A. Algebraic Varieties and Complex Analytic Spaces

In this Appendix we compile some results about algebraic varieties over C, complex analytic spaces and their interrelations, which are frequently used in this book.

To any algebraic variety X over the complex numbers one can associate a complex analytic space Xhol in a natural way (see Serre [1]). This construction is functorial: for any morphism f : Z → X of algebraic varieties over C there is a natural morphism of complex analytic spaces fhol : Zhol → Xhol. Moreover, many of the important notions in both categories are preserved under X → Xhol. For example: an algebraic variety X is complete if and only if Xhol is compact, and X is smooth if and only if Xhol is a complex manifold. Conversely, a complex analytic space X is called algebraic, if there is an algebraic variety X over C such that X Xhol. In both, the algebraic and the complex analytic category one has the notion of a vector bundle and its associated locally free sheaf. For our purposes it is convenient not to distinguish between these objects: we always speak of vector bundles, even if we mean the associated locally free sheaf. In particular, for a line bundle L on a complex analytic space (respectively algebraic variety) X we denote by H i(X, L), or H i(L) if there is no ambiguity about the base space X, the i-th cohomology group of X with values in the locally free sheaf associated to L. With a similar construction as above one can associate to an algebraic vector bundle (or more generally an algebraic coherent sheaf) F on the algebraic variety X over C a holomorphic vector bundle (respectively a holomorphic coherent sheaf) Fhol over the complex analytic space Xhol. Again this process is functorial: given algebraic vector bundles (respectively coherent sheaves) F and G on X, one can associate to any algebraic homomorphism f : F → G a holomorphic homomorphism fhol : Fhol → Ghol. Not every holomorphic vector bundle on an arbitrary algebraic complex analytic space comes from an algebraic one. However in the case of complete algebraic varieties we have the following comparison theorems due to Serre [1].

Theorem A.1. Let X be a complete algebraic variety over C. a) For any holomorphic vector bundle (respectively holomorphic coherent sheaf) F on Xhol there exists a unique algebraic vector bundle (respectively algebraic coherent sheaf) F such that Fhol F. 568 A. Algebraic Varieties and Complex Analytic Spaces

b) For every homomorphism f: F → G of holomorphic coherent sheaves F and G on Xhol there exists a unique homomorphism f : F → G of algebraic coherent sheaves on X such that fhol = f. Theorem A.2. Let X be a complete algebraic variety over C. For any coherent sheaf i i F on X the natural maps H (X, F ) → H (Xhol,Fhol), i ∈ Z, are isomorphisms of C-vector spaces. Thus in the case of a complete algebraic variety X we need not distinguish between algebraic vector bundles on X and holomorphic vector bundles on Xhol. Recall the following theorem due to (see Griffiths-Harris [1]). Theorem A.3. Suppose X is a complete algebraic variety over C and Z a closed analytic subset of Xhol, then there is an algebraic subvariety Z of X, such that Zhol Z. A consequence of Chow’s theorem is Corollary A.4. Suppose X and Z are complete algebraic varieties over C and f : Zhol → Xhol is a holomorphic map. Then there is an algebraic map f : Z → X with fhol = f. Moreover we need the following theorem due to Grauert-Remmert [1]. Theorem A.5. Suppose X is a normal algebraic variety over C and Z is a normal complex analytic space. If f: Z → Xhol is a finite morphism, then there exists a unique normal algebraic variety Z over C and a finite morphism f : Z → X such that Zhol = Z and fhol = f. Next let us recall a result about quotients of complex analytic spaces (see Cartan [1]). For this let G be a group acting as a group of isomorphisms on the complex analytic space X. The quotient X/G, endowed with the quotient topology, admits the structure of a ringed space in a natural way: denote by π : X → X/Gthe canonical projection. Then by definition OX/G(U), for U ⊆ X/Gopen, is the set of functions f : U → C, − for which fπ is an element of OX(π 1U). Moreover recall that the group G acts properly and discontinuously on the complex analytic space X, if for any pair of compact subsets K1,K2 of X the set {g ∈ G|gK1 ∩ K2 =∅}is finite. Then Theorem A.6. Suppose X is a complex analytic space and G is a group, acting properly and discontinuously on X. The quotient X/G is also a complex analytic space. Moreover, if X is normal, so is X/G. In the special case that X is a complex manifold there is a criterion for the quotient X/G to be also a complex manifold. Note that the action of G on X is said to be free, if gx = x for some x ∈ X and g ∈ G implies g = id. Corollary A.7. Let X be a complex manifold and suppose G is a group acting freely and properly discontinuously on X. Then X/G is also a complex manifold. Finally we give a proof of the so-called Seesaw Theorem, which we need in the following version. The same proof works also in the algebraic category. A. Algebraic Varieties and Complex Analytic Spaces 569

Seesaw Theorem A.8. Let X be a connected compact complex manifold, Z are- duced analytic space, and L a holomorphic line bundle on X × Z.

a) The set Z0 ={z ∈ Z| L|X ×{z} is trivial} is Zariski-closed in Z. b) If q : X × Z0 → Z0 denotes the projection map, then there is a holomorphic line bundle M on Z0 such that ∗ L X × Z0 q M.

