in the analytical topology) subset of (gx)given by those a E H2(X,C ) with
gx(a)= 0 and gx(a + 6) > 0.
(1.114)
Remark 1.22 (On the set Q x ) It is true that (gx) i s a n irreducible quadric and that Q X i s an open connected subset of this quadric. However, we shall not need this result. The polynomial gx and the quadric Q X do only depend o n the topological structure of X . Let 7r : X -+ S be a deformation of X over a simply-connected smooth base (S,O). As S is simply-connected, the local systems Ri7r,Z for i = 0,1,. . . are trivial. Therefore, there exist canonical ring isomorphisms Hi(&,Z) N HZ(X, Z) for all t E S. Via these isomorphisms, in particular a class ot E H2(Xt,C ) can be viewed as a class in Hi(X, C ) .
Lemma 1.11 (The period codomain) Let ot E H2i0(Xt) be a holomorphic symplectic form o n Xt. Then its class [ut] in projective space lies in Qx. Proof. As a? is a (4n,O)-form on Xt and therefore vanishes, we have that gx (ut) = 0. Recall that ut +8t is a Kahler form with respect t o another structure of a compact complex manifold on X. Therefore, sx(ut+at)2n> 0,i.e. ut E Qx. []
37
Compact hyper- Kahler manifolds
The map P : S -+ Q x that maps each t E S to the class [at]of the class at of the symplectic form of Xtviewed as an element in Q x c H2(X,C ) is called the period map of the family X -+ S. Remark 1.23 (Holomorphicity of the period map) The period map P is holomorphic.
As X is irreducible holomorphic symplectic, it has a universal smooth deformation space (Def(X),O). Moreover, we can assume that Def(X) is simply-connected. Theorem 1.5 (“Local Torelli”) The period map P : Def(X) the universal deformation X -+Def(X) is a local isomorphism. Proof. 1.5.2
--+
Q x of
0
See [Beauville (1983)l.
A v a n i s h i n g result for p o l y n o m i a l s on H2
Let X be a differentiable manifold. Let Cx be the set of classes [a]E H2(X,C ) such that there exists a structure of an irreducible holomorphic symplectic manifold on X for which a is the cohomology class of its symplectic form.
Proposition 1.21 (A vanishing result) The Zarislci closure of {a 8 : a E Cx} is either empty or the whole second cohomology group H2(X,C ) . I n other words, if EX # 0, i.e. there exists at least one structure of an irreducible holomorphic symplectic manifold o n X , every polynomial f : H2(X,C ) -+ C that vanishes on all elements of the f o r m a+@ with a E Cx vanishes o n the whole H2(X,C ) .
+
Proof. Let us assume that CX # 0 - in the other case there is nothing to proof. With a E C X also ta E CX for all t E C x . In other words, it suffices to proof that the Zariski closure of the set {[a a’]: a E C X } in P(H2(X,C ) ) is the whole projective space. Equip X with a structure of an irreducible holomorphic symplectic manifold. Consider the period map P : X 4 Q x c P(H2(X,C ) ) of X . By the Local Torelli Theorem, theorem 1.5, one knows that the image of P is an open subset (in the analytic topology) of (gx). This yields that its Zariski closure is the whole (gx)as (gx) is irreducible. As all fibres of the universal deformation X -+ Def(X) are diffeomorphic to X, we see in particular that
+
38
C h e r n Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
the Zariski closure of { [o]: o E Ex} in P(H2(X,C ) ) is the hypersurface
(gx). Let f : P(H2(X,C ) )+ C be a homogenous polynomial that vanishes on all [o+a] for o E C. We have to prove that the zero set (f) in P(H2(X,C ) ) defined by f is the whole projective space. With o E C x , also e - i 4 / 2 u E C x for 4 E R. Therefore f(a e24,) = 0 for all 4. From this, one concludes that f(o t8) = 0 for all t E C . In particular, f(o)= 0 for all o E EX.From this and the considerations about (gx)above, it follows that (gx)c (f). Furthermore, o t8 # (gx)for all t E C x , so the codimension of (f) is strictly less than the codimension of 0 (gx), which is one. This proves (f)= P(H2(X,C ) ) .
+
+
+
Chapter 2
Graph homology
2.1
The space of graph homology
In this section we shall introduce the space of graph homology as it is used in the construction of Rozansky-Witten invariants by giving a simple definition. In the following sections we focus on the universal property of this space, i.e. we answer the question: why these graphs and this space and not something else?
2.1.1
Jacobi diagrams
Definition 2.1 (Graph) A graph r = (F,V, E ) is a finite set F together with a partition V c p(F) of F into disjoint non-empty subsets and a partition E c p(F) of F into disjoint two-element subsets. The elements of F are called the flags or half-edges of r, the elements of V are called the vertices of I? and the elements of E are called the edges
ofr. A vertex u E V with # v = n is called n-valent. A 1-valent vertex is also called univalent, a 2-valent vertex is also called bivalent and a 3-valent vertex is also called trivalent. Two graphs are called isomorphic, if there exists a bijection between their sets of flags respecting the partitions into vertices and edges. To visualise a graph by a diagram, we draw a point for every vertex. Then we draw for every edge e consisting of the two flags fi and f 2 a line between the vertex having f l as an element and the vertex having f2 as an element. On the other hand, from a drawn diagram displaying a graph, we can recover the graph up to isomorphism. 39
40
Chern Numbers and R W-lnvariants of Compact Hyper-Kahler Manafolds
Example 2.1 (Flags, vertices and edges) The diagram
-4
(2.1)
displays a graph with eight flags, say f1,. . . , fs, and vertices = {{fl)t {fZ, f 3 1 7 {f4r f57f6) I {f7) > {fS}}
(2.2)
and edges
E
= {{fl,fZ)
( f 3 7 f 4 ) (f59f7) {fS,fS})
*
(2.3)
It has three uni-, one bi- and one trivalent vertex. Definition 2.2 (Orientation of trivalent vertices) Let I? be a graph and v = {fl, f2, f3) a trivalent vertex of F. An orientation of ‘u is a cyclic ordering of the three flags f1, f2, f3. Remark 2.1 (Possible orientations) A t each trivalent vertex v = {fi, f2, f3) there are exactly two choices of a cyclic ordering of the flags fl, fz, f 3 , namely and
(2.4)
The first choice is denoted by fl < f2 < f3 (or equivalently by f 2 < f3 < f l or f3 < f1 < f2), the second one by f2 < f1 < f3. Definition 2.3 (Jacobi diagram) A Jacobi diagram is a graph whose vertices have only valency one and and three and with a choice of an orientation for each trivalent vertex. The univalent vertices of a Jacobi diagram r are also called legs and the trivalent vertices internal vertices. An isomorphism of Jacobi diagrams is an isomorphism of the underlying graphs respecting the chosen orientations at the trivalent vertices. When we draw a Jacobi diagram as a diagram in the plane, we usually don’t mark the uni- and the trivalent vertices using dots. In such a drawing, “open ends” of edges correspond to the univalent vertices and points where three edges come together are the trivalent vertices. Due to the twodimensionality of the plane, there may also be crossings of edges. They are not to be interpreted as four-valent vertices (a Jacobi diagram has non of these!). Further, if no cyclic ordering at a trivalent vertex is marked in the
Graph homology
41
drawn diagram, we mean the cyclic ordering given by the counter-clockwise ordering of the flags induced by the drawing. For example, the two graphs
and
(2.5)
actually depict the same.
Definition 2.4 (Marked Jacobi diagram) Let I be a finite set. A Jacobi diagram over I is a Jacobi diagram together with a marking (over I ) , i.e. a bijection of the set of its legs to I . An isomorphism of marked Jacobi diagrams over I is given by an isomorphism of the underlying Jacobi diagrams that respects the given bijections of their set of legs to I .
Remark 2.2 (Relabelling of legs) Every bijection q5 : I
--f J of finite sets induces naturally a map 4*from the class of Jacobi diagrams over I to the class of Jacobi diagrams to J : If r is a Jacobi diagram, L its set of legs and m : L -+ I a marking over I , q5 o m : L -+ J is a marking over J .
If we draw a Jacobi diagram over I , we label the legs by elements of I . Example 2.2 (Marked Jacobi diagram) The Jacobi diagram 1 2‘
t
3-
5 4
(2.6)
is marked over the set {1,2,3,4,5}. 2.1.2
Chains of Jacobi diagrams
Remark 2.3 (“Set-theoretic nonsense”) I n what follows, we would like to talk about the set of all Jacobi diagrams. However, such a set does not exist because it would be way too big. A s we are mostly interested in isomorphism classes of Jacobi diagrams, there is an easy way to get rid of this annoying problem: We j i x an infinite set 5 and from now on, we only want to consider graphs whose set of Jags is a subset of 5. Jacobi diagrams built on those graphs constitute a set and the isomorphism classes of these Jacobi diagrams
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
42
are in bijection with the isomorphism classes of Jacobi diagrams whose flags do not necessarily belong to 5. Most of the time we won’t mention 5 any further and silently assume that all constructions are carried out in this set. Similarly, we want all our finite sets I with which we mark the legs of Jacobi diagrams to lie in a fixed infinite set 3.
From now on until the end of this section, let k be a field of characteristic zero, e.g. k = Q or k = C . We shall also consider the polynomial algebra k[O]over k in one variable “0”. Definition 2.5 (Chains of Jacobi diagrams) We denote by J’ the free k [Ol-module spanned by the isomorphism classes of Jacobi diagrams and by J I the free k[O]-module spanned by the isomorphism classes of Jacobi diagrams over a finite set I . Elements of these spaces are called chains of Jacobi diagrams (over I ) . Remark 2.4 (Functoriality) Let 4 : I + J be a bijection of finite sets. The map & from the class of Jacobi diagrams over I to the class of Jacobi diagrams over J induces an isomorphism q5* : J I -+ J J . I n this sense, I H 31becomes a functor from the category S of finite sets whose morphisms are bijections to the category of k[O]-modules. Remark 2.5 (Gradings) The k[O]-module 31 is graded by the number of internal vertices and the number of connected components. The k[O]-moduleJ is graded by the number of legs, the number of internal vertices and the number of connected components.
Notationally we shall not distinguish between a Jacobi diagram (over
I ) and its isomorphism class. Remark 2.6 (“Forgetful” map) There is an obvious ‘~orgetful”map 31+ J’ forgetting the marking. 2.1.3
Glueing legs and product of Jacobi diagrams
Let r be a Jacobi diagram over the finite set I and i,i’ E I two different elements. We define an element F / { i , i ’ } E J I / { + , } by the following procedure: Let us denote the set of flags of r by F , the set of trivalent vertices by V and the sets of edges by E. Further, let us identify the set of legs L with I (by means of the given bijection). Let f and f’ be the flags belonging
Graph homology
43
to i and i’, respectively (recall that a leg consists of exactly one flag!). Set F’ := F \ {f,f’}and L’ := L \ {i,i’}. There are flags g,g’ such that e := {f,g } and e’ := {f’,g’} are edges of r. Now we have to consider two cases: {g,g’} is not an edge of I?. It follows that f
# g’ and g # f’. Set E’
:=
E \ { e ,el} U { { g , g’}} and let I?’ be the Jacobi diagram over I\ {i, i’} with set of flags F’, set of trivalent vertices T , set of legs L’ and set of edges E’. Finally, let F / { i , i’} be the image of r’ in JI\{+). {g,g’} is an edge of I?. In this case, f = g’, g = f’ and e = e’. Set E’ := E \ { { g , g ’ } } and let I” be the Jacobi diagram over I \ {i, i’} with set of flags F’, set of trivalent vertices T , set of legs L’ and set of edges E’. Finally, let r/{i,i’} be 0. I” in JI\{i,i,l. Definition 2.6 (Glueing of legs) We say r/{i,i’}is the chain of Jacobi diagrams over I \ {i, i’} obtained f r o m glueing the legs i and a’ of r. If 7r = {{il, ii},.. ., {in, ib}} c @ ( I )is a set of pairwise disjoint twoelement subsets of I , we set
r/T := r/{il,i;}/..
./{in, i;}.
(2.7)
For y E J I ,we finally define y/n by Ic[O]-linearextension.
Remark 2.7 (Independence of the order of glueing) It is easy t o see that the definition of r/7r in (2.7) is in fact independent of the enumeration of the elements of 7 r . Example 2.3 (Glueing of legs) For example, 1 (2.8)
3 Definition 2.7 (Product of Jacobi diagrams) Let be a Jacobi diagram over a finite set I and r’ a Jacobi diagram over a finite set I’. By r U I” we want to denote the product of r and I?’: it is the Jacobi diagram over I k~I’ given by the disjoint union of r and I”. For y E 31and y’E 311the , product y U y’ E J I ~ is I defined ~ by Ic[O]linear extension of the product of Jacobi diagrams over finite sets I and I‘, respectively.
44
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
Example 2.4
1+2
(Product of Jacobi diagrams) For example,
u
l ’ k 3 t
l l’. -> 2/
=
2’
o
2
.
(2.9)
3’
Remark 2.8 (Product over an arbitrary family) W e can easily extend the notion of the product of two Jacobi diagrams to the notion of a product over a finite family of Jacobi diagrams: Let (I’j)jcJ be a finite family of Jacobi diagrams such that rj is a Jacobi diagram over the finite set I j . B y I’j we denote the disjoint union of the Jacobi diagrams rj. It is a Jacobi diagram over the set It)jE I j . Similarly, we define UjEJ yj for a family (yj)jEJ with yj E .7r,
ujE
+
2.1.4
Subspaces and ideals
Definition 2.8 (Subspace) By a subspace ( A I ) Iof the space of Jacobi diagrams we mean a subspace AI of J I for every finite set I such that for every bijection $J : I -+ J , we have #,(AI) = A J . Example 2.5 (Trivial subspace) The collection the space of Jacobi diagrams.
(J1)I
is a subspace of
Remark 2.9 (Intersection of subspaces) Let ( ( A ~ , j ) ~J ) jbee a family of subspaces of the space of Jacobi diagrams. The intersection
n( A I , j ) i
:=
j€J
(,?J
(2.10)
AI,.) I
is again a subspace of the space of Jacobi diagrams. Definition 2.9 (Ideal) A subspace ( A I ) I of the space of Jacobi diagrams is called an ideal of the space of Jacobi diagrams or simply an ideal if for all finite sets I and I’, y E A I , y’E J I , we have y U y’E A I ~and ~ ,for all finite sets I , i, i’ E I with i # i’ and y E AI we have y / { i , i’} E AI\{i,i,}. Example 2.6
(Trivial ideal) The subspace
(JI)I is an ideal.
Remark 2.10 (Intersection of ideals. Ideals generated by elements) A s usual, an arbitrary intersection of ideals is again an ideal, i.e. if ( ( A r , j ) l ) jJc is a family of ideals, J(A1,j)lis again an ideal.
nj,
Graph homology
45
Therefore, whenever given a collection G of chains of Jacobi diagrams, we can speak of the ideal generated by G: it i s the intersection of all ideals (AI)I such that for every E G with r E 31for a finite set I , we have
r E AI.
Remark 2.11 (Sum of ideals)
If (Ar)I and (A;)I are ideals, their sum (2.11)
i s again a n ideal.
Remark 2.12 (Equivalence relations) Every ideal ( A I ) I induces a n equivalence relation: y and yl in 31for a finite set I are equivalent modulo (AI)I i f they project t o the same element in the quotient space JI/AI. Definition 2.10 (Induced submodules) Let (AI)I be an ideal (or more generally a subspace of the space of Jacobi diagrams). Let A be the smallest k[O]-submodule in J’ such that for every finite set I the image of AI under the forgetful map 3 1 4 J’ is a submodule of A . We say that the submodule A is induced by (AI)I. 2.1.5
The graph homology spaces
Definition 2.11 (Anti-symmetry relation) Let AS be the ideal generated by I?+ r- where
+
r+=
‘>3 2
and
r-=
(2.12)
The generated equivalence relation is called the anti-symmetry or ASrelation. The anti-symmetry relations says that changing the orientation of one trivalent vertex is equal to changing the sign in JI/AS. Example 2.7 (Anti-symmetry relation) In g{,,/AS, we have for example
0 - 1
Definition 2.12
=o.
(2.13)
(Jacobi relation) Let IHX be the ideal generated by
46
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
where
1
2
3
4
and
3
4
(2.14)
The generated equivalence relation is called the Jacobi or IHX-relation.
Definition 2.13 (Induced relations) The submodule of J’ induced by the ideal AS IHX is again denoted by AS IHX. We call the relations induced by this submodule on J’ again AS- and IHX-relations.
+
+
Definition 2.14 (Space of graph homology) The quotient space
B
:= J’/(AS
+ IHX)
(2.15)
is called the graph homology space. For every finite set I , the quotient space
BI := J’I/(AS
+ IHX)
(2.16)
is called the graph homology space over I .
Remark 2.13 (Gradings) The different gradings o n J induce gradings o n B. Firstly, there is a grading on B given by the number of legs. Secondly, a grading o n B is given by the number of internal vertices. Lastly, B i s graded by the number of connected components. The default grading shall be the grading by the number of legs. The graph “0” has by definition no vertices and one connected component. The homogeneous (with respect to the number of legs) component of B with no legs is called ,130. Example 2.8 (Gradings) The following graph
0 (2.17) is homogeneous with respect to all gradings. It has four legs, two internal vertices and three connected components.
Graph homology
47
Remark 2.14 (Forgetful maps and lifts of diagrams) The forgetful maps 31-+ 3 induce forgetful maps BI -+ B for all finite sets I . They are surjective in the following sense: for everg y E 23 that is homogenous with respect to the number of its legs, there exist a finite set I and y‘ E BI such that yl is mapped to y via the forgetjul map. W e call yl a lift of y.
2.2
Symmetric monoidal categories
The notion of a Lie algebra object (to be defined later) generalises the notion of a Lie algebra. A usual Lie algebra will become a Lie algebra object in the category of k-vector spaces, k a field. More general Lie algebra objects will “live” in more general categories. However, all these categories will be symmetric monoidal categories. These categories will be introduced in LLcoordinate-free’’ language in this section.
2.2.1
Definition
Definition 2.15 (Symmetric monoidal category) A symmetric monoidal category C is a category with functors
@ : c’
-+
I
c : (XZ)ZEI H @ X i
(2.18)
G I
for every finite set I and isomorphisms functorial in (Xi)icl
(2.19) ZEI
j€J Ga-l({j})
for every map Q : I -+ J of finite sets I and J . These functors isomorphisms X ( Q ) have to satisfy the following conditions:
and
(1) If I consists of a single element, @I : C’ = C 3 C is the identity, and if : I J is a map of one-element sets, X ( Q ) is the identity automorphism of the identity functor. (2) Given three finite sets I , J and K and maps a : I 3 J and ,O : J -+ K , ---f
48
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
the diagram @iEI
X(PO.*)
Xi
@kEK
@iGa-
x(41
1
(p- 1 ((k)))
I@~LK
xi
X(ak)
@ie*-l({j})xi
@k€K @j€p-l({k))
@&I
@ZGa-l({j))
xi
(2.20)
of functorial isomorphism commutes (called the associativity condition). Here Qk : a-l(p-l({k))) -+
is the restriction of
Q
p ( { k } ) , iH Q(i)
(2.21)
defined for every k E K .
A functor between symmetric monoidal categories C and D is a functor from the category C to the category D that respects the functors @ I and the natural isomorphisms ~ ( a ) . The object BiGI X i is often called the tensor product of the X i . If a symmetric monoidal category is given, we usually write a* instead of X ( Q ) when Q : I -+ J is a map of finite sets. We shall also omit ~ ( aaltogether ) if no confusion can arise.
Example 2.9 (Permutations) Every permutation induces an isomorphism cT*
:
0x2 G I
--t
0
@X.l(Z)
of the elements of I (2.22)
i€I
for X i , i E I objects of a symmetric monoidal category. That's why a symmetric monoidal category is called symmetric.