Proof. a) Note ﬁrst that a holomorphic line bundle N on a connected compact complex manifold is trivial if and only if h0(N )>0 and h0(N −1)>0. Consequently % & 0 0 −1 Z0 = z ∈ Z | h (L|X × {z})>0 and h (L |X × {z})>0 and the assertion follows from the Semicontinuity Theorem (see Grauert-Remmert [2] Theorem 10.5.4). b) According to the Base Change Theorem (see Grauert [1] p.2 (2)) the sheaf M := q∗(L|X × Z0) is invertible on Z0 and the canonical base change homomorphism ϕ(z) : M(z) → H 0 LX × {z} =∼ C is an isomorphism for every z ∈ Z0. We have to show that the canonical map ∗ ∗ q M = q q∗ L X × Z0 → L X × Z0 is an isomorphism. For this it sufﬁces to show that the induced map ∗ H 0 q MX × {z} → H 0 LX × {z} is surjective for every z ∈ Z0, since L|X ×{z} is trivial. But this is a consequence of the surjectivity of ϕ(z).

The following statement is an immediate consequence of the Seesaw Theorem. It is often called the Seesaw Principle. Corollary A.9. Let X and Z be compact complex manifolds and L a holomorphic line bundle on X × Z.IfL|X ×{z} is trivial for all z out of an open dense subset of Z and L|{x0}×Z is trivial for some x0 ∈ X, then L is trivial. B. Line Bundles and Factors of Automorphy

In this section we outline how to describe certain line bundles by factors of automorphy. Though the results are valid in greater generality, we restrict ourselves to the cases needed in this book. Suppose X is a complex manifold (for us X will be a complex torus or the universal abelian variety over the Siegel upper half space), and let π : X → X be the universal covering. The topological space X inherits the structure of a complex manifold in a natural way. We denote by π1(X) the fundamental group of X → X.Inthe notation we omit the base point of the fundamental group. Our aim is to describe those holomorphic line bundles L on X, whose pullback π ∗L is trivial, in terms of the cohomology of the fundamental group π1(X) acting from the left on X.

Let us recall the deﬁnition of the cohomology group H 1(π (X), H 0(O∗ )) of π (X) 1 X 1 with values in the multiplicative group H 0(O∗ ) of nonvanishing holomorphic func- X tions on X: the action of the fundamental group π1(X) on X induces a π1(X)-module structure on H 0(O∗ ). A holomorphic map f : π (X) × X → C∗ satisfying the co- X 1 cycle relation f (λμ, x)˜ = f(λ,μx)f(μ,˜ x)˜ (1) for all λ, μ ∈ π (X) and x˜ ∈ X is called 1-cocycle of π (X) with values in H 0(O∗ ). 1 1 X Under multiplication these 1-cocycles form an abelian group Z1(π (X), H 0(O∗ )). 1 X For historical reasons we call the elements of the group Z1(π (X), H 0(O∗ )) fac- 1 X tors of automorphy or simply factors. The factors of the form (λ, x)˜ → h(λx)h(˜ x)˜ −1 for some h ∈ H 0(O∗ ) are called boundaries. They form the subgroup X B1(π (X), H 0(O∗ )) of Z1(π (X), H 0(O∗ )). The group H 1(π (X), H 0(O∗ )) is 1 X 1 X 1 X deﬁned to be the quotient

H 1(π (X), H 0(O∗ )) = Z1(π (X), H 0(O∗ ))/B1(π (X), H 0(O∗ )) . 1 X 1 X 1 X

Any f in Z1(π (X), H 0(O∗ )) deﬁnes a line bundle on X as follows: consider the 1 X holomorphic action of π1(X) on the trivial line bundle X × C → X

λ ◦ (x,t)˜ = (λx,f(λ,˜ x)t)˜ for all λ ∈ π1(X). This action is free and properly discontinuous, so the quotient 572 B. Line Bundles and Factors of Automorphy

L = X × C/π1(X) is a complex manifold (see Corollary A.7). Considering the projection p : L → X induced by X× C → X one easily checks that L is a holomorphic line bundle on X. By deﬁnition the group of holomorphic line bundles on X is canonically iso- 1 O∗ morphic to the group H (X, X). Hence the construction above gives a map Z1(π (X), H 0(O∗ )) → H 1(X, O∗ ). The following proposition shows, that this 1 X X map induces an injective homomorphism of cohomology groups. Proposition B.1. There is a canonical isomorphism