If all X i equal X, we shall write X@' := g i G I X for short. If I = (1,. . . ,n } , n E No, we write X@'" := X@{'y..+}. Remark 2.15 (Partitions) The maps a : I -+ J are in bijection with the partitions of I in (possibly empty) subsets labelled by the elements of J . For I= I3 we therefore have a natural isomorphism
ujE
@Xi i€I
-+@@xi. j € J i€Ij
Ojlen we shall write this isomorphism as an equality.
(2.23)
Graph homology
49
When defining a symmetric monoidal category we shall often only give the definition of @ I when there is no doubt what the isomorphisms x(a) shall be.
Example 2-10 (Categories with finite products) Let C be any category with finite products. Then @I := & : C' 3 C for every finite set defines on C the structure of a symmetric monoidal category. The morphisms x(a) are the natural ones. The commutativity of (2.20) follows from the fact that products are unique up to a unique isomorphism. Example 2.11 (The initial PROP) Let S be the category whose objects are all finite sets and whose morphisms are the bijections between finite sets. This category has a natural structure of a symmetric monoidal category. It is given by
@ : S' I
-+
Xi.
S, (xi)iel+-t
(2.24)
iE1
With this structure the category S is called the initial PROP (it is the initial object in the category of PROPS, see, e.g. [Hinich and Vaintrob (2002)l).
Example 2.12 (The category of modules over a ring) Let R be a commutative ring. The category of R-modules together with the ordinary tensor product @ i c I M iof a family ( M i ) i E 1of R-modules is a symmetric monoidal category. We can expand this example to ringed spaces:
Example 2.13 (The category of modules over a ringed space) Let ( X ,Ox)be a ringed space. Then, the category of Ox-modules forms a symmetric monoidal category under the usual tensor product of Ox-modules. 2.2.2
k-linear categories
Let k be a field (or, more generally, a ring). Let us denote by category of k-vector spaces.
v k
the
Definition 2.16 (k-linear category) A k-linear category is a Vk-enriched category, that is a category whose morphism sets are k-vector spaces and whose composition laws are k-bilinear.
50
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Remark 2.16 (Enriched categories) I n fact, one can define V-enriched categories whenever V is a symmetric monoidal category, see [Mac Lane (1998)] for more. Example 2.14 (k-linear categories) The category V k of k-vector spaces is obviously k-linear. So is the category of complexes of k-vector spaces, etc. Example 2.15 (Additive category) Let A be an additive category. By tensoring every hom-set of this category with k and extending the composition maps k-linearly we can make A into a k-linear category A @J k which has the same objects as A. Remark 2.17 (Functor of k-linear categories) There is a notion of a functor of k-linear categories: it is a functor of categories that respects the k-linear structure o n the hom-sets (better called hom-k-vector spaces). W e shall also call these functors k-linear functors. Similarly, one can define the notion of a k-linear symmetric monoidal category. Example 2.16 (k-linear symmetric monoidal categories) The category of k-vector spaces is a k-linear symmetric monoidal category. More generally, the symmetric monoidal category of Ox-modules on a ringed space (X, O x ) over a ringed space (Y,/c) where & denotes the constant sheaf with stalks k (e.g. Spec k) is a k-linear symmetric monoidal category. 2.2.3
Global sections
Let C be a symmetric monoidal category.
Definition 2.17 (Unit object) We call 1 := @() where () is the empty family of objects in C the unit object in C or the unit for the tensor product in C. Remark 2.18 (Unit object) I n fact, 1 is a unit as 1 @J X = X @ 1 = X , at least up to a natural isomorphism. Example 2.17 (Categories of modules) In the category of modules over a commutative ring, the unit is given by the ring itself. In the category of (X, O,y)-modules for (XIOX) a ringed space, the unit is the Ox-module OX.
Graph homology
Definition 2.18 (Global section functor) The natural functor to the category of sets given by the mapping
X
H
r ( X ) := Hom(1, X)
51
from C
(2.25)
is called the global section functor.
Example 2.18 (Global sections of a module over a ringed space) In the category of Ox-modules, the global section functor maps an Ox-module
3 exactly to its space of sections r ( X ,F)= Hom(Ox, F),hence the name “global section functor”.
Example 2.19 (Global sections of a module over a ring) In the category of modules over a commutative ring, the global section functor is naturally isomorphic to the forgetful functor to the category of abelian groups. As I? is a functor, every morphism f : X -+ X’ in C induces a morphism f* : r ( X ) r ( X ’ ) . In particular, every action of a group G on an object X induces an action of G on r ( X ) .
Example 2.20 (Action of the symmetric groups) Let I be a finite set and G I the group of bijections of I . As we have already seen, there is a natural action of G Ion X B 1 for every object X in I . This induces a natural action of 61 on
r(xB1).
Remark 2.19 (Global sections of k-linear symmetric monoidal categories) If C is a k-linear symmetric monoidal category, the global section functor takes naturally values in the category of k-vector spaces. Remark 2.20 (Product of global sections) Let X and Y be two objects in a k-linear symmetric monoidal category C . There exists a natural morphism
r(x)8 r(y) r(x8 Y ) .+
(2.26)
of k-vector spaces given by the functoriality of the tensor product an C .
2.2.4
External tensor and symmetric algebras
Let k be a field and C be a k-linear symmetric monoidal. Assume for a moment that C is the category of k-vector spaces. Let X be an object in C . We can form the direct sum X B n which is called the (internal) tensor algebra of X . For an arbitrary k-linear symmetric monoidal category C, however, infinite coproducts do not have to exist, so
52
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
the notion of an internal tensor algebra doesn’t make sense. Nevertheless, we can form what is called the external tensor algebra of an object:
Definition 2.19 (External tensor algebra) We call
(2.27) the external tensor algebra of X .
Remark 2.21 (Algebra structure) The external tensor algebra of X is naturally a No-graded k-algebra. Example 2.21 (Category of k-vector spaces) In the category of k-vector spaces] the external and the internal tensor algebra of an object X are naturally isomorphic, see example 2.19.
Definition 2.20
(External symmetric algebra) The quotient, i.e. the
coinvariants, (2.28)
of the external tensor algebra of X is called the external symmetric algebra of X . Remark 2.22 (External tensor/symmetric algebra functor) We can easi l y extend the definition of the external tensor and symmetric algebra to functors T and S , respectively, from the category C to the category of kvector spaces.
Remark 2.23 (Natural transformations) Let us denote the functors I?, T and S from the k-linear symmetric monoidal category C t o Vk also by rc, Tc and Sc, respectively. Let F : C 4 D be a functor of k-linear symmetric monoidal categories. This functor induces natural transformations from the functors rc,Tc and o F , TD o F and S D o F , respectively. S c to the functors 2.3 2.3.1
Metric Lie algebra objects
Definition
Let k be any field of characteristic zero and C be a k-linear symmetric monoidal category.
Graph homology
53
Definition 2.21 (Metric Lie algebra object) A metric Lie algebra object ( L ,[., .],b, c) in C is given by an object L in C and three morphisms
[.,.]:L@L-+L,b : L @ L + I
and c : l + L @ L
(2.29)
such that
(1) [., .] is antisymmetric, (2) b and c are symmetric, (3) b and c are adjoint, i.e. the map
L (4)
idL@c
L@L@L
b@idL
(2.30)
L
is the identity map of L , [.,-1 is compatible with respect to b, i.e. the two morphisms
(2.31) and
L@L@L
- id^@[.,.]
L@L
b
1
(2.32)
are equal, and (5) [-,-1 obeys the Jacobi identity, i.e.
L @ L @ LA L @ L @ LL [. id^ L B L -[.L .I L
(2.33)
is the zero map. Here the morphism
a
=
C o* : L@’ + L @ ~
(2.34)
U€%3
is the sum over all commutativity morphisms for even permutations. By a metric Lie algebra we mean a k-linear symmetric monoidal category C together with a metric Lie algebra object in C as defined above.
(Category of metric Lie algebra objects) The metric Lie algebra objects in the category C themselves f o r m a category with the obvious morphism between them. Furthermore, there exists the category of metric Lie algebras. The morphisms between two metric Lie algebras are given by functors between the underlying categories that respect the metric Lie algebra objects and all their data.
Remark 2.24
54
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Example 2.22 (Sum of two metric Lie algebra objects) Let C be a klinear tensor category. So the direct sum of two objects exists. For each pair ( ( L 1 ,a l , bl, c l ) , ( L 2 ,a2, b2, c2)) of metric Lie algebra objects in C the direct sum (L1 @ L2, a , b, c ) is naturally a metric Lie algebra object where
and
(2.35) Note that
(2.36) 2.3.2
Examples from the category of vector spaces
.I)
Example 2.23 (Lie algebras with invariant scalar product) Let (g, [., be a finite-dimensional k-Lie algebra and (., -) an ad-invariant scalar product, i.e.
(2.37) for all A, B,C E g. As the scalar product is per definitionem nondegenerate, it induces an isomorphism g -+ g*. Therefore, we can identify the dual of (., .) given by a map k -+ g* @ g*with a morphism c : k g @ g. The tuple ( L ,[-,-1 , (., -) ,c ) is a metric Lie algebra object in the category of finitely dimensional k-vector spaces. --f
Proof. By definition of a Lie algebra, [., .] is anti-symmetric. The scalar product (., .) is symmetric and therefore its “dual” c as well. In fact, if A1, . . . ,A, is a orthogonal basis of g with respect to (., .), we have (2.38)
Graph homology
55
The morphisms (., -) and c are adjoint, as
n
=
C (A, Ai) Ai = A
(2.39)
i= 1
for all A E g. The ad-invariance of (., .) is exactly the compatibility of [.,.] with (-, .). Finally the Jacobi identity of a metric Lie algebra object is fulfilled as this is exactly the Jacobi identity of the Lie algebra [g, [., .]) in this case. []
Remark 2.25 (Metric Lie algebra objects in the category of finitely dimensional vector spaces) O n the other hand, every metric Lie algebra object in the category of finitely dimensional k-vector spaces gives rise to a k-Lie algebra g together with a n ad-invariant scalar product.
Remark 2.26 (“Super-mathematics”) Of course, the whole thing works also with the category of metric super-lie algebras in the category of superk-vector spaces, i.e. Z/2Z-graded vector spaces.
2.3.3 Morphisms between tensor powers of metric Lie algebra objects
Lemma 2.1 (Cyclic invariance of the “Lie bracket”) Let ( L ,[-,-1 ,b, c) be a metric Lie algebra object in the category C . Then the morphism Q
:= b o
([., .] @ idL) : LB3 .+ 1
(2.40)
is antisymmetric, i.e. a o u* = -a for every transposition u E 6 3 . I n particular, Q is cyclicly invariant, i.e. Q o cr* = Q f o r o E !2l3, which a cycle of length three. Proof. 63.
It is enough to show that
Q
o o* = -a for o E
{ (1 2) , (1 3)) c
We have (2.41)
56
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
and a 0 (2 3), = b o ([.,-I
id^) o (2 3),
= b o (idL 8
[., .]) 0 (1 2)*
= -b o
([., -1 @ idL) = -a.
(2.42)
0 Let ( L ,a , b, c) be a metric Lie algebra object in the category C. Let r be a Jacobi diagram over the finite set I . We define a morphism F: : L@I -+ 1 as follows: Let G be the set of flags of r,T be the set of trivalent vertices and E the set of edges. Further, let us identify the set of legs and I by the given bijection. As c and b are symmetric by the axioms of a metric Lie algebra object, they define for every two-element set 2 canonically a morphism c" : 1 + L@" and b" : L@'" -+ 1, respectively. Similarly by the cyclic invariance of LY (see lemma 2.1), for every cyclicly ordered threeelement set y, we have a canonical morphism a9 : 1 -+ L@Y. We set cry := (a" @ idBy)o c@Y: 1 4 L@Ywhich can be viewed as the dual to QY with respect to b. We use these morphisms to construct the morphism:
-
F;
: L@'
(1)
L@'
@
L@V L, aVET @
-,
~ @ e (3)
eEE
1 (2.43)
where (1) is the map idLer @ BVET 3 (note that each E T is a cyclicly ordered set of three elements!), (2) is given by the associativity of the tensor product (note here that H I HT = G = H E ! ) and finally (3) is the map BeEE be. It is clear that this construction only depends on the isomorphism class of r. For later use, let us denote the composition of (1) and (2) by pk. Definition 2.22 y =
(Morphisms given by chains of Jacobi diagrams) For
zrEgI Cn=Oar,n. 0" . I? a chain of Jacobi diagrams over I with M
ar,n E k we set M ..
Fy" :=
C C a r l n .Fk 8 ( b r€.?I
0
c ) @: ~L@'
-+
1.
(2.44)
n=O
Proposition 2.1 (Functoriality) Let $ : I sets and y E 31.T h e n
Fy" = $*F:*?
+
J be a map between finite (2.45)
:= F$*-, o $*.
Proof. This follows at once from the definitions of q5* : $*y E JJ and FL.
L@I -+
LBJ, cl
Graph homology
Recall the definitions of Proposition 2.2
57
r+,r-, rI,r H and rx.
(Morphisms induced by special graphs) W e have
FL r+ = Q : L@' + 1 and FF- = -a :
4
1.
(2.46)
Set
,B := b o (([.,.I
o
([.,.I
id^)) @idL) : LB4
---f
1.
(2.47)
Then and
(2.48)
Proof.
This
follows directly
from the
definitions of F.L and
0
r+,r-,rI,rH,rX.
Corollary 2.1 (Compatibility with the AS- and the IHX-relation) W e have
F,L++,r= 0 = Fk-rH+rX.
(2.49)
Proof. The claim Fk++r- = 0 follows immediately from proposition 2.2. The second assertion follows from this proposition as well, together with the fact
(2.50)
due to the Jacobi identity.
0
Lemma 2.2 (Compatibility with the cup-product) Let y and y' be two chains of Jacobi diagrams over finite sets. Then
F,,u.,f L
= Fy" @ F$.
(2.51)
Proof. We may assume that y and y' are given by Jacobi diagrams I? and I?' over finite sets I and I,, respectively. By the associativity of the tensor product %I" in the symmetric monoidal category C , the maps Ftur, and F; @ Fb coincide by definition. 0
58
Chern Numbers and R W - I n v a r i a n t s of Compact Hyper-Kahler Manifolds
Lemma 2.3 (Compatibility with glueing) Let y be a chain diagrams over I and i ,i’ E I two different elements. Then
of
Jacobi
(2.52)
commutes. Proof. We can assume that y is given by a Jacobi diagram I? over I . Let f , f ‘, g , g‘, e, e’ and E and I” be as in Subsection 2.1.3. Again we have to distinguish two cases: Let {g,g’} be not an edge of r. The diagram
(2.53)
commutes due to the adjointness of b and c. Recall the definition of
F;f. : L@I
La
@
4
@
L@‘(f>f’,S,9’)
(2.54)
PEE\{e,e’}
and
F$ : L@I\{V’}
@
LZ
@
L@{S>9’)*
(2.55)
ZE\{e,e’}
We have F; = V o F;f. and F&{L,l,l = V‘ o
Fk.
Furthermore, Fk =
fib @ c { ~ ? ~ €+om ’}. this and the commutativity of (2.53) the commutativity of (2.52) follows. Now let’s consider the case when ( g , g ‘ } is an element of r. Here the diagram (2.56)
Graph homology
59
commutes. Analogously to the first case, we have Fk = V o Fk and Again, the commutativity F/,{i,i,l = V' o Fk and = pk o [] of (2.52) follows.
fik
Proposition 2.3 (Induced map on the level of graph homology) Let y and y' be two chains of Jacobi diagrams ouer the finite set I that are equivalent modulo the AS- and the IHX-relations. Then
FYL = F$ : L@'
-+ 1,
(2.57)
i.e. it is makes sense to set FLl := F[ for every graph homology class
lrl E D I . Proof.
For every finite set I , let AI be the kernel of the map
Ff
: 31---f Hom(L@', 1),y H F$.
(2.58)
Note that ( A I ) Iforms a subspace of the space of Jacobi diagrams over I . By lemma 2.2 and lemma 2.3 the subspace ( A I ) Iis in fact a n ideal. Corollary 2.1 yields that r++r-E A{l,2) and TI -I'a+I'x E A{1,2,3,4).In other words, ( A I ) Icontains the ideal AS IHX which proves the compatibility of FL with the AS- and IHX-relation. 17
+
2.3.4
The PROP of m e t r i c Lie algebras
Let k be a field of characteristic zero. Definition 2.23 (Set of morphisms) Let I and J be two finite sets. Let us set
L ~ ( IJ ,) := ~
I t y j .
F'urther, we abbreviate L M ( m , n ):= L'({l,.
(2.59)
. . , m } ,(1,. . . , n } ) for m,n E
No. Let I , J, K be finite sets with J = {jl,. . . , j n } , n = # J . In IUJW J U K , containing two copies of J , we denote by jl,. . . j, the elements in the first copy of J and by ji,. . . ,j ; the corresponding elements in the second copy of J . Set 7r := {{jl,ji},. . . , {jnljk}} c p ( 1 J~U J U K ) . Definition 2.24 (Composition) We define a k[O]-l'inear map
LM( J ,K ) @ LM( I ,J )
---f
L'(I, K ) ,y' @ y ++ y' 0 y
(2.60)
60
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
as follows: Note that the set of legs of y 8 y' is given by I u J k~J Then define y' o y := (y U y')/r.
Example 2.24 example,
@
K.
(Composition of marked graph homology classes) For
(2.61)
with I = {l},J = {2,3} and K = (4).
Definition 2.25 (The category L M ) Let L M be the following k-linear category: The objects of LM are all symbols of the form L' where I is a finite set. The set of morphisms from an object L' to an object LJ is given by L M ( I J, ) . The composition of the morphisms y' E L M ( J ,K ) and y E L M ( I ,J ) is given by y' o y,see definition 2.24. We shall write Ln := L{l-n) for n E NO. Remark 2.27 (Associativity and the identity morphisms) It is easy to see that the composition is in fact associative. The identity of the object L' in LM is represented by the Jacobi diagram over I u I given by il -i;
in -i:,
(2.62)
with I = (21,. . . , i n } , n = # I .
Definition 2.26 (The k-linear symmetric monoidal category L M ) We make LM into a k-linear symmetric monoidal category by setting
@ L 4 := L k J j , J
'3
(2.63)
j€J
for every finite family
( I j ) j €J
of finite sets and
(2.64)
for finite families ( I j ) j €J and (I;)?€J of finite sets and morphisms yj E
L~ (rj ,I;).
Remark 2.28 (Unit object) The object Lo is the unit for the tensor product in the category L M .
Graph homology
61
Example 2.25 (Tensor product of morphisms) One example for forming a tensor product of morphisms LM({ll, (4)) x L'({2,
31, (5,611
+
L'({1,2,
(2.65)
31, (4,5761)
is given by
'x; ": l
41-
@
3
o
4
()2.66
=
3
.
(2.66)
Remark 2.29 (PROP of metric Lie algebras) The category LM is a PROP, i.e. we have a functor of symmetric monoidal categories from the symmetric monoidal category of finite sets with bijections between finite sets as morphisms to LM being bijective on objects and injective o n morphisms. It is caEled the PROP of metric Lie algebras. For more on this, see [Hinich and Vaintrob (2002)] and below. Example 2.26 (Symmetry) Let # : I + J be a bijection between the finite set I = ( 2 1 , . . . ,in} with # I = n and J . The morphism +* E L'(I, J ) is then given by il
-i;
in -i; 2.3.5
(2.67)
The universality of the PROP of metric Lie algebras
Definition 2.27 (Algebra over LM) An algebra over LM is a k-linear tensor category C together with a functor F : LM + C of k-linear symmetric monoidal categories. Remark 2.30 (Category of L'-algebras) category with the obvious morphisms.
The algebras over LM form a
Let us define the morphisms a E LM(2,1), b E LM(2,0) and c E
LM(0,2)by a=
'>it
2
, b=
and
c=
(2.68)
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
62
Remark 2.31
(Decomposition of morphisms) Every morphism in LM can be written as a k-linear combination of compositions of tensor products of the morphisms a , b, c and idL1.
Example 2.27 (Decomposition of a morphism) The morphism
(2.69)
in LM({ 1,2,3}, { 4,5}) is given by the composition (idL1 8 a) o (idL1 8 a @ idL1) o ( c 8 idL1 8 idL1) o (a 8 idL1)o (idL1 8 a 8 idL1) o ( c 8 idL1 @ idL1) o (a 8 idL1).
(2.70)
(We have suppressed the “relabelling morphisms”.)
Proposition 2.4 (The “universal” metric Lie algebra object) The tuple (L’,a,b, c ) i s a m e t n c Lie algebra object in the category LM.
Proof. We have to check if the axioms for a metric Lie algebra object are fulfilled: The antisymmetry of a follows from the AS-relation; the symmetry of b and c is evident (the edges of a Jacobi diagram are not oriented!); that b and c are adjoint means diagrammatically that 2-1
(” 1’ -4
O
31
=1’-1
(2.71)
4
in L M ( l ,l),the compatibility of a with b is again evident (we fix only cyclic orders on the internal vertices, not total ones!); and the Jacobi identity is in fact the IHX-relation
:+ 3
+
!+
+ +’
=o (2.72)
in LM(3,1).
[]
Proposition 2.5 (L1 as the initial object) The metric Lie algebra (L1,a,b, c ) is a n initial object in the category of metric Lie algebras, i.e. for
Graph homology
63
every metric Lie algebra there exists exactly one morphism of metric Lie algebras from (L1,a, b, c ) to this metric Lie algebra. W e call that morphism the classifying functor.
Proof. If there is a functor F : LM -+ C representing a morphism of metric Lie algebras mapping the Lie algebra object in LM to ( L ,a, b, c), it is necessarily unique. It has to map the object L’ of LM with I being a finite set to L@’ as it is a functor of symmetric monoidal categories, in other words it is uniquely defined on objects. Furthermore, it has to fulfil F(a) = a , F ( b ) = b and F ( c ) = c. Now, every morphism in LM can be written as a sum of compositions of tensor products of the morphisms a, b, c and idL1. This shows that F is uniquely defined on morphisms as well as F is k-linear and respects tensor products and composition. To construct such a functor F , recall the definition of FL for the given metric Lie algebra object ( L , a , b , c ) . Let F map L’ to L@’. For y E LM(I,J ) with I and J being finite sets, set
F ( y ) := F (’
@ id@’) o
: L@’
(idgr 8 c*’)
4
L@’.
(2.73)
By proposition 2.2 and the following lemma, F is a functor of k-linear symmetric monoidal categories from LM to C. This functor fulfils 0 (F(L1),F(a),F ( b ) ,F ( c ) ) = ( L ,a, b, 4.
Lemma 2.4 (Compatibility with composition) Let ( L ,a, b, c ) be a metric Lie algebra object in a k-linear symmetric monoidal category C and I , J, K finite sets. Then (2.74)
F(7’ O 7) = F ( r 9 0 F(y)
for all y E LM(I,J ) and 7’ E LM(J,K)where F is defined as in (2.73).
Proof.
Let
7r
F(y’ 0 y) =
be as in Definition 2.24. Then @ idFK) o (id:’
@c
@ ~ )
= (F&,7,),r 8 idFK) o (id?’ 8 c g K ) = ((F(&, 8 c@’) 8 idfK) o (id:’
8c @ ~ )
= ( ( ( F t8 F$) 8 c@’) 8 idFK) o (id?‘ 8 c @ ~ )
= (F: 8 idZK) o (idEJ 8 c @ ~o F ()’
8 id?’) o (id?’ 8 c*’)
= F(y’) O F ( y )
(2.75)
by several applications of lemma 2.3 and one of lemma 2.2.
0
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
64
The following corollary of the previous proposition is the reason why
LM is called the PROP of metric Lie algebras. Corollary 2.2 (Category of metric Lie algebras) The category of metric Lie algebras is naturally isomorphic to the category of algebras over LM. The isomorphism is given by sending a functor F : LM + C that defines an algebra over LM to the metric Lie algebra object (F(L1),F(a),F(b),F ( c ) ) in C. Proof.
The isomorphism is well-defined, i.e. (2.76)
is in fact a metric Lie algebra object in C. This follows directly from the fact that (L1,a, b, c) is a metric Lie algebra object in LM and F is a functor of k-linear symmetric monoidal categories. The rest follows from [] proposition 2.5.
Lemma 2.5 (Gradings) Let u s assume that the homomorphism groups of C are graded as k-vector spaces, i.e. a graded k-linear symmetric monoidal category. Let u s further assume that [., .] is homogeneous of degree 0, and b and c are homogeneous of degree -2 and degree 2, respectively. Then F is a functor of graded k-linear symmetric monoidal categories i f the grading on the homomorphism group LM(I,J ) is given by k+#J-#I f o r homogeneous elements y E LM(I,J ) with k internal vertices. Proof. First note that the degree of CY is 4. From this and the construction of F(y) one shows that the degree of F ( y ) is -2e 4k 2 j where 2e is the number of edges of y. Finally recall that for a Jacobi diagram with 1 := # I + # J legs and k internal vertices, the equation 2e = 3k 1 holds.
+ +
+
0
2.4
Weight systems
To prove naively that two chains of Jacobi diagrams are not homologous, i.e. equivalent with respect to the AS- and IHX-relation, can be a quite tedious work. A better way to distinguish between graph homology classes is to calculate invariants of them. This is one purpose of the invention of the so-called weight systems.
65
Graph homology
2.4.1
Definition
Let k be a field of characteristic zero. Definition 2.28 (Weight system) Let W be a No-graded k-vector space. A weight system (on the space of graph homology) with values in W is a k-linear map
f:B--+W
(2.77)
that is degree-preserving with respect to the grading on number of legs.
B given by the
Definition 2.29 (Morphism of weight systems) Let f : B 4 W and f' : x3 -+ W' be two weight systems. A morphism between f and g is a degree-preserving k-linear map 4 : W -+ W' with 4 o f = f'.
Remark 2.32 (Category of weight systems) With this notion of a morphism, all weight systems form a category. Example 2.28 (The universal weight system) The identity map ida : B --+ B is a weight system, called the universal weight system.
Remark 2.33 (The universal weight system is initial.) The universal weight system is the initial object in the category of weight systems. 2.4.2
Constructions of weight systems
Proposition 2.6 (The graph homology space as a weight system) There exists exactly one isomorphism B --+ SL making the following diagram in the category of graded k-vector spaces commutative: @,"=o
-
B,l ,.4, 63,"=ovJn)
1 B
-
1
(2.78)
SL. Here the vertical arrows are given by the forgetful map and the projection onto the external symmetric algebra, respectively. The top arrow is just the . identity (recall that r(Ln) = L"(0, n ) = B{,,...,,})
Proof. Both vertical maps are surjective and the kernel of both maps is the same. It is given in both cases by the subspace of 23 generated by elements of the form y - (T* o y where y is an arbitrary element in B and 0 E 6, a permutation of (1,. . . ,n}. This proves the claim.
66
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Now we can understand the role of the vector space B better: it is just the external symmetric algebra of the universal metric Lie algebra object.
Example 2.29 (Construction of metric Lie algebra weight systems) Every metric Lie algebra object L in a k-linear symmetric monoidal category C gives rise to a weight system: Let F : Lc 4 C be the unique functor that maps the Lie algebra object L in LM to the metric Lie algebra object L. This functor induces by naturality of S a k-linear map SL S L . Composed with the natural isomorphism B -+ SL this yields a weight system -+
w L: B
-+
SL,
(2.79)
called the L-weight system. For example, every metric Lie algebra in the category of finitely dimensional vector spaces (e.g. gI, with the inner product (A, B ) = tr(A o B ) ) gives rise t o a weight system. For more on these weight systems, see for example [Bar-Natan (1995)l.
Modules of metric Lie algebra objects
2.4.3
Let ( L ,[., .], b, c) be a metric Lie algebra object in a k-linear symmetric monoidal category C.
Definition 2.30 (Modules of metric Lie algebras) A module (R,4) of ( L ,[., b, c ) or simply an L-module is an object R in C together with a morphism 4 : L 8 R -+ R such that
.I,
(2.80)
Let ( R ,4) be an L-module that is a rigid object in C (for the definition of rigidity, see e.g. [Deligne (1990)l). In particular, there is a trace morphism tr : R -+ 1and we can view 4 as a morphism L -+ W ( R ) .We use this t o define a morphism Q, := tr(4gn) : L@, -+ 1
for each n = 0,1,. . . . Let c : SL symmetrising.
+ T L be
(2.81)
the canonical lift given by totally
Graph homology
67
Definition 2.31 (Weight system associated to modules of metric Lie algebras) The weight system
wLJ :=
wL): a -+ q i )
(2.82)
is called the ( L ,R)-weight system.
Remark 2.34 (Sawon and Thompson) I n the case of Rozansky-Witten invariants (see next chapter), this weight system was defined by Sawon and independently by Thompson.
Operation with graphs and special graphs
2.5 2.5.1
Special gmphs
In this section we list some graph homology classes that will play an important role in the applications of graph homology to hyper-Kahler geometry that we have in mind. Let k be a field of characteristic zero. Definition 2.32 (Empty graph) The graph homology class induced by the empty graph (the graph whose set of flags is the empty set) is denoted by 1. Definition 2.33 (l-graph) The graph homology class induced by the Jacobi diagram with two legs, one edge and no internal vertices is denoted by e. Definition 2.34 (The wheels) Let k = 0,1,. . .. We denote by wk the graph homology class induced by the Jacobi diagram
(2.83) 7
k spokes
if k
> 0 and set w o := 0. The element wk for any k is called the k-wheel.
Remark 2.35 (“Odd” wheels) B y the AS-relation, the “odd” wheels WZk+l,lc = 0,1,. . . vanish.
C h e r n Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
68
Definition 2.35 (Double-wheels) For k l , k 2 = 0,1,. . . we denote by W k l , k 2 the graph homology class of the Jacobi diagram
-
(2.84)
kg spokes
called a double-wheel. In particular, we call 0 := WO,O the Theta-graph (homology class). Definition 2.36 (Universal wheel) Let us call (2.85)
the universal wheel. 2.5.2
Operations
In this subsection, we shall define different operations that produce new graph homology classes in terms of old. In the next subsection these operations are related t o weight systems. Let k be a field of characteristic zero. Let y,yI E I3 be two homogeneous graph homology classes. Let ,B E B, and ,BJ E BJ be lifts of y and y’,respectively. The image of p u ,B’ E B I ~ under J the forgetful map is independent of the chosen lifts. We denote this image by y U y’. Definition 2.37 (Disjoint union of graph homology classes) The induced map (by k-linearity)
B@Ba,(YlY?- + Y U Y ’
(2.86)
is called disjoint union of graph homology classes or simply the cup-product. Often, the “U”-sign will be omitted, i.e. we shall write yy’ instead of
u Remark 2.36 (Gradings and the cup-product) The cup-product is klinear and respects the diflerent gradings o n B.
Graph homology
69
By mapping 1 E k t o 1 E B,the space B together with the cup-product becomes a graded k-algebra. From now on, we will view B in this way as a k-algebra.
Example 2.30 (Subalgebras of subalgebras of B.
B) The spaces k[O] and BO are k-
Let again y and y’ be two homogeneous graph homology classes and E BI and p’ E B J , respectively, lifts of these classes. Let us denote by ?(?I) the image of
P
b(P’)=
@
(P U P ’ ) / { ( i t f(2)): E 1) E
(2.87)
J’cJ # J’ = # J - # I
f:I-J
injective
under the forgetful map. This element in B does not depend on the specific choices for the lifts p and p’. Further, let E B J ~i ,= 1, . . . ,n, be lifts of homogeneous graph hornology classes ~i E 13, and let us denote by rjrl ,..., ,.,(~l,. . . ,yn) with ri E NO the image of
(2.88)
under the forgetful map. This element in B does not depend on the specific choices for the lifts p and , . . . ,pn.
Definition 2.38 (Differential operator “?’) By every y E B the “differential operator”
3 : B -+ B we denote for
? : yf H ?(y’).
(2.89)
Here, we have extended the previous definition by k-linearity. Further, we define the “multidifferential operator” ?rl,...,r,,
: (71,. ..,Tn)
?r1 ,...,T,,(T~,.-.
, ~ n )
(2.90)
by k-linearity. Pictorially, the operator of y’in all possible ways.
3 glues all the legs of y t o
(some of) the legs
70
Chern Numbers and R W-Invariants of Compact
Example 2.31 (Operator
w2)
Hyper-Kahler Manafolds
For example,
2.91()
With this picture in mind, the statements in the following remark should be clear. Remark 2.37 (Compatibility with the Ic-algebra structure) The map : 8 4 End(B), y H b is a map of k-algebras, i.e. h
yy' = 9 0 y for y, y' E i.e.
B. The image
(2.92)
lies in the set of &-linear endomorphisms,
of
(2.93)
W 7 ' 0 =TwY')
for y,y' E B and y" E a,. For every y E Bo,we have T = y U .. If y is homogeneous with 1 legs, we have 3 = T l . Further, 90 = y U .. The operator TTl,,,,,r,, is symmetric. The next proposition gives the reason why we call T a differential operator. Proposition 2.7 (Multi-differential operator) The operator Trl ,...,,-, is a multi-differential operator of multi-degree ( q ,. . . ,rn). With respect t o the number of legs, it is homogeneous of multidegree ( - r l , . . ., -rn).
Proof. The claim on the multidegree follows directly from the definition. The other claim will be proved by induction. Firstly we have %,rz
,...,r,(Y~t...,yn) = ~ 1 9 r z,...,r , ( c ~ 2 1 . . - i ~ n ) -
(2.94)
Secondly Trl,...,r,
(71yi1721. . .,yn) =
C
Tp,q,rz ,...,r, (711
. . y n ) . (2.95) i
P+q=Ti
From this and the symmetry of +... the claim follows by induction. Example 2.32
0
(Glueing of two legs) We set
a := -i : a -+ a. 2
(2.96)
71
Graph homology
Acting by d on a graph homology class means to glue two of its legs in all possible ways. For example, we have 2k-2 dW2k
=k
Wn,2k-2-n
(2.97)
n=O
f o r k = 1 , 2 , .... We continue to denote by y and y‘ two homogeneous graph homology classes and @ E 231 and @$, respectively, lifts of these classes. Let us denote ’ ) image of by ( 7 , ~ the (2.98)
under the forgetful map. This element in B does not depend on the specific choices for the lifts @ and p’. Definition 2.39 (Inner product) The induced map (by k-linearity) 2.99()
is called the inner product of B. Again it helps a lot if we view the inner product pictorially: (y,y’)is the graph that we get when we glue all the legs of y to all the legs of y’.
Remark 2.38 (Inner product) The inner product is &-linear. If y and y’ are homogenous with respect to the number of legs, (y,7‘)= 0 whenever the number of legs of y diflers from the number of legs of yl. The map ( I l .) : B + B is exactly the projection o n the summand BO of B. Using this, the inner product can be expressed in terms of :. W e have (2.100)
Example 2.33 (C/2 and
a) In particular, we have (2.101)
Remark 2.39 (Completions) As the space 23 is graded, we can complete this space with respect to the different gradings. A s long as the previously
72
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
defined operations respect the gradings, they can also be completed to the completed spaces. For example, the element
(2.102) lives in completions of B that are complete with respect to the grading given by the number of legs (the default grading of B). In what follows, we shall make plenty use of these “limits” like exp(l/2), also to simplify the notation. However, we won’t name the completion in which these limit elements live as it should always be easy to see which completion is needed. Therefore, by abuse of notation, we shall denote all completions by the same symbol,
ti. Proposition 2.8 (On exp(d)) For y,y‘E
B, we have
(exp(Wyyf)71) = (exP(a)r,e x P ( m f ).
(2.103)
Proof. We may assume that y and yf are homogenous of degree 1 and 1’ such that 1 I‘ = 2 n with n E No. So, we have to prove
+
d“ -(yy’) n!
=
g (
)
’
(2.104)
m,m’=O 1-2m=1’-2m’
since ( - , l )means to remove the components with at least one univalent vertex. By the definition of (., .) and the fact that applying $ on a graph homology class means to glue all subsets of 2k of its univalent vertices to k 0 pairs in all possible ways, this is true. 2.5.3
Closed and connected graphs
Recall that the space I3 of graph homology classes is in particular graded by the number of connected components and the number of legs. Definition 2.40 (Closed graphs and closure) A graph homology class y is called closed if it is homogeneous with respect to the number of legs and has no legs. The closure of y is defined by (7):= (7, exp(l/2)) *
(2.105)
Remark 2.40 (The closure is closed) The closure of a graph is in fact closed, see remark 2.39.
73
Graph homology
Example 2.34 (Taking the closure) Taking the closure means connecting all legs of a graph homology class pairwise in all possible ways. For example, (w*)=4.
@
+2.
@
(2.106)
For every finite set, let P2(L) be the set of partitions of L into subsets of two elements. Let 9 be a graph homology class over L that is mapped to y via the forgetful map. Remark 2.41 (Closure) B y the definition of the inner product (., .), we have that (y) = (y,exp(C/2)) is the image of (2.107) X€PZ(L)
under the forgetful map l3, 4 Bo.
Definition 2.41 (Connected graphs and connected components) A graph homology class y is called connected if it is homogenous with respect to the number of connected components and has exactly one connected component. The connected component of y is the homogeneous (with respect to the number of connected components) part of y that has degree 1 with respect to the number of connected components. Example 2.35 (Connected component) The number of connected components of the graph homology class 1 is zero. An example for a connected component is for example w2 as a connected component of 1 w2 w4w6.
+ +
Definition 2.42 (Connected closure) The connected closure ((7)) of a graph homology class y is defined to be the connected component of its closure (7). Example 2.36 (Connected closure) We have (w2) = ( ( ~ 2 ) )= 0 and (w,”)= 2 0 2 + O2 and ((w,”))= 2 0 2 , where 0 2 is the graph (2.108) Let L1,. . . , L, be finite sets. We set L := +=l:J partition of L in 2-element-subsets.
Li. Let
?r
E Pz(L) be a
Definition 2.43 (Linked legs) A pair 1,l‘ E L is linked by ?r if there is an i E (1,. . . , n } such that 1,l’ E Li or {1,1’} E ?r. We say that n
74
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
connects the sets L1,. . .,L, if and only if for each pair I , 'I E L there is a chain of elements 1 1 , . . . ,lk such that 1 is linked to 11, 1; is linked to li+l for i = 1,.. .,k - 1 and lk is linked to 1'. The subset of P2(L) of partitions connecting L1,. . . ,L, is denoted by P2({L1,.. . ,Ln}).
Remark 2.42 have
P2(L) =
(Decomposition of the set of connecting partitions) W e
u {u
3={1,...,n }
7rI : 7rI E
1
P2({Li : 2 E I}) .
I67
(2.109)
Let 71,. . .,yn be connected graph homology classes over L1,. . .,L,, respectively. We denote by y := uy=l yi the product over all these graph homology classes. It is a graph homology class over L := Li.
Lemma 2.6 (Decomposition of the closure) For every partition T E P2(L) the graph y/n is connected if and only zf n E Pz((L1,.. . ,Ln}). Further (2.1 10)
Proof. As y is the disjoint union of the connected graph homology classes yi, the graph y / is~connected if and only if, by the process of glueing the legs of y,every yi will be linked to another yj. This is by definition 2.43 of course the case if and only if 7r links L1,. . . ,L,. Using the decomposition (2.109), we calculate
Remark 2.43 (Extension to graph homology classes) Via the forgetful maps, equation (2.110) is also true if y and the yi are graph homology classes in B.
Proposition 2.9
(Closure and connected closure) For every connected graph homology class y,we have exp((4) = (exp7).
(2.112)
75
Graph homology
Proof.
By (2.110) we have
( y n )=
c
n((r#I)) =
k J 3={1, ...,n } I E 3
c n n!
XP(n)
&(((yi))/i!)A*
i=l
for each n = 0 , 1 , . . . . Using this, we finally calculate
2.5.4
Polywheels
Definition 2.44 (Polywheels) For each n = 0,1,. . ., we set Let X be any partition. Recall that we have set
W,
= -wn.