π∗ φ : H 1(π (X), H 0(O∗ )) → H 1(X, O∗ ) −→ H 1(X, O∗ ) . 1 1 X ker X X

Proof. Step I: Existence of φ1. It is not difﬁcult to see that the map

Z1(π (X), H 0(O∗ )) → (H 1(X, O∗ ) → H 1(X, O∗ )) 1 X ker X X deﬁned above is a homomorphism and factorizes via the group H 1(π (X), H 0(O∗ )). 1 X However in Chapter 2 we need its explicit description in terms of cocycles in 1 O∗ { } ∈ Z (X, X). For this let Ui I be an open covering of X, such that for every i I −1 there is a connected set Wi ⊆ π (Ui) with πi := π Wi : Wi → Ui biholomorphic. 2 For every pair (i, j) ∈ I there is a unique λij ∈ π1(X) such that

−1 = −1 πj (x) λij πi (x) . (2) for all x ∈ Ui ∩ Uj .

λij Wj

πi

πj

Uj Ui B. Line Bundles and Factors of Automorphy 573

This implies that λij λjk = λik (3) for all i, j, k ∈ I. Suppose f ∈ Z1(π (X), H 0(O∗ )).Fori, j ∈ I and x ∈ U ∩ U define g (x) = 1 X i j ij −1 { } 1 O∗ = f(λij ,πi (x)). Then Ui,gij I is a 1-cocycle in Z (X, X), since gij gjk gik for all i, j, k ∈ I which is an immediate consequence of (3) and the cocycle relation (1) for f .SowegetamapZ1(π (X), H 0(O∗ )) → H 1(X, O∗ ). Obviously this is 1 X X a homomorphism of groups. In order to define φ it remains to show, that B1(π (X), H 0(O∗ )) is in the ker- 1 1 X nel of this homomorphism. Let h ∈ H 0(O∗ ). Our homomorphism maps the fac- X tor h(λx)h(˜ x)˜ −1 ∈ B1(π (X), H 0(O∗ )) to the cocycle {g } with g (x) = 1 X ij I ij −1 −1 −1 ∈ ∈ ∩ h(λij πi (x))h(πi (x)) for every i, j I and x Ui Uj . By (2) we have = −1 −1 −1 { } 1 O∗ gij h(πj )h(πi ) ,so gij I is an element of B (X, X). Since the definition of φ1 does not depend on the choice of Ui and πi, this completes the proof of Step I. Step II: The inverse map. Suppose L ∈ ker(H 1(X, O∗ ) → H 1(X, O∗ )), in other words π ∗L is the trivial line X X ∗ bundle on X. Let α : π L → X × C be a trivialization. The action of π1(X) on X induces holomorphic automorphisms of π ∗L over this action. Via α we get for every λ ∈ π1(X) an automorphism φλ of the trivial line bundle X ×C. Necessarily φλ is of the form φλ(x,t)˜ = (λx,f(λ,˜ x)t)˜ with a map f : π1(X)×X → C, holomorphic in X. The equation φ = φ φ implies that f is a 1-cocycle in Z1(π (X), H 0(O∗ )). λμ λ μ 1 X Suppose α : π ∗L → X×C is a second trivialization. Then there is an h ∈ H 0(O∗ ), X such that αα−1(x,t)˜ = (x,h(˜ x)t)˜ for all (x,t)˜ ∈ X × C. Denoting by φ the λ automorphism of X × C associated to λ ∈ π1(X) with respect to the trivialization α, we obtain: ˜ = −1 −1 −1 ˜ = ˜ ˜ ˜ −1 ˜ φλ(x,t) (α α )φλ(α α ) (x,t) (λx, h(λx)f(λ,x)h (x)t) . Hence the class of the cocycle f in H 1(π (X), H 0(O∗ )) does not depend on the 1 X ∗ → × C 1 O∗ → trivialization π L X and we get a canonical map ker(H (X, X) H 1(X, O∗ )) → H 1(π (X), H 0(O∗ )). It is easy to see that this map is the inverse X 1 X of φ1, and this completes the proof. Remark B.2. With an analogous proof one can show that there are canonical homomorphisms

n 0 ∗ n φn : H (π1(X), H (X, π F)) −→ H (X, F) for all sheaves F of abelian groups on X and all n ≥ 0 (see Mumford [2] Appendix to § 2). In particular, using the notation of Step I in the proof of Proposition B.1 the homomorphisms φ1 and φ2 are given as follows: = −1 = −1 (φ1f)ij f(λij ,πi ) and (φ2F)ij k F(λij ,λjk,πi ) 574 B. Line Bundles and Factors of Automorphy

1 0 ∗ 2 0 ∗ for f ∈ Z (π1(X), H (X, π F)), F ∈ Z (π1(X), H (X, π F)) and i, j, k ∈ I.