(2.113) i= 1
The closure (Wx) of Wx is called a polywheel. The subspace in BO spanned by all polywheels is denoted by W and called the polywheel subspace. The subalgebra in BO spanned by all polywheels is denoted by C and called the algebra of polywheels. The connected closure ((Wx)) of w x is called a connected polywheel.
Remark 2.44 (The size of the space of polywheels) First recall that all “odd” wheels vanish, i.e. only the polywheels of the form (Wzx) do not vanish. Therefore, there are at most p ( n ) linear independent polywheels with 2 n internal vertices where p ( n ) := # P ( n ) is the partition function. O n the other hand, calculations in BO show that the dimension, say b ( n ) , of the subspace in 130 spanned by homogeneous graphs with 2 n vertices grows quicker than p . Therefore definitely, W # Bo. O n the other hand, it is unknown i f the inclusion C c BO is proper. Remark 2.45 (Subalgebra spanned by connected polywheels) The subalgebra in l30 spanned by all connected polywheels equals C . This is true since we can use proposition 2.9 to express every polywheel as a polynomial of connected polywheels and vice versa.
76
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
Example 2.37 (Expressions of connected polywheels in terms of polywheels) Using proposition 2.9 one is able to calculate expansions of the connected polywheels in terms of polywheels. We have done this up to eight internal vertices:
and
2.5.5
The Hopf algebra structure on the space of g r a p h ho-
mology Let k be a field of characteristic zero. The space of graph homology B has also a coproduct over k :
Definition 2.45 (Coproduct) Let A : B
-+
23 @ B be defined by (2.115)
With respect to Ul this is a “very nice” coproduct, i.e. we have:
Proposition 2.10 (Hopf algebra structure) The space of graph cohomology together with the product u and the coproduct A becomes a Hopf A) algebra, i.e. (a,u) is a commutative associative k-algebra with unit, (B, is a cocommutative, coassociative k-coalgebra with counit and
A : ( B 1 U )+ (Blu)@ ( B I U ) is a homomorphism of k-algebras.
(2.116)
77
Graph homology
Proof. The counit : B -+ k is given by projecting a graph homology class to its homogeneous (with respect to the number of components) part with no components, which is always a multiple of 1. It is easy to verify the claims about the algebraic structure. Explicitely we shall only show that A is a homomorphism of k-algebras: Let y,y' E f3. Then
(2.117)
Let C be a k-linear tensor category and (L1,a l , b ~c1) , and (L2,a2, b2, c2) be two metric Lie algebra objects in C. Note that there is a natural morphism
SL1@SL2
+
S ( L @ L2)
(2.118)
given by the following procedure: Given a1 E l?(Ly1)/6inland a2 E r ( L p ) / G n zfor 721, 722 = 0,1,. . . , we map a1 8 a2 as follows: let &I and &2 be lifts of a1 and a2 to I'(LFnl) and r(Ly2), respectively. We then and finally inject &1 8 b2 E I'(L?l 8 L f n 2 ) into r ( ( L 1 @ L2)@(nlfnz)) / 6 ~@~L2). +~~ project down to r ( ( L 1 @ L ~ ) @ ( ~ l + ~ z )E) S(L1
Proposition 2.11 (Coproduct and Lie algebra weight systems) The diagram
f3
.l BOB
-
S(Ll@L2)
-
T
(2.119)
SL18SL2
where the horizontal maps are given by the weight system associated to L1 @ L2 (see example 2.22) and L1 and L2, respectively, is commutative.
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manafolds
78
Proof. First, let us note that the coproduct can be “lifted” to a coproduct on the morphism sets L’(I, J ) , namely:
r1%-Y2=r Now, consider the functor F from the category LM to C given by mapping the object L@’ to (L1 @ L2)@”and that maps morphisms in LM(I,J ) via the composition
LM(I,J ) -5
@
L M ( L51) €9 LM(12,5 2 )
I1YI2=I J1YJz= J
Homc(Ly”, LyJ1) €9 Homc(Lf12, LFJ21
@
-+
I1W I 2 =I J 1 Y J 2 =J
-
Homc((LI@&)‘I, (LI @ Lz)@’) (2.121)
where the second map is given by the two functors classifying the metric Lie algebra objects L1 and L2, respectively, and where the last map is the natural one. The functor $’ is a functor of k-linear symmetric monoidal categories and it maps the metric Lie algebra object LM together with the structure morphisms to the metric Lie algebra object L1 @L2 in C. In other words, is the classifying functor for L1 @ L2. This more general result has the proposition as an easy corollary: Just “project down”. 0 Obviously, A(?) = y €9 1 + 1 €9 y for connected y. There is also a nice formula for A(expy) for such a y:
Proposition 2.12 (Connected graphs and the coproduct) For each connected graph homology class y, A(exp y) = exp y €9 exp y.
Proof.
(2.122)
This is straight-forward:
A(expy) = n=O
= n!
PA=O
p ! . q!
=expy@expy.
(2.123)
0
79
Graph homology
The Wheeling Theorem
2.6
2.6.1
The wheeling element st
Let k be any field of characteristic zero. Every power series f E k [ [ t ]with ] constant coefficient 1 defines a graph homology class yf by setting (2.124) with log f = a 1 z
+ a22’ + . . ..
Lemma 2.7 (“Only the ‘even’wheels matter.”) For the given f ( t )E k [ [ t ] ] define g(t) = ( f ( t > f ( - t i) .) Then
37 = 7 g -
(2.125)
Proof.
This follows from the fact that logg = $(logf(z) and that the “odd” wheels vanish.
+
logf(-z))
0
Set (2.126) Definition 2.46 (Wheeling element) We call the graph homology class
R
(2.127)
:= Yf
the wheeling element.
In other words, (2.128) where the
bk
are defined via sinh f (T) . 1/2
00
= In
(2.129)
k= 1
The
bk
are called the modzjed Bernoulli numbers due t o the following:
80
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manafolds
Remark 2.46 (Modified and ordinary Bernoulli numbers) W e have bk = 0 f o r k = 1 , 3 , . . . , bo = 0 and (2.130)
where the
Bk
are the ordinary Bernoulli numbers for each k = 1 , 2 , . . . .
Recall that the Bernoulli numbers are defined via (2.131)
For later usage, let us set (2.132)
where R k is the homogenous (with respect to the number of internal vertices) part of R whose number of internal vertices is k. 2.6.2
Wheeling and the Wheeling Theorem
Let y,y’E 23 be homogeneous with respect to the number of legs such that y has n legs and y’ has n’ legs. We define s(y) E B~l,,,,,n) by taking a lift 9 E B{l,,,,,n}and mapping this lift to $CDE6,o,9. The graph s(y’) E B{l,,,,,nt}is similarly defined. We define the graph y x y’E B to be the image of
47)u s(7’) E B{l,...,n}kY{l, ...,n’} = B{l,...)n+n’}
(2.133)
under the forgetful map.
Definition 2.47 (Cross-product) We call the induced product
B €9 B + B 7 y€9 y’--t y X y/
(2.134)
the cross-product.
Remark 2.47 (A new algebra structure) With respect to this product, B becomes a &-algebra. W e have (7 x 7’71) = (7 u 7’7 1)
f o r all y7y‘ E B.
(2.135)
Graph homology
81
The wheeling element intertwines the two algebra structures (B, U) and (4x):
Theorem 2.1
(Wheeling Theorem) The morphism
fl : (a,u)
--t
(a,x )
(2.136)
is a morphism of algebras.
Proof.
cl
See the original proof in [Thurston (2000)].
Corollary 2.3
((a,.) is multiplicative.)
For all y,y’E B , we have (2.137)
(0,77’) = (Q, 7) u (R, 7’).
Proof.
We have
(Q,YY’)= (fl(rr’), 1) = (QY) x fl(Y’), 1) = (wdu
fw), 1)
= (0,y) U (Q, 7’).(2.138)
0 Corollary 2.4
(Application of
? on R) For y,y’E a, we have
(?(Q>,7’) = (?(Q),
Proof.
Note that (?(R), y’)
Remark 2.48
=
1) u (a,7’).
(h,yy‘).
(2.139)
0
(Thurston) I n fact, already ?(Q) = (?,Q)Q
(2.140)
is true; see [Thurston (2000)].
Corollary 2.5 (Wheeling element as a n eigenvector to 6’) The wheeling element is an eigenvector to the operator d , i.e.
t2Q d Q ( t ) = -Q(t). (2.141) 48 Proof. We can assume that t = 1 as the equation respects the grading with respect to the number of internal vertices. Now recall that d = j / 2 and note that (2.142)
0
82
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Remark 2.49 (Elementary proof) For an elementary proof of this fact (and this will be the only corollary f r o m the Wheeling Theorem we shall actually be needing), see [Nieper (2003)].
Chapter 3
Rozansky-Witten theory
3.1
The Rozansky-Witten weight system
Now we are ready combine what we have learnt in the previous two chapters: the geometry of compact hyper-Kahler manifolds and the combinatorial data of the Jacobi diagrams. For this approach, see also [Roberts (2001)]. 3.1.1
The derived category
Let X be a compact complex manifold.
Definition 3.1 (Derived category of coherent sheaves) We denote by D(X) the derived category of bounded complexes of coherent sheaves on
X. Recall the definition of the (bounded) derived category of an abelian category A: First, one considers the category Kom(A) of bounded complexes over A. Identifying two maps whenever they are homotopic yields the category K(A) whose objects are bounded complexes of objects A and whose morphisms are maps between bounded complexes defined up to homotopy. In this category, a morphism that induces an isomorphism on the level of cohomology, is called a quasi-isomorphism. Now, the derived category of A is the category K(A) localised by the quasi-isomorphism, i.e. its objects are again bounded complexes of objects of A and among its morphisms are not only maps up to homotopy but also formal inverses of the quasi-isomorphisms. For the construction of D(X) the abelian category of coherent sheaves on X plays the role of A. Note that the derived category of an abelian category is in general not abelian anymore. However, it is additive and moreover possesses a structure 83
84
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
of a triangulated category (for this, see e.g. [Gel'fand and Manin (2003)l). We strongly advise those readers who are not familiar with derived categories to consult a textbook on this topic first. We recommend [Gel'fand and Manin (2003)l and also the beginning of [Hartshorne (1967)l.
Remark 3.1 (Coherent sheaves as complexes) Let us denote the abelian category of coherent sheaves o n X by Coh(X). There is a canonical additive functor Coh(X) -+D(X), mapping a coherent sheaf 3 to the complex that has only one non-vanishing component, namely .F in degree 0. In what follows, we shall suppress this functor in the notation] i.e. we shall freely view coherent sheaves as objects in the derived category of X.
Definition 3.2 (Shift functor) The autofunctor of D(X) that moves the objects, i.e. complexes of coherent sheaves, by one degree to the left, is called the shift functor and denoted by T : D(X)
(3.1)
+ D(X).
We abbreviate T"(-)by .in] for n E 2, e.g.
T"(X)
(3.2)
=X[n]
for X an object in D(X).
Definition 3.3 (Extension groups) For every n E Z and XIY objects in the derived category, we set Ext"(X,
Y):= HomD(x)(X,Y [ n ] )
(3.3)
where the abelian group of morphisms on the right hand side is to be understood as the group of morphisms in the derived category.
Remark 3.2 (Extension groups) For E and 3 two coherent sheaves o n X, we have
EXt",,,(,)(E,
n1
3)= Ext"(E1
(3.4)
where the Ext-group on the left hand side is the derived functor in the category of sheaves on X and the Ext on the right hand side is the one previously defined in definition 3.3. This identification identifies the Yoneda product with the composition in the derived category. For a proof, see e.g. [Gel 'fund and Manin (2003)].
Rozansky- W i t t e n theory
85
Definition 3.4 (The graded derived category) By D(X) we denote the graded derived category of X , that is the category whose class of objects is the class of objects of X, The set of morphisms between two objects K and L , however, is given by the graded abelian group Homg(x,(K, L ) := @ Ext”(K, L).
(3.5)
nEZ
Remark 3.3 (D(X) as a subcategory) Obviously, D(X) can be viewed as a subcategory of D(X). Example 3.1 (Cohomology classes as morphisms) Let a E H k ( X , F ) be a cohomology class on X. This cohomology class can be viewed as a morphism in D(X). We have Q
E H k ( X , 3 )= EXt&,,(,)(ox,F)
c H o m f i ~ x ) ( ~ x , ~ ‘ [ ~ (3.6) ])-
The category of coherent sheaves on X is a tensor category with respect of coherent sheaves, i.e. 8 is an additive bifunctor to the tensor product “8” and Coh(X) together with @ becomes a symmetric monoidal category. The bifunctor @ is right-exact (in both arguments) and it is acyclic with respect to, e.g., flat sheaves. Therefore, there exists the derived functor f
@L
- : D(X) x D(X)
--f
O(X)
(3-7)
with
Tor”(€, F)= Hn(€ 8L3)
(3.8)
for n = 0 , -1,. . . . In what follows, we shall just write “8”instead of “@L’r. Important for us is that D(X) together with @ becomes a tensor category. In fact, it is a graded C-linear tensor category. With respect to this structure, the shift functor T is an odd operator.
3.1.2
A metric Lie algebra object in the derived category
Let ( X ,0) be a holomorphic symplectic manifold. The cohomology class c E Ho(X,Qg)induces an alternating map
5 : o p 4 ox.
(3.9)
If U is an open subset of X and 01,. . . ,e2” are holomorphic 1-forms over U with 0 = eai-I A can, we have 8 = C? 2=1 (82i-1 @ @2i- Oai @ e2aP1). The map 8 induces an isomorphism V, : O x + Ox.
cyZl
86
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Definition 3.5 (Dual of the symplectic form) We call the map 6 := -(Vu-l8 vu-’)(b): Ox +
@?’
(3.10)
the “dual” of b. Here, we have viewed 6 as an element in O x -+ O?’. The reason why we call it the dual is the following lemma:
Lemma 3.1 (“The dual is dual.”) The map Ox
idex @a
>@idex
O X ~ @ X ~ @ X
Ox
(3.11)
i s the identity.
Proof. The proof is just general nonsense. Just note that the minus sign 0 in (3.10) is crucial here because b is alternating.
Definition 3.6 (Definitionof ax, bx and CX) If we view 3 as an extension class and therefore as a morphism in the graded derived category, instead of b we write bx E HomT2 (@x[-l]@’, O x ) D(X)
= EXtg,h(X)(@$2,O x ) .
(3.12)
Similarly for 6 viewed as an extension class, we write
OX[-^]@') = EXt&(x)(OX,
cx E Hom&,)(Ox,
@y).
(3.13)
Finally, we use the symbol
ax E H ~ m & ~ ) ( @ x [ - l ] @@x[-l]) ’,
(3.14)
when we think of the Atiyah class (ug, as an extension class.
-
So, we have constructed the three maps
ax : Ox[-1]@’2
bx :@x[-l]@’ and
c x : Ox
[-21
PI
@x[-1], Ox
(3.15)
@~[-1]@~
in D(X). (The bracketed numbers on the arrows denote the degree of the morphism.) With these preparations we can formulate the following theorem that will finally lead to the construction of the Rozansky-Witten invariants.
Rotansky- Wztten theory
87
Theorem 3.1 (Ox[-l] as a metric Lie algebra object) Let ( X , a ) be a holomorphic symplectic manifold. Then (@x[-1l,ax,bx,cx)
(3.16)
is a metric Lie algebra object in the graded derived category D(X) of coherent sheaves on x .
Proof. We have to verify the axioms of a metric Lie algebra in D ( X ) : The symmetry of bx follows from the fact that 8 is alternating and the Koszul sign rule. Similarly, cx is symmetric. Lemma 3.1 expresses the fact that bx and cx are adjoint. To see that ax is anti-symmetric, recall that the Atiyah class "lives" in fact in the summand H1(X,S2Rx @ 0x1 = HomD(x)(A2(@x[-11),O x ) -+ H~m&~)(Ox[-l]@'~,O x ) . (3.17) The fact that the Atiyah class "ex of the symplectic tangent sheaf lies in the direct summand H1(X,R x 8 .End(Ox, a)) translates directly into the compatibility of ax with bx. Just recall the definition of OX, a) and note that a sign change occurs due t o the shift by -1 and the Koszul sign rule. Finally, we have t o show that ax obeys the Jacobi identity. This is 0 exactly the following lemma, applied to & = O x . Let & be a locally free sheaf of Ox-modules. Let us view its Atiyah class a&E H1 (X, R, @ End(Ox)) as a morphism ab E Homg(x)(E 8 Qx[-l],&).
(3.18)
(This construction is totally analogous t o the construction of ax out of
"ex Lemma 3.2 (The Bianchi identity reviewed) We have
(3.19) in H ~ m & ~ , ( Q x [ - l ] B @E~, € ) .
Proof. S2Rx
Recall the definition of S : R x 8 End(&) 8 R x @ End(@x)-+ End(&) from Subsection 1.4.3.
@a(&) and L : ( R x @.End(€))@2
---f
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
88
Now let us translate (3.19) back into cohomological terms, i.e. expressing it first in terms of the ordinary Ext-groups and the Yoneda-product and then translating it further into terms of sheaf-cohomology and the cup-product. The first summand yields
S*(oe U o e x ) E H2(X,S2Qx@End(€)) + H o m ~ ~ x ) ( Q x [ - 1 ] * 2 ~ € , € ) (3.20) (after having been factored over (Qx)*’ -+ S2Rx), and the sum of the second and third summand is equivalent to
L*(CYE U a € )E H2(X,S 2 n x @End(€)) + H ~ m & ~ , ( Q x [ - l ] @ @I €,€). ~ (3.21) In other words, the lemma states nothing else than the Bianchi identity, 0 proposition 1.16.
C o r o l l a r y 3.1 (Locally free sheaves as 8 x [-11-modules) Every locally free sheaf & of Ox-modules o n X is a module of the metric Lie algebra (Ox[-l],ax,bx,cx). The operation of Ox[-l] o n E is given by a€ : Ox[-l] @ € + €. Proof.
This is just lemma 3.2 in view of theorem 3.1.
17
Remark 3.4 (All objects as Ox[-11-modules) The Atiyah class viewed as an extension class does not only make sense for locally free sheaves but for any coherent sheaf and moreover for complexes of coherent sheaves. I n this sense, in fact every object of D(X) is a module of the metric Lie algebra object (Ox[-11, ax,bx, cx).
3.1.3
Rozansky- Witten weight systems
As Ox[-1] is a metric Lie algebra object in the graded derived category D(X) of a holomorphic symplectic manifold (X, c),we have an associated weight system 00
a
+
S(QX[-I]>
@ ~ o m f i j ( x , ~( ~~ ~[ -, l ] @ ~ > / e n ,
(3.22)
n=O
which we shall investigate in this subsection. Proposition 3.1 (The external symmetric algebra over Q x [-11) There is a natural isomorphism of C-algebras from the external symmetric algebra
Rozansky- Witten theory
89
over Ox[-l] t o the Dolbeaut cohomology ring H*(X,R>). This isomorphism is induced degree-wise by isomorphisms S"(Ox[-1])
-+ H*(X,R?).
(3.23)
Proof. First note that for every metric Lie algebra object L in a C-linear tensor category, we have an isomorphism of C-vector spaces
S"L = Hom(1, L@'")/Gn --t Hom(L@",l.)/an.
(3.24)
This morphism is induced by the morphism Hom(1, LBn) + Hom(L@",l),4 ++ bBn o (id:"
o
4).
(3.25)
This together with H ~ m $ ( ~ ) ( O x [ - l ] @Ox)/B, ",
= Horn*-" ( O F , Ox)/G; D(X)
= H*-,(X,
RF)/6;
= H*-"(X,
R>)
(3.26)
yields the claim where the superscript E shall indicate that G, acts via the alternating operation.
Definition 3.7 (Rozansky-Witten weight system) We denote the so constructed weight system by RW(x'") : B
4
H*(X,R*)
(3.27)
and call it the Rozansky- Witten weight system associated to ( X ,u).
Remark 3.5 (Construction of the Rozansky-Witten weight system) The Rozansky-Witten weight system as the composition of 23 -+ S(@x[-l]) with the identification S(Ox[-l]) 4 H*(X,R;) allows the following description. Let y E B be a homogeneous graph homology class with n legs. Let E LM(O,n ) be a lift of y. I n other words, we just number the legs of y from 1 to n in an arbitrary way. W e can view y also as a map in LM(n,0 ) . This corresponds to the isomorphism (3.25). Let the functor F : LM D(X) be the classifying functor of the metric Lie algebra object @x[-l]. Applying this functor to y yields an element
+
--f
F(y) E H ~ r n ~ ( ~ , ( O x ( - l ] OX) @ " , = H*-"(X, O F ) . Finally, apply the natural projection R p
H*-" (X, a;).
-+
(3.28)
a>. This yields RW(xf"'(y)
E
90
3.1.4
C h e r n Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
Properties of the Rozansky- Witten weight system
Let us summarise the most obvious properties of this weight system first: Proposition 3.2 (Rozansky-Witten weight system as a graded algebra morphism) The map RW(x3") : B .+ H*(X,a*) is a morphism of Calgebras. It maps the space of homogeneous graph homology classes with 1 legs and k internal vertices to Hk(X,a",,.
Proof. The cup-product on t3 is the product on the exterior symmetric algebra S ( Q x [ - l ] ) , which in turn corresponds to the cup-product on H*(X,a*). To show the claim about the grading, let y be a graph homology class with k internal vertices and, say, 1 legs. It corresponds to a morphism LM(O,l). By lemma 2.5, which is not by accident exactly applicable to this ]@ x ) .~ , case, this morphism is mapped to a morphism in Homk'l ( Q x [ - - ~ O D(X) The rest follows from the explicit description of the isomorphism of 3.1.
0 Proposition 3.3 (Rozansky-Witten class of l ) We have
RW(X'")(C)= 20.
(3.29)
Proof. Here and in later similar proofs we shall make use of the description as given in remark 3.5: First, the graph C viewed as an element in LM(2,0) is mapped to B E Ho(X, Projecting to Ho(X,a$) yields 20. (Recall the definition of B and recall that the natural map -+ as24 a$ is just multiplication by 2! = 2.) 0 Lemma 3.3 (Hitchin and Sawon) The diagram
End(Qx)
tr
of Ox-modules commutes exactly up to a -1-sign.
OX
T.
(3.30)
Rozansky- Witten theory
Proof.
91
First note that tr
&&(OX)
OX
d
T.
1
*@idm(eX)
ex 8 ex 8 Rx 63 ex of Ox-modules with
:=
'
idex @(.,.)@idexOQ.
(3.31)
Qx8Qx
-
( i 3" 2) E G4commutes due the trace formula
tr:End(Bx)--+Rx8,8x
(.,.)
Ox
(3.32)
and lemma 3.1. As 6 factors over AZQx the result follows by swapping two Ox-factors. 0 Corollary 3.2 (Hitchin and Sawon) The Rozansky- Witten weights of the wheels are given by
RW(x'")(~k)=
-n,(*$k),
(3.33)
i.e. RW(x>")(Wj,)= Tr,(aek),for each k = 0,1,. . . . In particular, = - dim X.
RW(x+')(w~) Proof.
(3.34)
Let us view wk as an element in L M ( k ,0). Then, Wk
= b o (idL 8 p) o
( c 8 idLn)
(3.35)
where
p = a o (a18 idL) o . . . o (a8 id:-')
: Ln+l + L.
(3.36)
n terms
Rom this description, it follows that RW(x7")(wk) is the image of the element
erg: E H"(X, R
p 8M(Bx)@'"
(3.37)
under the map induced by
8' 8 * . * 8 8" 8 A1 8 A, H (6" A . . . A 6,)
@bo
(idex(u) 8 (A1 0 . '. o A,)) o 8 (3.38)
where ,. . . ,8" E R x ( V )and A1, . . . ,A, E M ( Q x ) (V )are local sections over U ,an open subset of X. Therefore the result follows from the previous lemma and the definition of Tk.
92
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
Definition 3.8 (Rozansky-Witten classes) Let us call the classes that lie in the image of the Rozansky-Witten weight system Rozansky- Witten classes. Corollary 3.3
(Characteristic classes) All the characteristic classes of
X are Rozansky- W i t t e n classes. More precisely, (3.39)
where w ( t )E B[[t]] is the universal wheel, see definition 2.36.
Proof. 3.1.5
0
This is proposition 1.20. An inner product o n the cohomology of a holomorphic symplectic manifold
Let (X, c) be a holomorphic symplectic manifold.
Definition 3.9 (Inner product of morphism
a>) Let
for each m, n = 0,1,. . . the
p : c2F @a; + OX
(3.40)
be defined by the following procedure if n = m: Firstly, there is a morphism @ -+ OX given by 6mn. Secondly, by restriction this yields a morphism Anax @ fly OX which by definition of the coinvariants factors over AnQF @ 0; --f Ox. Lastly, compose this morphism with the identification A,Ox -+ a;. If n # m, p shall be the zero map. The map defined by p on the level of cohomology is denoted by
s2y Cly
(.,.) : H*(X,C?>)@H*(X,52>)4 H * ( X , O x ) .
(3.41)
Example 3.2 (Inner product and determinant) Let U be a n open set of X and al, . . . ,an and a”, . . . ,a’n local 1-forms over U , i.e. sections of O x ( U ) . Then,
p(a’ let
A
... Aan,a” A . . . A a I n ) =det((c?(aa,a’j))i,j).
(3.42)
Let the dimension of the holomorphic symplectic manifold be 2 n , and : A*Rx -+ a$’ be the projection of the forms of maximal degree.
Rozansky- Witten theory
93
Lemma 3.4 (Inner product and the symplectic form) For all a E A*Rx, we have (3.43)
Proof. It suffices to prove this formula locally over small open subsets U . . . A a 2 P E R z ( U ) with p E No and
of X . Let us assume that Q = CY' A aiE R x ( U ) . So we have to prove
(3.44)
. . , e2" be a basis of Rx(U) such that
Let
el,.
6=
CZl(e2i @I e2i-1-
e2i- 1@1e2i)for
u=
CZl 62i-1 A e2i. Then
a dual basis e l , . . . ,ezn of 01,. . . ,02n.
We have un = n! . O1 A . . . A 02" and therefore ( a A uneP,8)
(3.45)
0 The relation of this inner product to the "inner product" of graph homology classes is subject to the following proposition: Proposition 3.4 (Rozansky-Witten classes and the inner product) Let y and y' be two graph homology classes. Then
RW(x9")((y,y'))= (RW(X+"'(r),RW(X7")(y')).
(3.46)
Proof. Let us assume that y and y' are homogenous graph homology classes, both with n legs. View y and y' as elements in LM(n,1). Let us totally symmetrise y: this yields the element 7 := CDEB, y o u*. Now consider the element p := (T @ 7')@I cn E L'(O,O). On the one hand, under the Rozanskythis element is mapped to (RW(x>")(y),RW(x+')(y')) Witten weight system, one the other hand, p is just (r,7')when viewed as a closed graph homology class. 0
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manafolds
94
3.1.6
Roaansky- Witten invariants
Let ( X ,a ) be holomorphic symplectic manifold of complex dimension 2n. Definition 3.10 (Rozansky-Witten invariant) Let y E 23 be any graph homology class. The integral
(k) s, 2n
b,(X, a ) :=
RW(xl")(y) exp(a + b)
(3.47)
where E H2(X,Ox)is the complex-conjugate to a is called the RozanskyWitten invariant of (X,a) associated to y.
Remark 3.6 (Hitchin and Sawon) By corollary 3.2, every Characteristic number is a Rozansky- Witten invariant. (Every characteristic number can be written as an integral over X of a polynomial in the chk). In the next subsection we shall investigate this in terms of complex genera. Every Rozansky-Witten invariant is an invariant of a closed graph. By this, we precisely mean: Proposition 3.5 (Rozansky-Witten invariants of non-closed graphs) For every graph homology class 7,we have
b,(X,a) = b ( , ) ( X , 4 . Proof.
(3.48)
We use lemma 3.4 to calculate:
b,(X, a ) =
(k) s, 2n
RW(x>u)(y)exp(a
=
(&)'"
=
(k)'"k
+ 8)
k(RW(X.u)(y),exp(a)) exp(a
+ 8)
+
(RW(Xiu)(y),RW(Xtu)(exp(l/2)))exp(a 8)
(3.49)
0 Proposition 3.6 (Scale invariance) Let y be a graph homology class. For every t E Cx, we have by(X,ta) = by(X,a),
(3.50)
Rozansky- Witten theory
95
i.e. Rozansky-Witten invariants are scale-invariant.
Proof. Reviewing the construction of Rozansky-Witten invariants one sees that RW(Xytu)(y)= t(l-k)/2RW(X,") E Hk(X,
nk)
(3.51)
if y is homogeneous with 1 legs and k internal vertices. For such a graph
3.52()
0
due to degree reasons. From this the scale-invariance follows at once.
Remark 3.7 (Invariance under deformations) In fact the RozanskyWitten invariants of holomorphic symplectic manifolds are invariant under deformations of the complex structure. W e refer the reader to [Sawon (1 999)]. Let (XI,(TI) and (X, ( ~ 2 )be two holomorphic symplectic manifolds. What can we say about the Rozansky-Witten invariants of their product (X,a ) := (XI x X2, prTul pr;c2)? Here pri is the canonical projection from X1 x X2 to Xi.
+
Proposition 3.7 (Rozansky-Witten invariants of the product of two holomorphic symplectic manifolds) I t is
by(X,(T)= (b(x1,"l)@ b(x2,"2))(A(Y)) where the "transpose"
b(xi+'i)
:
B
(3.53)
+ C is defined by b(x""i)(y)
=
by(Xi,gi).
Proof. First note that forming the Atiyah class is compatible with taking products of complex manifolds. Therefore the metric Lie algebra object Ox[-1] is the sum of pr;Oxl[-l] and pr;Qx2[-1] in the category D(X). From this and proposition 2.11 it follows that RW(x,")(y) = (pr;RW(X1,"1)
@
pr3RW(X27"2) ) ( N y ) > .
Integrating over X yields the claim.
3.1.7
(3.54)
0
Complex genera and Rozansky- Witten invariants
Let R be any commutative Q-algebra.
Definition 3.11 (Complex genus) A complex genus 4 (with values in R) is a map from the class of all compact complex manifolds to R such that:
96
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
(1) For each pair of compact complex manifolds X and X’, we have
#(XI = #(X’)
(3.55)
whenever X and X’ are cobordant, i.e. have the same set of Chern numbers. (2) The value of # on a point is 1. (3) For each pair of compact complex manifolds X and X I , we have
# ( X Id X’)
=#(X)
+ #(X’).
(3.56)
(4) For each pair of compact complex manifolds X and X’, we have
# ( X x X’)
= # ( X ) .# ( X I ) .
(3.57)
Remark 3.8 (Complex cobordism ring) Equivalently, a complex genus is a Q-algebra homomorphism from the rational complex cobordism ring RU 18 Q to R. Example 3.3 (Multiplicative power series) Let # E R [ [ c lc, 2 , . . .]] be a non-vanishing power series in the “universal Chern classes” such that q5 is multiplicative with respect to the Whitney sum of locally free Ox-modules, i. e.
#(& fr3 F)= #(&) . #(F)
(3.58)
for two locally free Ox-modules E and 3 (here, #(&) := # ( c ( E ) ) . Any 4 with this property induces a complex genus, also denoted by 4, by setting # ( X ) := Jx # ( e x ) for X a compact complex manifold. Let us call such a # multiplicative.
Remark 3.9 (Hirzebruch) B y Hirzebruch ’s theory of multiplicative sequences and complex genera (see [Hirzebruch (1966)]), at is well-known that (1) each complex genus is induced by a unique multiplicative # and (2) the multiplicative elements in R [ [ c lc, 2 , . . .]] are exactly those of the f o r m e x p ( C E l akk!chk) with uk E R.
Definition 3.12 (Complex genera for pairs of compact complex manifolds and cohomology classes) Let # be the complex genus that is given by the multiplicative power series with the same name #. We then set (3.59)
Rozansky- W i t t e n theory
97
for all compact complex manifolds X and classes a E H2(X,C) and call 4 ( X ,a ) the complex genus 4 of the pair ( X ,a). Every complex genus can be seen as a Rozansky-Witten invariant: Let q!~ = e x p ( C k m _ l u k k ! c h k ) be a multiplicative sequence. Set f := exp(U k t k ) E R[[t]]. Recall the definition of yf E 8 8 R from subsection 2.6.1.
xEl
Proposition 3.8 (Genera as Rozansky-Witten invariants) The genus + ( X ,(i/27r)(a+ 5))is a Rozansky- Witten invariant: (3.60)
Proof.
By 3.3, we have
(3.61)
Recall the definition of the Todd genus of compact complex manifolds: it is the genus given by /oo
t d := exp
xUkk!Chk (k=l
\
)
oo
with
x U k t k k= 1
t = log -
l-et‘
(3.62)
Let us set
(3.63) Obviously, ( t d ; ( 1 ) ) 2= t d .
Corollary 3.4 (The wheeling element and the Todd genus) We have bfi(t)(X,a)=tdi(t) in C [ [ t ] ] .
(3.64)
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
98
Proof.
Recall the definition of f l ( x ) . It is R ( x ) = yj with (3.65)
Now, f ( t )= ( g ( t ) g ( - t ) ) - 1 / 2 with (3.66) as one can easily see by expanding sinh y = (e” - e-y)/2. By lemma 2.7,
we have O ( x ) = yg-l, i.e.
with so the claim of the corollary follows from proposition 3.8.
0
Remark 3.10 (The square root of the Todd genus and the modified Bernoulli numbers) It follows in particular that 00
t d 1 j 2 ( X )= exp
1bak(%)!chak(X) (k=l
)
(3.68)
where the b2k are the modified Bernoulli numbers. (Recall that the “odd” components of the Chern character vanish on a holomorphic symplectic manifold.) Some more Rozansky-Witten invariants behave “genus-like”, i.e. are multiplicative.
P r o p o s i t i o n 3.9 (Rozansky-Witten invariants and connected graphs) Let y be a connected graph homology class and ( X , a ) and (X’,d) be two holomorphic symplectic manifolds. Then bexp7(X x X’, pr*a
+ pr‘*a’) = bexp7(X,a) .bexpy(X’,a’).
(3.69)
Proof. This follows from proposition 3.7 together with proposition 2.12.
0 Remark 3.11 (Rozansky-Witten invariants of connected graphs and genera) A s bexp7 is only defined for holomorphic symplectic manifolds and not for all compact complex manifolds, it doesn’t follow from the last proposition that bexpy is (the restriction of) a complex genus. However, i f one could
Roransky- Witten theory
99
prove this, this would have a huge impact o n the theory of the RozanskyWitten weight system as this would show that bexpy = bzF=*a k w k f o r some ak, i.e. “wheels sufice”.
Some applications
3.2 3.2.1
Chebyshev polynomials
Definition 3.13 (Chebyshev polynomial) For each n = 0,1,. . . the nth Chebyshev polynomial is the polynomial Tn with Q-coefficients that fulfils
T,(z) = cos(narccos2).
(3.70)
(First Chebyshev polynomials) We have
Example 3.4
and
T ~ ( x=)8x4 - 8x2 + 1. (3.71)
Remark 3.12 (Symmetry of Chebyshev polynomials) The nth Chebyshev polynomial Tn is a polynomial of degree n. It is even if n is even, and it is odd if n is odd.
We shall need the Chebyshev polynomials due to the following fact: Lemma 3.5 such that q
(Main property of Chebyshev polynomials) Let q, z E C be = z . Then
+ q-l
for
each k = 0 , 1 , . . . ,
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
100
Proof. We may assume that z E [-2,2]. arccos $. Therefore, qk
+ 4-k
- eikx
+ e-ikx
Then q = ehiX with
= 2 cos(kz) = 2 cos(k arccos(z/2)) = 2Tk
=
2
z (-) 2 (3.73)
0 Corollary 3.5 (Generalisation) Let a,p, A, p E C be such that (ap)2
+
+ p2 + A .
= a2
(3.74)
Then
for each k = 0,1,. . . .
We may assume that a, /? # 0. Set q := ( a / p ) p 2 . It follows that q q-' = a/@ /3/a A/(@). Now use lemma 3.5 and multiply both sides with ( ~ y p ) ~ .
Proof.
+
3.2.2
+
+
An application of the Wheeling Theorem
Recall the definition of the Wheeling element Q ( t ) .
Proposition 3.10 (A consequence of the Wheeling Theorem) Let a,,B,A E C. The following relation holds in the (sufzciently completed) graph homology space:
Proof. Without loss of generality, we can assume that a,,B # 0. Let us choose a p E C with (ap)2
+ ( p / p ) - 2 = a2 + P2 + A.
Rozansky- Witten theory
101
Note that ( ~ ( L Y )R(0)) , = (Q(crp),n(p/p)) due to degree reasons. By example 2.33, corollary 2.5, and corollary 3.5:
(3.77)
3.2.3
O n the genus td$(a)td;(P) of a n irreducible holomorphic symplectic manifold
Let (X,a) be an irreducible holomorphic symplectic manifold of complex dimension 2n. Definition 3.14 (Nieper-Wiflkirchen) For every w E H 2 ( X ,C ) ,we set
X(W)
:=
if well-defined otherwise
(3.78)
For L E Pic(X), we set X(L) := X(c,(L)). Lemma 3.6 (Rozansky-Witten weight of Q) The weight of the Thetagraph under the Rozansky- Witten weight system is given by
102
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
Proof. Due to the irreducibility of X, i.e. H2'((X,Ox) k = 0,1,. . ., we can write
Jx w exp(a w = n Jx exp(o
+ 5) + 8) . [el
=
C.
[@Ik
for
(3.80)
for all w E H2(X,0x). Using this, we have
27T
= --(2!ch2(X),expo)
i
47T
= -(~2(X),expa) a
0
which proves the proposition due to lemma 3.4.
Corollary 3.6 (Nieper-WiBkirchen) W e have X
(z)
+ a)) = RW(x>n)
7(;T -((a
.
(3.82)
Proof. This is just a reformulation of the previous lemma and uses the 0 last statement of the following proposition. Proposition 3.11 (L2-norm of the Riemannian curvature tensor) Let g be a hyper-Kahler metric on the real 4n-dimensional manifold X that is compatible with the structure of an irreducible holomorphic symplectic manifold o n X. Then
(3.83)
where IlRll is the L2-nonn of the Riemannian curvature tensor of g. I n particular, Jx c2(X)exp(a a) > 0.
+
Proof.
Let w be a Kahler form of g. Then we have (3.84)
For this see e.g. [Besse (1987)]. Now let ( I ,J , K ) be the quarternionic structure on (X,g) such that I is the original complex structure of X. Then WJ = (c 8)/2. From this (3.83) follows if we set w = WJ in (3.84). For the last claim note that llRl1 > 0 as (X,g) cannot be flat; if it were, it would be a torus and therefore not irreducible.
+
Rozansky- Witten theory
103
We want to denote by t d i ( a ) t d i ( P ) ( X ) the genus defined t d i ( a ) ( X ) t d i ( P ) ( X ) . In particular, t d i ( a ) t d i ( P ) ( X , w ) = td'(a)(X)tdi(P)(X)exp(w) for a pair (X,w) of a compact complex manifold X together with a cohomology class w E H2(X, C). by
sx
S.;
Theorem 3.2
(Nieper-Wiflkirchen) For all
a,P
E C and all
w E
H2(X,C),we have tdfr(a)tdi(P)(X,w)
Proof.
Let us first prove the claim for w = & ( u
tdi(+d+
(P)
(x,
-(u T ;
+8)
)
+ a). We have
.I
= bn(a)n(P)(X,U)= b(n(a)n(p))(X,
(3.86)
Now, we use proposition 3.10 with X = X ( i / ( 2 ~ ) )This . yields b(n(a)n(p)) (X,
= b,(X,
0)
(3.87)
with
Let us now express P7(X,u ) in terms of ch(X),u and B by using corollaries 3.2 and 3.6:
. exp(-a) exp(u + 8 )
(3.89) In the last line we could ignore the factor exp(u) due to degree reasons (look at the holomorphic and antiholomorphic degree of the form to be
104
Chern Numbers and RW-Invariants of Compuct Hyper-Kahler Manifolds
+
integrated!). Thus we have proved the theorem for w = &(o 8). The general case follows from proposition 1.21. For this note that the characteristic classes C h k and ck do not depend on the holomorphic structure, i.e. are well-defined on X as a differentiable manifold. 0
Corollary 3.7 (Nieper-Wifjkirchen) For all w E H2(X,C),we have
= (1
+ X ( w ) ) n t d i( X ) .
+
Proof. Set cx = 1, p = 0 in (3.85). Then use the fact that T k = 2 k - 1 ~ k terms of lower degree for k = 1,2,. . .. This proves the first equality. The second follows due to degree reasons.
Corollary 3.8 (Nieper-Wifjkirchen) For all w E H 2 ( X ,C ) , we have
k
tdiw2 = 2nX(w)tdi(X),
c ~ ( X ) W ~= " -24n(2n ~ - 2)!A(w)"-ltd;(X)
(3.91) (3.92)
and
s,
w2n = ( 2 n ! ) X ( w ) 9 d (i X ) .
(3.93)
In particular, t d t ( X ) # 0. Proof. Substitute w in (3.90) by tw where t is a formal parameter. Expand the left and the right side and compare coefficients. For the second equation also note that 1 td' = 1 + - ~ 2 + .... 24
(3.94)
In the next three subsections we want to deduce three important facts from what we have proved in this subsection.
Rozansky- Watten theory
3.2.4
105
The Lz-norm of the Riemannian curvature tensor of a compact hyper-Kahler manifold
The following theorem, relating the differential-geometric quantity )IRII to a topological quantity of X was the first major application of the RozanskyWitten theory to the geometry of compact hyper-Kahler manifolds. Recall that for a hyper-Kahler manifold the Chern classes are welldefined.
Theorem 3.3 (Hitchin and Sawon) Let ( X , g ) be an irreducible compact hyper-Kahler manifold of real dimension 4 n and IlRll the L2-norm of the curvature tensor of g . Then (3.95) I n particular, t d * ( X ) > 0.
Proof. Let ( I ,J,K ) be a quarternionic structure on X. The volume vol(X) of X is given by (3.96)
+
because of u d = 2 W J where u is the holomorphic symplectic form associated to the complex structure I . Let us view X as an irreducible holomorphic symplectic manifold with respect to this structure. By proposition 3.11, it follows that vol(X) --
1
X(a
) ) R ) ) 2 768.rr2n
+ a).
(3.97)
+
We plug this in (3.93) and use again sx(o a)2" = (2n)!4"vol(X). Thus we arrive at the claim.
3.2.5
The beauville-Bogomolov form
Proposition 3.12 (Uniqueness) Let X be a 4n-dimensional compact differentiable manifold. Then there exists at most one quadratic form q x : H2(X,R)+ R with the following properties: (1) qx is induced by a primitive quadratic form on H2(X,Z).
(3.98)
106
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
(2) There exists an a E H2(X,R) with qx(a) # 0 , and for all a E H2(X,R) with qx(a)# 0 , we have
(3.99) (3) There exists a positive real constant c > 0 such that qx (a)n= c
sxa2".
Proof. First we note that up to a sign there exist at most one qx that fulfils (3) with c = 1. This can be seen by the following consideration: In the ring S*H2(X,R)", there are at most n solutions x of the equation xn = y
(3.100)
for a fixed y E S2nH2(X,R)V.Since all these solutions differ only by a nth-root of unity, there is actually up to a sign at most one x with xn = y. We apply this to (3.101) Thus we know that qx - if it exists - is unique up to a rational constant. Property (1) fixes this constant up to a sign, property (2) fixes this sign if n is even, property 3 fixes it if n is odd. 0
Definition 3.15 (Beauville-Bogomolov quadratic form) Such a form qx on an irreducible compact hyper-Kahler manifold is called BeauvilleBogomolov quadratic form. Theorem 3.4 (Beauville, Bogomolov) Every irreducible compact hyperKahler manifold possesses a Beauville-Bogomolou quadratic form. It is given by a multiple of X : H2(X,C) -+C.
(3.102)
Proof. In order to show that such a qx exists, it is enough to show that there exists a quadratic form q : H2(X,Q) + Q that fulfils properties (2) and (3). By (3.91) we see that X defines a quadratic form on H2(X,Q) that is induced by a rational quadratic form (3.103) since t d i is a rational characteristic class. By (3.93), this rational X has property (3). It remains to show that it also has property (2). But this
Calculations for the example series
109
is (3.92) because tdi(X) > 0 and the first Pontrjagin class is given by 0 pl(X) = -2cZ(X) on a hyper-Kahler manifold. We should remark that the proof we gave for the existence of the Beauville-Bogomolov form is quite non-standard. However, we think it is a nice application of the Rozansky-Witten theory.
3.2.6
A Hirzebruch-Riemann-Roch formula
Let (X,a ) be an irreducible holomorphic symplectic manifold.
Theorem 3.5 (Nieper-Wiakirchen) There exists a polynomial PX E Q[z] of degree n such that the holomorphic Euler characteristic of every invertible sheaf C E Pic(X) o n X is given by
X(X, C)= PX(W).
(3.104)
The polynomial i s given by co
2 cb2k(2k)!ch2kTk (1 k=l
+
z))
.
(3.105)
(The existence of the polynomial has already been observed by D. Huybrechts.) Proof.
Using the Hirzebruch-Riemann-Roch formula x(X, C) =
/
td(X)ch(C) = td(X,cl(C))
(3.106)
X
(as ch(L) = exp(cl(L))), the theorem follows immediately from the following lemma. 0
Lemma 3.7 (Twisted Todd genus) For all w E H2(X,C), we have 00
td(X,u ) = Proof.
Set CY =
exp (-2
bzk(2L)!chzkTk
k=l = 1 in (3.85).
(1+ 7 A(w))).
(3.107)
0
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Chapter 4
Calculations for the example series
In this chapter we shall give an algorithm to calculate the Chern numbers and Rozansky-Witten invariants for the example series for irreducible holomorphic symplectic manifolds, the Hilbert schemes of points on a K3 surface and the generalised Kummer varieties. To be able to use the methods of algebraic geometry, we shall assume that all our manifolds are projective. This isn't really a restriction as in every deformation class of a holomorphic symplectic manifold, there is also a projective one (see e.g. [Huybrechts (1999)]) and the Chern numbers and Rozansky-Witten invariants are invariant under deformation.
4.1 More on the geometry of the Hilbert schemes of points on surfaces We shall investigate the geometry of Hilbert schemes of points on surfaces a little bit more, which is an interesting topic by itself. We shall mostly state results without proofs. A good reference for these things is [Ellingsrud et. al. (2001)l or [Lehn (1999)l.
4.1.1
The universal family
Let X be a projective complex surface. Recall that the Hilbert scheme X["I of zero-dimensional subspaces of length n on X represents a functor for each n = O , l , . . . . Definition 4.1 (Universal family) The universal family over XI"]is the family of zero-dimensional subschemes of X of length n parametrised over X["1 that is represented by the identity in Hom(X["I, XI"]). 109
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
110
So Z" is a subscheme of Xi"] x X which is of finite degree n over XI"]. The fibre over a E E Xi"] is just 5 viewed as a subscheme in X. Let us denote the two projections by p : X["I x X -+ XIn] and q :
XI"] x 4.1.2
x -+ x.
The incidence variety XInyn+'l
Let X be a projective complex surface and n = 0,1,. . . .
Remark 4.1 (Incidence variety) There exists an irreducible complex variety X["~"+'] of dimension 2n + 2 that parametrises all pairs ( [ , < I ) with E XI"] and E' E X["+l] such that E c E'. Definition 4.2 dence variety.
(Incidence variety) We call the space XIn*"+'] an inci-
X["J+'] comes equipped with three holomorphic maps: Let us denote : X[","+'] -+ X["+'] the map that maps ((,<') to <' and by 4 : X["~"fl]+ X["] the map that maps (E,E') to [. Finally, there is a map x : X["~"+'l -+ X that maps (6, <') to 5 if is obtained by extending 5 at by
+
c'
the closed point 2. Let Z" be the ideal sheaf of the universal family F in XIn] x X.
Remark 4.2 (Construction of the incidence variety) One can show that X[">"fl]= P ( P ) where P ( P ):= Proj(S*(I")). Let us denote by L E Pic(X[">"+ll) the tautological quotient line bundle of
Xl","+ll = P(I"). For later usage let us state the following lemma from [Ellingsrud et. al.
(aool)]: Lemma 4.1 (On the tautological quotient line bundle) In H*(X["I x x,fi>[,,l&y) we have (4,X)*(sl(qk> = .k(z").
(4.1)
Here, so, s1,. . . denote the Segre classes.
Proof.
See [Ellingsrud et. al. (2001)].
0
We shall need the incidence variety due to the following reason. Via the maps 4, and x we can relate X["+'I to X["] and the surface X itself. This will enable us to use induction over n to prove structural results on the Hilbert schemes of points on surfaces. One example for such a relation
+
Calculations for the ezample series
111
between data on X["+'l on the one hand and XIn] and X on the other hand is given by the following proposition. More relations follow in the next subsections. The reader should remind themselves of the definition of a["]for a class (Y E H 2 ( X ,C ) . Proposition 4.1 (Induction and a["])For a n y a E H 2 ( X ,C ) , we have $*&+lI
= 4*&1 + x*(Y
(4.2)
in H2(Xln++'l, C ) .
Proof. Let p : X ( " ) x X -+ X ( " ) and q : X ( " ) x X + X denote the canonical projections. Let T : X ( " ) x X -+ X("+') be the obvious symmetrising map. The following diagram
-x[n,n+l]
x[n,n+lI
is commutative. (Note that we have primed some maps to avoid name clashes.) We claim that 7*dn+') = p*a(") q*a. In fact, since
+
this follows from the definition of a(n).Finally, we can read off the diagram that
112
4.1.3
C h e r n Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
Calculations in various K-groups
For convenience of the reader we want to recall the Grothendieck groups of coherent sheaves. Let X be a projective variety. The Grothendaeck group K ( X ) of coherent Ox-modules is the group spanned by equivalence classes [F]of coherent sheaves on X where the equivalence relation is given by
+
[3l= [3'] [PI for every short exact sequence
of coherent Ox-modules on X . The Grothendieck group is already generated by locally free Ox-modules as every coherent sheaf admits a finite resolution by locally free ones. Note that the characteristic classes are well-defined on the level of the Grothendieck groups. We shall make K ( X ) into a co- and a contravariant functor: Let f : X -+ Y be a morphism of projective varieties. Then there is a pushforward f! : K ( X ) + K ( Y ) mapping a class of a coherent sheaf to C,"==,(-l)"[Rif+FI and there is a well-defined pull-back f ! : K ( Y ) -+ K ( X ) that maps a class of a locally free sheaf t o [ f *3l. Furthermore, K ( X ) becomes a ring by setting [F] . [G] := [F@G] whenever 3 and are locally free Ox-modules. This ring carries an involution given by [fl" = [3"] for locally free sheaves.
[A
[A
Remark 4.3 (Grothendieck group and the derived category) The Grothendaeck group K ( X ) sits somewhere in between the (bounded) derived category D ( X ) and the cohomology ring H*( X ,Q;): there as a ring homomorphism f r o m D ( X ) to K ( X ) that maps a complex ... + A-l + do+ dl -+ . . . to Cn,z(-l)"[An], and there is a ring homomorphism from K ( X ) to H * ( X ,a>) mapping [3lto c h ( 3 ) . Please be aware that we won't distinguish notationally between a coherent sheaf 3 and its class in the K-group anymore and shall write F in both cases. Let us denote by Q n the tangent sheaf of the Hilbert scheme X [ " ] . Let us set
[A
0
:=
(f#l,x): X["~"+ll+ Xi"] x x.
(4.8)
For inductive arguments on X["1 we shall need the following relation later:
113
Calculations for the ezample series
Proposition 4.2 (Ellingsrud, Gottsche and Lehn) The class of the tangent sheaf in K(X["I) fulfilsthe following relation: ?$0"+1 =
Proof.
+ L: . &""+ c" . x!w); - p!(l - e x + &&.
(4.9)
0
See [Ellingsrud et. al. (2001)].
So, the ideal sheaf 1" appears here as well. Therefore, we shall also need the following proposition: x X) Proposition 4.3 (Ellingsrud, Gottsche and Lehn) I n K(X["Y"+~] we have (q5 x idx)!Z"+l = (c x idx)!Z"
-
p ! C . ( p x idx)!Oa
(4.10)
where OA is the structure sheaf of the diagonal in X x X .
Proof. 4.1.4
17
See [Ellingsrud et. al. (200l)l.
Chern numbers of the Hilbert schemes
Let X be a projective complex surface and m, n = 0 , 1 , . . .. Let a E H2(X, C ) be an arbitrary cohomology class. Let us fix some notations: Set z := ~ [ ~ * x~ x". + ~ 1 We foIlow [Ellingsrud et. al. (2001)] and write pr, for each I c {0,1,. . . ,m} for the projection from X["+'] x X" to the product of the factors indexed by I . Recall the definition of the maps 4, ?b, and p. These induce maps Qj:=
4 x idx-
: 2 -+ X["+l] x
X"
(4.11)
and @ := ( 4 x idx-, pr:p) : 2 --f XIn] x XmS1.
(4.12)
Lemma 4.2 (Ellingsrud, Gottsche and Lehn) Let f be polynomial over a commutative R-algebra in the following cohomology classes o n XIn+'] x X": pr&@+'l
and
p r f a for each i = 1,.. . ,m
and the Segre classes of the following sheaves o n
pr(;W+',
pr&T+',
prfjOa,
(4.13)
x X":
and p r f 8 x f o r each i , j = 1,.. . , m. (4.14)
114
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Then there exists a polynomial depending only on f (and not on X!) i n the analogously defined cohomology classes on XIn] x X["fl] such that
J
x["+'lxxm
f=J X["lxXm+l
f.
(4.15)
Proof. We follow the proof given in [Ellingsrud et. al. (2001)]: First note that the morphism $ (and thus @) is generically finite of degree n 1, so that
+
(4.16)
Obviously, we have @!prfQx= +!prf@x and
Q!pr:jOA = +!pi$O* for i , j = 1 , . . . ,m. (4.17)
Now by proposition 4.3, Q'prGiT+' = +!prG,+T- p r k + + l l ~. @!pr;,,+,OA
(4.18)
and by proposition 4.2,
(4.19)
And finally, by proposition 4.1,
From this we conclude that there are polynomials fo, f1,. . . depending only on f in the cohomology classes pr&["]
and pr:a
and in the Segre classes of the sheaves
and on Xi"] x Xm+l such that
(4.21)
115
Calculations for the example series
By lemma 4.1 this integral equals
Now let
j
0
be the integrand in this expression.
Theorem 4.1 (Ellingsrud, Gottsche and Lehn) Let R be a commutative Q-algebra. Let P E R[cl,c2, . . . ,a] be a polynomial over R in the "universal Chern classes" and a. There exists a polynomial P E R(z1, z2,z3, zq) such that for each projective complex surface X , a E H 2 ( X , Q ) and n E No we have
=P
(s,g1s,
C 1 W )
. a,
s, 2, s, ) c4(x)
c 2 ( X ) . (4.25)
Proof. First, we apply lemma 4.2 repeatedly. This yields a polynomial j over R depending only on P in the cohomology class p r f a and in the characteristic classes of the sheaves of the form prH@x and prfjC?A
(4.26)
on X" such that (4.27)
(Recall that the Segre classes span the whole ring of characteristic classes.) Finally note that any such expression Sxn$ can universally (i.e. only depending on $) be written as
Remark 4.4 (A usable recursion formula?) I n principle one can use the constructive proof of the last theorem to explicitely calculate the Chern numbers of the Hilbert schemes of points o n surfaces in terms of the Chern numbers of the surface itself. However, the recursion formula one may
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
116
naiirely extract from the proof is far too complicated to be applicable even for low values of n. Nevertheless we shall see soon how we can use the theorem to calculate Chern numbers of the Hilbert schemes of points o n surfaces. 4.2
Genera of Hilbert schemes of points on surfaces
4.2.1
Two decomposition results
Let XI and X2 be two projective surfaces and let X = X1 Xz be the disjoint union of XI and X2. In particular X is again a (non-connected) projective surface. What is the Hilbert scheme of n points on X ? It parametrises zero-dimensional subschemes of length n on X . Now zero-dimensional subschemes on X are given by unordered pairs of zero-dimensional subschemes of X1 and X2, respectively. So, we expect the following decomposition result:
Proposition 4.4 (Decomposition of the Hilbert scheme) Then nthHalbert scheme of X is given by
X["I
xp11
=
xpz1.
(4.29)
n*+nz=n
Proof.
Let S be a connected projective surface. Then
Hom(S,
u
XPl1 x X p z l )=
n1 +nz=n
Hom(S,Xp'] x X p z l ) nl+nz=n
(Hom(S,XP']) x Hom(S,Xp1)). (4.30)
= n1+nz=n
Now Hom(S, Xi[n,') is the set of families of zero-dimensional subschemes of Xi of length n parametrised over S. Therefore, Hom(S,X?']) x Hom(S,Xp'I) is the set of families of zero-dimensional subschemes of X of length n such that the length of {[Xi is ni parametrised over S. Together with (4.30) this yields Hom(S,
u
Xi"'] x XFz1)= Horn(S,X["])
(4.31)
nl+nz=n
for all connected projective surfaces. FYom this the statement follows.
0
117
Calcdataons for the example series
Remark 4.5 (Compatibility with the Hilbert-Chow morphism) The isomorphism (4.29) is compatible with the Hilbert-Chow morphisms, i.e. if we denote by i : X?'] x X p l + XIn] with n = n1 + n2 the natural inclusion map and by j : Xin') x X p ) -+ X(")the natural symmetrising map, the diagram
(4.32)
commutes. We use this to prove the following proposition: Proposition 4.5 (Decomposition of cohomology classes) If a E H2(X,C) decomposes as alx, = a1 and aIx2 = a 2 , then *In] decomposes as (4.33) Proof.
By definition of ojni)and a("),we have j*&)
= pr;a(;"')
+ praap2).
(4.34)
This together with the commutativity of (4.32) yields the formula we want to be proven. 0 We can formulate these results also in the following way: Let X a very large set ("something like a universe"). We shall assume implicitly that all our complex manifolds are members of this set. This allows us to define the following set
X is a smooth projective manifold of dimension n and a E H2(X,C) (4.35) of isomorphism classes of pairs (X, a ) for each n = 0,1, . . . . Here two pairs
( X , a ) and (X',a') are isomorphic if and only if there is an isomorphism X' i with $*a' = a. (Due to the Whitney embedding theorem this set is independent of X as long as X is big enough.)
Cp : X
118
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
Definition 4.3 (Ring of smooth projective manifolds together with cohomology classes) We define a structure of a graded ring on the graded set 03
K:=
Kn
(4.36)
n=O
by setting
[x,a]. [ X I ,a'] := [ X x X ' , pr;a
u pr&']
(4.37)
and for [ X , a ] [X',a'] , E K n , we set
[ X ,a]+ [ X ' ,a'] = [ X kJ X ' , a kJ a'] with ( a kJ a')Ix
=a
(4.38)
and ( a kJ a ' ) j ~=, a'.
Reformulating our results, propositions 4.4 and 4.5 yield Corollary 4.1 (Hilbert schemes and the ring K ) The map
n=O is a homomorphism of abelian groups.
Proof. sitions.
Just write out both sides and use both of the mentioned propo-
0
Now recall the definition of $ ( X , a ) for a genus $ with values in the Qalgebra R and a pair of a compact complex manifold X and a cohomology class a E H2(X I C ) .
Definition 4.4
(Genera of elements in K ) We set (4.40)
for ( X n lan)E K" and extend this t o a group homomorphism
4:K-R. Remark 4.6 (Compatibility with product structure) In fact, becomes a ring homomorphism. This is easy to see.
(4.41)
4 :K
+R
119
Calculations for the ezample series
4.2.2
A structural result o n genera of Hilbert schemes of points o n surfaces
Recall the notion of a multiplicative power series 4 E R [ [ c l , c 2 , - . . ] ]in the "universal Chern classes" over a Q-algebra R. Further we employ the notion of H [ X ,a]for an element [X,a]E K 2 again. Theorem 4.2 (Ellingsrud, Gottsche and Lehn) For each multiplicative E R[[cl,c2,. . .]] there ezist unique power series A b ( p ) ,B b ( p ) ,C+(p),D+(p) E pR[[p]] with vanishing constant coefficients such that for all smooth projective surfaces X and a E H2(X, C) we have:
The first t e r n s of A+(p),B+(p),C+(p) and D+(p) are given by
A&) = P + 0 ( P 2 ) , BdP) = 41P + 0 ( P 2 > , C+(P>= 4llP + 0(P2> and D+(P) = 42P + 0 ( P 2 ) (4.43) where 41 is the coefficient of the coefficient ofc2 in 4.
c1
in
4, 411 the coefficient of c f / 2 in 4 and
42
Proof. The first equation is just the definitional equation of the left hand side. Now define a map y : K 2 + Q4, [X, a]H (a2/2, c ~ ( X *) a,~ 1 ( X ) ~ /c2~, ( X ) )
(4.44)
sx
where we have suppressed the integral signs and we interpret the expressions a2,etc. as intersection numbers on X. As one can easily see by constructing explicit generators, the image of y spans the whole Q4. By theorem 4.1 we know that the map 71 : K 2
+
R"p1l [ X ,4
d(H[X, a]@))
(4.45)
factors through the additive y : K 2 + Q4 and a map h : Q4 + R[[p]]. As the image of y is Zariski dense in Q4 and log h is a linear function because 71 is a ring homomorphism, we conclude the first part of the theorem.
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
120
To calculate the first terms of the power series, we expand both sides of (4.42). The left hand side expands to 1
+ ( Q 2 / 2+ 4 l C l ( X ) Q+ 4llC?(X)/2+ 42CZ(X))P+ 0 ( P 2 ) ,
(4.46)
while the right hand side expands to 1
+ (A1a2/2 + Blcl(X)a + C 1 c l ( X ) 2+ D l c g ( X ) ) p+ O ( p 2 )
(4.47)
where All B1, C1, D1 are the linear coefficientsof A+, B+,C+and D+, which 0 can therefore be read off by comparing the expansions. Recall the definition of a""]]E H2(A""]1,C) for an abelian surface A and Q E H2(A, C ) . Corollary 4.2 (Sawon) Let X be a n y smooth projective surface, a H 2 ( X ,C ) , and n = 0,1,. . . . Then
E
(4.48)
For X = A a n abelian surface and n 2 1, we get
Proof.
By theorem 4.2, the following equation holds in C[[p]][q]:
which proves the first part of the corollary by comparing coefficients of q. For the Kummer case, we calculate
using lemma 1.2, which proves the rest of the corollary.
0
Definition 4.5 ("Universal genus") Let ch be the "universal Chern character". We set
.]I.
E Q[a2,a4,. . . , ~ ] [ [ c I , c. .~ ,
(4.52)
121
Calculations for the example series
This is a multiplicative power series and thus gives rise to four power series A,(t)(p),B,(t)(~),c,(t)(p) and D a ( t ) ( ~E) P Q [ ~ ~ , Q , - 1tI"PII .according to theorem 4.2.
Definition 4.6 (Power series A(t) and D ( t ) ) The power series A(t) and D ( t ) in Q[az, a4,. . . ] [ [ tare ] ] defined by
A(t) := A+(,)(l) and D ( t ) := D+(t,(l).
(4.53)
Remark 4.7 (Constant terms) The constant terms of these power series in t are given by
A(t) = t + O ( t ) and D(t) = O ( t ) .
(4.54)
Remark 4.8 (Chern numbers are given by values of genera) Theorem 4.2 is very important for the calculation of Chern numbers of the Hilbert schemes of points o n surfaces. The main reason for this is the well-known fact that the information on all Chern numbers in included in the knowledge of all genera. I n principle, we just have to calculate A4, B4, C4 and D4 for a suficiently general genus 4. (For holomorphic symplectic manifolds, +(t) is general enough as all "odd" Chern classes vanish.) How this can be achieved will be shown in one of the next sections.
4.2.3
Genera of the generalised Kummer varieties
Theorem 4.2 gives a quite good description of the structural form of genera of Hilbert schemes of points on surfaces, in particular of Hilbert schemes of points on a K3 surface, our first example series for irreducible holomorphic symplectic manifolds. However, we are still missing a description for genera on the generalised Kummer varieties, making up the second example series. To find such a description is the purpose of this subsection. Let 4 be a complex genus with values in the Q-algebra R. First, we note the following:
Lemma 4.3 (Genera and products and coverings) The following result holds true: (1) For two complex manifolds X and Y together with cohomology classes cx E H2(X,C ) and ,B E H2(Y,C ) , we have
4 v x y,Pr;a u Pr;,B) and
=
4w,
. 4 KP )
(4.55)
122
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
(2) for every finite unramified covering v : X -+ Y and any /3 E H2(Y, C ) , it is
4 ( X ,.*PI = deg(v) 4 v , PI. *
(4.56)
Proof. The statement (1) is just the fact that 4 induces a homomorphism of rings K --f R. The second statement follows from the fact that
so (4.58)
0
The second statement follows as v,l = deg v. Let A be any abelian surface.
Lemma 4.4 (Genera of the generalised Kummer varieties) For each a E H2(A,C ) and each n = 1 , 2 , . . . , it is (4.59)
Proof.
We make use of lemma 4.3 and lemma 1.2. This yields
+(A,na) . 4(A[ln]l,
= +(Ax = $(A x
n . pr;a
+ pr;a""ll)
v*a["]) = n44(A["1, a["]),(4.60)
which proves the proposition as
4(A,na) = n2 since all non-trivial Chern classes of an abelian surface vanish.
(4.61)
0
Definition 4.7 (Generalised Kummer varieties and the ring K ) For each a! E H2(A,C ) we set 00
A [ a ] ( p ):= C[A"nll,a"nll]. p n E K[[p]].
(4.62)
n=l
We are interested in the value of the genus 4 on A[a](p), which incorporates all generalised Kummer varieties at once.
123
Calculations for the example series
Theorem 4.3 (Nieper-Wiakirchen) For each a E H2(A, C),w e have
In particular,
(4.64) Proof.
By lemma 4.4 and theorem 4.2, we have
(4.65)
Finally, (4.64) follows by expanding the exponential series and setting a =
I7
0.
Thus, we have expressed the value of a genus 4 on the generalised Kummer varieties in terms of A4. In the next section, we finally show how t o calculate this one and the other power series connected t o a genus. 4.3
4.3.1
Calculations of the power series A+, B+, C+ and D d
Bott's residue formula
The Bott residue formula is a very effective tool t o calculate Chern numbers of compact complex manifolds X that admit vector fields with isolated simple zeroes 2: (Polynomial) We call a map P : ~ ( Q x --+) OX of C-modules a polynomial of degree n if it is induced by an Ox-linear map
Definition 4.8
S,End(Bx)
-+
ox.
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manifolds
124
Let P be a polynomial of degree n. By P, we denote the map H1(X, S2x 8 OX)) + Hn(X,R$) induced by P. The map P, is homogeneous of degree n. Let the dimension of X be n.
Remark 4.9 (Characteristic numbers and polynomials) Every characteristic number of X is of the form (4.66)
This is easy to see. Just recall that ch = Tr*exp(&agx) and note that
Tr o (.)k is a polynomial of degree k. Let V be a vector field on X with isolated simple zeroes. It induces a morphism O X 4 O X of C-modules given by
V : Ox
4
Ox,6
H
[V,6]
where 6 E O x ( U ) is a vector field on U where
(4.67)
U is an open subset of X.
Lemma 4.5 (Induced linear map on the zero set) Restricting the morphism V to Z yields an Oz-linear map
Vz := VIz E End(Bx1z). Proof.
(4.68)
Let be 6 E O x ( U ) and f E Ox(U).We have
v(f6)lz = [ V , f 6 ] l z=flz[v,6]lz+(V.f)lz.6=fV(B))Iz as Vlz = 0.
(4.69)
0
Remark 4.10 (Representation at the isolated zeroes in terms of a basis) Let z E Z be an isolated simple zero of V and (21,. . . ,zn) a local coordinate system around z and set A, := (%)i,j with a l , . . . ,an those local functions
around z such that V = CZl ai&.
Then
V z ( z ) = -A, E E n d ( O x l z ) ( z )21 End(Cn)
(4.70)
with respect to the basis & ( z ) , . . . , & ( z ) of OX,*. This can be seen by giving a description of V in coordinates. The map
E n d ( O x l z ) 4 0 2 = @OZJ = @ c ZEZ
ZEZ
(4.71)
Calculations for the example series
125
induced by P is also denoted by P,. It is 02-homogeneous of degree n. Now we can state Bott's residue formula:
Theorem 4.4 (Bott's residue formula) It is (4.72)
Proof. A very clear proof in the algebraic setting can be found in [Carrell and Lieberman (1977)]. 17
Example 4.1 (Topological Euler characteristic) Let be P := det : M ( @ x ) OX. ---f
(4.73)
This is a polynomial of degree n, the dimension of X . We have c n ( X ) = P,(as,), so theorem 4.4 in this case yields
(&)"
ZEZ
det Vz( z ) = #Z, det Vz(z)
(4.74)
which is the well-known GauB-Bonnet formula.
4.3.2
How t o calculate C+ and Db
In general, the manifolds XIn] don't admit a non-trivial holomorphic vector field. For example, H0(X["],@x[n]) = Ho(X'nI,CIx[n~) = 0 if X is a K3 surface. Therefore we can't use Bott's residue formula directly to calculate the Chern numbers of, say7the Hilbert schemes of points on a K3 surface. However, by taking a look at theorem 4.2 we see that in order to calculate all Chern numbers of xin] it suffices to calculate C b ( p ) and D 4 ( p ) for all genera given by 4. (Recall that knowledge of all genera is in one-to-one correspondence with the knowledge of all Chern numbers of a compact complex manifold.) These in turn - again by using theorem 4.2, this time in the other direction - can be computed by calculating the Chern numbers of the Hilbert schemes of points on two surfaces X and Y such that (J, c : ( X ) / 2 ,Jx c2(X)) and (J" c:(Y)/2, Jr cz(Y)) are linearly independent in Q2. Such a pair of surfaces is given by P2 and P1 x P'. And there is a reason why we have chosen these projective surfaces. Their associated Hilbert schemes admit holomorphic vector fields with isolated simple zeroes:
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
126
Remark 4.11 (Torus action on the Hilbert schemes of points on the projective plane) Let be XO, XI, X2 E Z. The torus action
cxx P2 + P2, (t,[zo : z1 : z2]) H [tXOzo : t X 1 Z 1 : t%z]
(4.75)
induces naturally a torus action o n the Hilbert schemes (P2)InIofpoints o n P2.
Let u s denote by V E Ho(X,( P 2 ) [ n the ] ) holomorphic vectorfield that is induced by this action. For a general three-tuple (A*, A1, A), it has a discrete and simple zero set Z which can be given explicitely. The eigenvalues of V z ( z ) f o r z E Z can be calculated as well. A n explicit calculation is given in [Ellingsrud and Stramme (1987)l. See [Ellingsrud and Strflmrne (1996)l as well. This enables us to calculate all Chern numbers of the Hilbert schemes of points on the P2 effectively. Analogous statements hold true for P1x P I . Thus one can effectivelycompute the power series C,(p) and D,#,(p)for any complex genus 4.
4.3.3
The calculation of A+ and B+
In this subsection we shall see how to calculate the power series A+ and
BdJ.
Let 4 be a complex genus with values in a Q-algebra R .
Definition 4.9 (“Twisted genus”) Let t be a formal parameter. For a compact complex manifold, we “formally” set
4t(X) := 4(X,w y )E R[t]
(4.76)
where wx is the canonical sheaf of X.
Remark 4.12 (Well-definedness) Though w i B t is no well-defined expression for a formal parameter t (and only well-defined if the parameter takes values in Z), the expression ch(wSBt)= exp(t. c l ( w : ) )
(4:77)
is and so 4 t ( X ) makes perfect sense. Example 4.2 (Untwisted genus) Obviously, (4.78)
127
Calculations for the ezample series
Proposition 4.6 (The twisted genus is a genus.) The map that maps any compact complex manifold X t o &(X) E R[t] is a complex genus +t.
Proof. We have to check the axioms for a complex genus. Let us denote the multiplicative power series in R[[cl,cz, . . . ]] associated to 4 also by 4. Then (4.79) Now, 4 . exp(tc1) E R[[cl,cz,.. .]] is again multiplicative, i.e. of the form e x p ( C E l a&!ch) and therefore induces a genus cbt.
Proposition 4.7 (Power series associated to twisted genera) The power series Ab(p), B,#,(p)E R[[p]] are given by
c,, (P) = AdP) . t2 + 2BdP) . t + Cf#J(P). In particular C+,(p) is a polynomial of degree at most two. D4t (PI = W P ) *
Proof.
(4.80) Further,
By theorem 4.2, we have
On the other hand,
so again by theorem 4.2:
Finally compare coefficients.
0
The statement of the proposition has also been used in [Nieper-Wif3kirchen (2002b)) where the author computed the elliptic genus of the generalised Kummer varieties from the previously known elliptic genus of the Hilbert schemes of points on surfaces (see [Borisov and Libgober (2002)l). The effective computation of A$(p) and B+(p) is now straight-forward: We know that we can compute C4(t)by means of Bott's residue formula.
128
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
The proposition above enables us to compute the series A + ( p ) and B+(p) from that data. 4.3.4
Chern numbers f o r the example series
Though we have described the algorithm to compute A+, B+, C+ and D+ and thus the algorithm to compute arbitrary genera of Hilbert schemes of points on surfaces and (via theorem 4.3) of the generalised Kummer varieties, we won't give any explicit calculations here as they are better done by a computer program. However, we want to give some results in terms of Chern numbers of our main examples for irreducible holomorphic symplectic manifolds. The results are taken from [Ellingsrud et. al. (2001)l for the case of Hilbert schemes of points on surfaces and from [Nieper-WiBkirchen(2002a)l for the case of generalised Kummer varieties.
Remark 4.13 (Chern numbers of the example series) Let X be a K3 surface and A a complex torus of dimension teo. The Chern numbers for the first four examples of each example series of irreducible holomorphic symplectic manifolds are given by
k
c
1 2
c2
c; c4
c;
3 ~
2
c6
4
ci C;C~
ci c2c6
ca
4.4
c 24 828 324 36800 ~ 14720 4 3200 1992240 813240 332730 182340 25650 SXPl
SA""+III
24 756 108 30208 6784 448 1470000 405000 111750 37500 750.
Calculations of Rozansky-Witten invariants
Now that we have seen how one can calculate all Chern numbers of our main examples for holomorphic symplectic manifolds, we shall show how one can deduce from that the values of a huge class of Rozansky-Witten
Cakulations for the example series
129
invariants.
4.4.1 A lemma from umbral calculus The following lemma seemed to be quite surprising to the author when he found it in the first place. We shall need in a specific calculation later on. However, it may be usable for other calculations involving power series as well so we state it independently here. The proof makes heavy use of the so called umbral calculus. Everything we need from this calculus can be found in [Roman (1984)l. In particular the reader who wants to understand the proof is strongly advised to take a look in such an introductionary text. Lemma 4.6 (Nieper-WiBkirchen) Let R be any commutative Q-algebra and A(t) E R[[t]] and B ( t ) E tR[[t]] two power series. Let the polynomial sequences ( p , ( z ) ) n c N and ( s , ( z ) ) ~ E N be defined by (4.84)
and m
1
tk
sk(z)G = A(t) exp(zB(t))-
(4.85)
k=O
Let WB(t) E tR[[t]] be defined by WB(texp(B(t))) = t . Then we have (4.86)
and (4.87) Proof.
It suffices to prove the result for the field
R = Q(ao,ai,. . . , b i , b 2 , . . .)
(4.88)
and m
m
A(t) = x a k t k and B(t) = k=O
bktk. k= 1
(4.89)
130
Chern Numbers and
RW-Invariants of Compact Hyper-Kahler Manifolds
So let us assume this special case for the rest of the proof. Let us denote by f ( t ) the compositional inverse of B ( t ) ,ie. f ( B ( t ) )= t. We set g ( t ) := A - l ( f ( t ) ) .For the following we will make use of the terminology and the statements in [Roman (1984)]. Using this terminology, (4.84) states that (p,(x)) is the associated sequence to f ( t )and (4.85) states that (s,(z)) is the Sheffer sequence to the pair ( g ( t ) , f ( t ) ) (see theorem 2.3.4 in [Roman (1984)l). Theorem 3.8.3 in [Roman (1984)l tells us that (s,(rc - n)) is the Sheffer f(t)) with sequence to the pair (@(t),
The compositional inverse of f(t) is given by
B(t) := B(WB(t)):
Further, we have
A(t) := ij-’(B(t))
-
+A(t)
1 tB’(t)
o WB(t), (4.92)
which proves (4.87) due to theorem 2.3.4 in [Roman (1984)I. It remains to prove (4.86), i e . that (*) is the associated sequence to f(t). We already know that (pn(z- n ) ) is the Sheffer sequence to the pair (1 # , j ( t ) ) . By theorem 2.3.6 of [Roman (1984)] it follows that the associated sequence to f ( t )is given by (1 f ( & ) / f ’ ( & ) ) p n ( X - n). By theorem 2.3.7 and corollary 3.6.6 in [Roman (1984)], we have
+
+
= p,(x - n)
which proves the rest of the lemma.
- n ) - Icp,(x - n ) + np,(a: x-n x-n
7
(4.93)
0
131
Calculations for the example series
4.4.2
More on Rotansky- Witten invariants of closed graph homology classes
By proposition 3.5 we can restrict ourselves to the calculation of RozanskyWitten invariants for closed graph homology classes when we are interested in the Rozansky-Witten invariants for all graphs. Let y be a homogeneous (with respect to the number of internal vertices) closed graph with 2k internal vertices. Let (X, a ) be a holomorphic symplectic manifold of complex dimension 2n with n 2 k. Then RW(xl")(y) E HZk(X,Ox).
(4.94)
If X is irreducible, we therefore have RW(x7u)(y)= ,by. [a]k with a s H2k (X , O x ) = C
,ByE C
(4.95)
[aIkdue to the irreducibility. We can express Py as
This formula also makes sense for non-irreducible X, so we define:
Definition 4.10 ( "Rozansky-Witten number") We set
By k-linear extension, let graph homology classes y.
&(X,c) also be
defined for non-homogeneous
Recall that Bo is the graph homology space of closed graphs.
.>
Remark 4.14 (Homomorphic properties of p.) The map a0
is k-linear.
If X
c,7
H
MX,
(4.98)
is irreducible, it is a homomorphism of k-algebras.
From now on we shall restrict our attention to the example series: Let X be a smooth projective surface that admits a holomorphic symplectic form
132
Chem Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
(e.g. a K3 surface or an abelian surface). We further fix a holomorphic symplectic form CJ E HO(X, R$) that is normalised such that
s,.
= 1.
(4.99)
Definition 4.11 (The numbers h,(n)) For every homogeneous (with respect to the number of internal vertices) closed graph homology class y of degree 2k and each n = 0,1,. . . , let us set
hq(n) := j3r(X[k+nl,
(4.100)
By k-linear extension, we define h: (n)for non-homogeneous graph homology classes y as well.
Proposition 4.8 (The series built up by the h,(n)) For all closed graph homology classes y,we have
in C"91I. Proof. Then
Let us assume that y is homogeneous with 2k internal vertices.
In the last equation we have used corollary 4.2. Summing up and introducI7 ing the counting parameter q yields the claim.
Definition 4.12 (Some universal graphs built from wheels) We set (4.103) k= 1
W ( t )= exp(w(t)) and W Remark 4.15 nected.
.]I.
:= W(l) E B [ [ u ~ , u . .~ ,
(Connectedness of the wheels) The graph w ( t ) is con-
133
Calculations for the example series
Recall the definition of A(t) and D(t). These power series are closely related to the Rozansky-Witten invariants of w ( t ) : Proposition 4.9 (Rozansky-Witten invariants of elements built from wheels) The Rozansky- Witten invariants of the w(t) are encoded by
Proof.
Using proposition 4.8 yields
Ekill
= z=o
= exp(qA((2~/i)~t)) e~p(c2(X)D((2n/i)~t)).(4.105)
0 Corollary 4.3 (Nieper-Wii3kirchen) For every n = 0 , 1 , . . . we have hfW(t))(n)= exp(c2(X)D((2n/i)t)) exp(nlogA((2nli)t)).
Proof.
(4.106)
Comparison of coefficients in (4.104) gives h&(t)) (n) = A ( ( 2 ~ / i ) ~ tex~(cz(X>D((2n/i)~t)). )"
(4.107)
Lastly, note that A is a power series in t that has constant coefficient one.
0 Definition 4.13 with (4.106), we set
(Extending
the
definition of
h) Compatibly
h$,W(t),(n):= A((27~li)~t)" exp(c2(X)D((27~/i)~t))
(4.108)
for n E Z and n < 0. Proposition 4.10 (Calculations on the generalised Kummer surfaces) Let A be an abelian surface. Let us f i x a holomorphic symplectic form E Ho(X,Qi) that is normalised such that J, a5 = 1. Let y # 0be a homogeneous connected closed graph of degree 2k. Then we have (4.109)
134
Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
for any n > k. Proof.
The proof is a straight-forward calculation:
-
n
- -Py(A["I,~["I), n-k
(4.110)
where we have used lemma 1.2 and the following two lemmata.
Lemma 4.7 (Rozansky-Witten classes and coverings) Let Y : (X,Y*.-) --+ (Y,a)
(4.111)
be an unramijied covering of holomorphic symplectic manifolds. For every graph homology class y then follows
RW(x7"*")(y)= ~*(Rw('>~)(y)).
(4.112)
Proof. We have 8 x = v * @ yand therefore cre, = v*aeY.By definition of the Rozansky-Witten weight system, the lemma follows. 0 Lemma 4.8 (Rozansky-Witten classes and the product with a flat manifold) Let ( X , u ) and ( X ' , a ' ) be two holomorphic symplectic manifolds. If the tangent sheaf of X' is trivial, we have RW(xX x ' , p * U + p ' * O ' ) (7) = P * R W ( ~ ' ~ ) ( ~ )
(4.113)
i f y # 0 is a graph homology class. Here, p and p' are the two canonical projections X x X' -+ X and X x X' -+ X', respectively.
Proof. This result follows directly from 2.11 and the fact that RW(x'>u')(y') = 0 if y' has at least one trivalent vertex. 4.4.3
A structural result on the Rozansky- Witten weights of closed connected graphs on the example series
The following theorem will play a key role in the effective calculation of Rozansky-Witten invariants from data given by the Chern numbers of the
135
Calculations for the example series
main examples of holomorphic symplectic manifolds.
Theorem 4.5 (Nieper-WiBkirchen) For any homogenous (with respect to the number of internal vertices) connected closed graph of degree 2k lying in the algebra C of polywheels there exist two rational numbers a,, c, such that for each K3 surface X together with a symplectic form CT E Ho(X,52%) with C T = ~ 1 and n 2 k we have
sx
,D,(x["], PI)= a,n + c,
(4.114)
and for each abelian surface A together with a symplectic form = 1 and n > k we have HO(X,0%)with
sx
pY (A""]], ,[["]I)
= ayn.
d
E
(4.115)
Proof. All references to homogeneity and degree are to be understood with respect to the grading given by the number of internal vertices. Let ( X , a ) be a K3 surface or an abelian surface together with a symplectic a?= 1. Let W 2 k be the homogeneous component of degree form with 2k of W(1). Then W ( t )= EEoW 2 k t k . Thus we have by (4.106):
s,
c 00
h?W(t))(4=
h?W2,)(4tk = % ( X ) ( t ) exp(nV(t))
(4.116)
k=O
with Uc,(x,(t) := e x p ( c 2 ( X ) D ( ( 2 7 ~ / i ) ~and t ) ) V ( t ):= l o g A ( ( 2 ~ l i ) ~ t ) . Let us consider the case of a K3 surface X first. Note that c z ( X ) = 24. By definition of h:(n) we have =
P(w2k)(x'"1,ff["') h?wzk)(n
(4.117)
- Ic)
for all n 2 Ic. For n < k we take this equation as a definition for its left hand side. Let the power series T ( t )E Q[a2,a4,. . .][[t]] be defined by T ( texp(V(t))) = t , and set := V ( T ( t ) and ) o(t):= By lemma 4.6, we have
v(t)
m
hfW2,)(n- k)tk = o(t)exp(nv(t)).
,D(wct))(X["I,c7["1)=
(4.118)
k=O
Note that W ( t )is of the form exp(y) where y is a connected graph. By proposition 2.9 and remark 4.14 we therefore have
P((W(t)))(X? @["I)
= a o g (W(t))(XI"], ,["I)
=kP(W(t))
+
= nv(t) log U ( t ) . (4.119)
136
C h e r n Numbers and RW-Invariants of Compact Hyper-Kahler Manifolds
Finally, let X be any partition and set (4.120)
It is P((iVzx))= d 2 X P ( ( W ( t ) ) )= n a 2 m
+ 3 2 x 1%
m,
(4.121)
so the theorem is proven for K3 surfaces and all connected graph homology classes of the form ((Gzx)) and thus for all connected graph homology classes in C. Let us now turn to the case of a generalised Kummer variety, 2.e. let X = A be an abelian surface and n 2 1. Note that c2(A) = 0. Here, we have due to proposition 4.10:
p(wzk)(A""l],
n
=-
- k h?WZk) ( n -
(4.122)
for n > k. For n 5 k we take this equation as a definition for its left hand side. As Uo(t) = 1, lemma 4.6 yields in this case that
P(w(t))(A"nll,a"nll) =
n -hTWZk)(n k=O n - k
- k ) t k = expnv(t).
(4.123)
We can then proceed as in the case of the Hilbert scheme of a K3 surface to finally get P ( ( O z x ) )= n
. d2X(V'(t)).
(4.124)
0 Remark 4.16 (Origins of the previous theorem) I t has been conjectured by J. Sawon that the preceeding theorem holds true for all homogeneous connected closed graphs and not only those lying in C . This conjecture is older than the theorem itself and was one motivation for the author to prove the theorem. 4.4.4
Explicit calculation
Once we have calculated the constants a y and cy of the last theorem, we know the Rozansky-Witten invariants associated to y on our main examples of holomorphic symplectic manifolds as all information is encoded in ,By
Calculations for the example series
137
Again all references to homogeneity and degree are to be understood with respect to the grading given by the number of internal vertices. So now we have to find a way to calculate the constants a, and
c, for any homogeneous connected closed graph homology class y of degree 2k lying in C. By the previous theorem, we can do this by calculating ,Byon (X, a ) for ( X ,c) being the 2k-dimensional Hilbert scheme of points on a K3 surface and the 2k-dimensional generalised Kummer variety. We can do this explicitely by recursion over k: Let the calculation having been done for homogeneous connected closed graph homology classes y of degree less than 2k in C and both example series. Let X be any partition of k. We can express ((wax)) as ((WZX))
= (W2A)
+P
(4.125)
where P is a polynomial in homogeneous connected closed graph homology classes y of degree less than 2k in C (for this see proposition 2.9). Therefore, P((WZX)) ( X , is given by
where P’ is a polynomial in terms like &(X, a) with y’E C and deg y’ < 2k. However, these terms have been calculated in previous recursion steps. Therefore, the only thing new we have to calculate in this recursion step is /3(GZA)(X, a). We have:
As we have seen all the Chern numbers therefore able to calculate ,B(*zX)( X ,a). we have given an algorithm to compute connected closed graph homology class y
of X can be computed. We are This ends the recursion step as a, and cr for any homogeneous of degree 2k in C.
138
Chern Numbers and
RW-Invariants of Compact Hyper-Kahler Manifolds
Let us work through the recursion for k = 1,2,3. Firstly, we have ((W2)) = @2),
((w;)) = (W4)
- ((W2H2,
Now let X be a K3 surface and A an abelian surface. Let us denote by o either a holomorphic symplectic two-form on X with fx oo = 1 o r on A .with =1 Remark 4.17 (Calculation of Rozansky–Witten invarriants for the example series) Going through the recursion, we arrive at the following table:
k
y/ (&)2k
1
{{w2}}
2
((W1))
3
((W4)) ((W3)) ((w2w4)) ((w6))
a7
c,
12 -96 120
–36 -96 120
-1280 1600 -1400
-256 320 -280.
,B,(A"k+l]l) & ( X [ k ] ) –24 -2 88 360 -5120 6400 -5600
–48 -288 360 -4096 5120 -4480
Now, we would like to turn to Rozansky–Witten invariants: Let be any homogeneous closeed graph homology class of degree 2k. For any holomorphic symplectic manifold (X,) of dimension 2n, the associated Rozansky–Witten invariant is given by 2n
b,(X, c) =
(&)
RW(xIu)(I')exp(a
2n
( n - k)!
+ a)
/
& ( X , a)
X
exp(a
+ a).
(4.129)
To know the Rozansky-Witten invariant associated to closed graph homology classes, we therefore just have to calculate the value of p,. On an irreducible holomorphic symplectic manifold, y H /?, is multiplicative
Calculations for
the example series
139
with respect to the disjoint union of graphs, so it is enough to calculate Py for connected closed graph homology classes. However, we have just done this for the Hilbert schemes of points on a K3 surface and the generalised Kummer varieties - as long as y is spanned by the connected polywheels. By the procedure outlined above, theorem 4.5 therefore enables us to compute all Rozansky-Witten invariants of the two example series associated to closed graph homology classes lying in C.
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Bibliography
Bar-Natan, D. (1995). On the Vassiliev knot invariants, Topology 34, 2, pp. 423472. Beauville, A. (1983). Variete Kahleriennes dont la premikre classe de Chern est null, J. Diff. Geom. 18,4,pp. 755-782. Beauville, A., Bourguignon, J.-P. and Demazure, M. eds. (1985). GQombtrie des surfaces K3: modules et periodes, Astkrisque 126,Socihth Mathhmatique de France. Besse, A. L. (1987). Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 10,Springer-Verlag, Berlin. Borisov, L. and Libgober, A. (2002). McKay correspondence for elliptic genera, arXiv:math. AG/0206241. Carrell, J. B. and Lieberman, D. I. (1977). Vector fields and Chern numbers. Math. Ann. 225, 3, pp. 263-273. Deligne, P. (1990). Categories tannakiennes, The Grothendieck Festschrift, Vol. 11, Progr. Math. 87, pp 111-195, Birkhauser Boston, Boston. Ellingsrud, G., Gottsche, L. and Lehn, M. (2001). On the cobordism class of the Hilbert scheme of a surface, J. Alg. Geom. 10,1, pp. 81-100. Ellingsrud, G., S t r ~ m m eS. , A. (1987). On the homology of the Hilbert schemes of points in the plane, Invent. Math. 87,2, pp. 343-352. Ellingsrud, G., StrGmme, S. A. (1996). Bott’s formula and enumerative geometry, J. Amer. Math. Soc 9, 1, pp. 175-193. Deligne, P. and Freed, D. S. (1996). Sign manifesto. Quantum fields and strings: a course for mathematicians, Vol. 1, AMS, Providence, RI, pp. 357-363. Gel’fand, S.I. and Manin, Y. I. (2003). Methods of homological algebra, Second Edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin. Grothendieck, A. (1957). Sur quelques points d’algbbre homologique, T6holcu Math. J. 9, 2, pp. 119-221. Gross, M., Huybrechts, D. and Joyce, D. (2003). Calabi-Yau manifolds and Related Geometries, Springer-Verlag, Heidelberg Hartshorne, R. (1966). Residues and duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin-New York. Hinich, V. and Vaintrob, A. (2002). Cyclic operads and algebra of chord diagrams, 141
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Chern Numbers and RW-Invariants of Compact Hyper-Kahler Manafolds
Selecta Math. (N.S.) 8 , 2, pp. 237-282. Hirzebruch, F. (1966). Topological methods in algebraic geometry, Die Grundlehren der Mathematischen Wzssenschaften 131, Springer-Verlag New York. Huybrechts, D. (1999). Compact hyper-Kiihler Manifolds: Basic Results, Invent. Math. 135, 1, pp. 63-113. Lehn, M. (1999). Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136, 1, pp. 157-207. Mac Lane, S. (1998). Categories for the working mathematician, Second Edition, Graduate Texts in Mathematics 5,Springer-Verlag, New York. Nieper-WiBkirchen, M. A. (2002). On the Chern numbers of generalised Kummer varieties, Math. Res. Let. 9, 5 & 6, pp. 597-606. Nieper-WiDkirchen, M. A. (2002). On the Elliptic Genus of Generalised Kummer Varieties, arXiv:math.A G/0208077. Nieper, M. A. (2003). Hirzebruch-Riemann-Roch formulae on irreducible symplectic Kahler manifolds, J. Alg. Geom. 12,4, pp, 715-739. O’Grady, K. (1999). Desingularized moduli spaces of sheaves on a K3, J. Reine Angew. Math. 512,pp. 49-117. O’Grady, K. (2000). A new six-dimensional irreducible symplectic manifold, arXiv:math.A G/0010187. Roberts, J. (2001). Rozansky-Witten theory, arXiv:math. QA/O112209. Roman, S. (1984). The umbra1 calculus, Pure and Applied Mathematics 111, Academic Press, Inc., New York. Salamon, S. M. (1996). On the cohomology of Kiihler and hyper-Kahler manifolds, Topology 35,1, pp. 137-155. Sawon, J. (1999). Rozansky-Witten invariants of hyperkahler manifolds, PhD Thesis, University of Cambridge. Wakakuwa, H. (1958). On Riemannian manifolds with homogeneous holonomy groups Sp(n), TGhoku Math. J. 10,2, pp. 274-303. Thurston, D. (ZOOO), Wheeling: A diagrammatic analogue of the Duflo isomorphism, PhD Thesis, U. C. Berkeley.
Index
I3, 46
(1,), 71 1, 50
b, 61 Bo, 46, 69 b-if ( X ,a h 9 7 b,(X, a),94 Br, 46 Bk,80 bk, 79 b(n(a)n(p)) ( X ,4, 103 b n ( t ) ( X , a ) ,97 B d P ) , 119, 127 b x , 86 b", 56 Beauville, A., x, 106 Beauville-Bogomolov quadratic form, 106 Bernoulli number, 80 modified, 79, 98 P,(X,a), 131 Betti number, 10, 26 Bianchi identity, 30, 34, 87 bivalent, 39 Bogomolov, F., 5, 106 Bott's residue formula, xii, 125
a, 61
A""]],16 A[ffI(P), 122 a t , 87 a?, 135 ( A I ) I 44 , a aa,,xv a x , xiv Q,, xiii A&.@),119, 127 A ( t ) , 121 a x ,86 adjoint, 53 adjunction formula, 8 algebra of polywheels, 75 f f , 55 a("),14 &II, 17 ff["+l], 111 a["],14, 111 a + ,48 ay,56 alternating action, xvii alternating group, xiii AS, 45 associativity condition, 48 Atiyah class, xi, 27, 86, 87 naturality of the, 27 symmetry of the, 32, 33
C, xiii
c, 75 c , 61
%, 135
C&.(P),119, 125
86 56
CX,
c",
143
144
Chern Numbers and R W-Invariants of Compact Hyper-Kahler Manafolds
C-valued points, 11 Calabi Conjecture, x, 7 Calabi-Yau theorem, 7 canonical sheaf, 4, 13, 126 Carrell, J. B., 125 category enriched, 50 k-linear, 49 of complexes, xv of metric Lie algebra objects, 53 of metric Lie algebras, 53, 64 of weight systems, 65 Cech cocycle, 28 Cech cohomology, 28 ch, 92 chains of Jacobi diagrams, 42 characteristic class, 92 characteristic number, 24, 124 Chebyshev polynomial, 99 Chern class first, 5, 14, 29 odd, 21 rational, 23 Chern number, xi, 24, 96, 121 Chern numbers of the example series, 128 Chern-Weil theory, 34 X I 110 X(ffh47 x ( x , ~ ,107 XY(X), 24 0146 classifying functor, 63 closure, 72 connected, 73 cobordant, 96 Coh(X) , 84 cohomology class, 14, 85 invariant, 9 cohomology group second, 14 coinvariants, xvi compatibility, 53 completion, 71 complex cobordism ring, 96 rational, 96
complex genus, 95, 96, 121 of a pair, 97 twisted, 126 complex numbers, xiii complex structure, 3 almost, 3 complex torus of dimension two, 9, 15 composition of marked graph homology classes, 59 connected component, 73 connected graph homology class, 74 coordinate-fi-ee language, 47 coproduct, 76, 77 in the category of sets, xiii cotangent sheaf, xvii counit, 77 counter-clockwise ordering, 41 covariantly constant tensor, 2 covering, 121 cup-product, 57, 68 cycle, xiii cycle notation, xiii cyclic invariance, 55 cyclic ordering, 40
D(X), 85 D+((P), 119, 125 D ( t ) , 121 D(X), 83 decomposition of the Hilbert scheme, 116 Decomposition Theorem, 5 Def(X), 7 (Def(X),0 ), 7 deformation, 7 invariance under, 95 semi-universal, 7 universal, 8, 37 A, 12, 76, 95 8, 70, 71 A ( e x p 4 , 78 8Q(t),81 derived category, 112 bounded, 83 graded, 85 of coherent sheaves, 83
Index
descend, 13 diagonal, 12 big, 12 differential operator, 69 disjoint union of graph homology classes, 68 divisibility of the Euler number, 25 Douady space, 11 Douady-Barlet morphism, 11
E, 12 edge, 39 Ellingsrud, G., xii, 115 elliptic genus, 127 End(€), xvii End(&,a), 20 equivalence relation, 45 77, 12 Euler characteristic holomorphic, 107 topological, 10, 125 exceptional divisor, 12 exP(a), 72 exP(Y)>74 exP((Y)), 74 exp(ff), 93 Extl(E,Rx 8 E ) , 27 Ext&-h(,,(E,F), 84 Ext"(X, Y),84 extension class, 28 extension group, 84
5, 41
Ft, 56 F:, 56 F t , 56 f!,112
f!,112 first-order jet bundle, 27 flag, 39 forgetful map, 42 functor between symmetric monoidal categories, 48 k-linear, 50 of k-linear categories, 50
145
fundamental group, 15 g x , 36 Gottsche, L., xii, 115 GAGA principle, 11 %Y% 69 Y x Y 1 ,80 y U y', 43, 68 Y f , 79 YYI, 68 (Y,YO, 71 r H , 46 .i.>69 "la), 81 % ,...,m(71 ,. . ' 7 ~ n, 69 ) ,...,r,, , 69, 70 rI,46 r/{i,ill, 42, 43 r-, 45 riT,43 r+,45 Y l O Y,59 W),51 rx,46 GauSBonnet formula, 125 global section functor, 51 of Ic-linear symmetric monoidal categories, 51 glueing, 58, 70 glueing of legs, 43 grading, 42, 46, 64, 68, 90 default, 46 graph, 39 closed, 72 connected, 73 empty, 67 e, 67 Theta, 68 visualisation of a, 39 graph homology space, 46, 65 Gritzenko, N., 25 Grothendieck group of coherent modules, 112 Grothendieck's construction. 11
H 2 ( X ,C), 37