Next we want to describe the group of global sections of a line bundle L in ker(π ∗ : H 1(X, O∗ ) → H 1(X, O∗ )) using a factor of automorphy. For any line X X ∼ ∗ bundle L on X there is a natural isomorphism H 0(X, L) −→ H 0(X, π L)π1(X). Suppose that π ∗L is trivial. A trivialization α : π ∗L → X × C induces an isomor- ∗ ∼ ∗ phism H 0(X, π L)π1(X) −→ H 0(X, X × C)π1(X).Iff ∈ Z1(π (X), H 0(O )) 1 X is the factor of automorphy associated to L with respect to the same trivialization ∗ α : π L → X×C (see proof of Proposition B.1), the elements in H 0(X, X×C)π1(X) are just the holomorphic functions ϑ : X → C satisfying

ϑ(λx)˜ = f(λ,x)ϑ(˜ x)˜ (3)

∼ 0 for all x˜ ∈ X and λ ∈ π1(X). Via the composed isomorphism H (X, L) −→ ∗ ∼ H 0(X, π L)π1(X) −→ H 0(X, X × C)π1(X) the sections of L over X may be considered as holomorphic functions on X, satisfying this functional equation. Note that the isomorphism depends on the trivialization: choosing another trivialization exactly means identifying the elements of H 0(X, L) with holomorphic functions satisfying (3) with respect to an equivalent factor of automorphy (to see this, look at the deﬁnition of the maps in Step II of the proof of Proposition B.1).

Finally we show that the homomorphism of Proposition B.1 is functorial. For this suppose ϕ : Y → X is a holomorphic map of complex manifolds. Let Yand X denote the universal covering spaces of Y and X and π1(Y ) and π1(X) the corresponding fundamental groups. The map ϕ induces a natural homomorphism ϕ∗ : π1(Y ) → π1(X) (of course the base points for π1(X) and π1(Y ) have to be chosen compatibly) and a holomorphic map ϕ : Y → X such that the following diagram commutes

ϕ / Y X

πY πX ϕ / Y X.

Pulling back cocycles we obtain homomorphisms between the cohomology groups of ∗ ∗ X and Y : the map f → (ϕ∗×ϕ) f induces the homomorphism H 1(π (X), H 0(O )) 1 X 1 0 ∗ 1 ∗ 1 ∗ → H (π1(Y ), H (O)), whereas H (X, O ) → H (Y, O ) is given by {gij }I → ∗ Y X Y {ϕ gij }I . One easily veriﬁes that the diagram

H 1(π (X), H 0(O∗ )) /H/1(π (Y ), H 0(O∗ )) 1 X 1 Y

φ1 φ1 1 O∗ //1 O∗ H (X, X) H (Y, Y ) B. Line Bundles and Factors of Automorphy 575 commutes. In particular, if L is any line bundle on X deﬁned by a 1-cocycle f in Z1(π (X), H 0(O∗ )), the line bundle ϕ∗L on Y is determined by the cocycle 1 X ∗ ∗ (ϕ∗ × ϕ) f : (λ, y)˜ → f(ϕ∗(λ), ϕ(y))˜ of Z1(π (Y ), H 0(O )). 1 Y C. Some Algebraic Geometric Results

In Chapter 15 we applied some more recent results of higher dimensional Algebraic Geometry, which we compile here for convenience of the reader.

C.1 Some Properties of Q-Divisors

Let X be a normal projective variety of dimension g. In this section we collect some properties of divisors on X. A Q-divisor (respectively an R-divisor) on X is a finite formal linear combination D = riDi of irreducible subvarieties Di of X of codimension 1 with rational coefficients ri ∈ Q (respectively real coefficients ri ∈ R). The round-up -D., the round-down /D0 and the fractional part {D} of D are defined as follows -D.:= -ri.Di, /D0:= /ri0Di, {D}:=D −/D0 where as usual for r ∈ Q, -r. denotes the smallest integer ≥ r, and /r0:=[r] denotes the largest integer ≤ r.AQ-divisor (respectively an R-divisor) D on X is called nef ,if (D · C) ≥ 0 for every irreducible curve C on X. Here the definition of the intersection numbers (D · C) is an obvious generalization of the usual one.

Theorem of Kleiman C.1. For a Q-divisor D on X the following statements are equivalent i) D is nef, ii) (Ds · Y) ≥ 0 for any normal subvariety Y ⊂ X of dimension s and all integers s with 0 ≤ s ≤ g. 578 C. Some Algebraic Geometric Results

For a proof see Hartshorne [2] Theorem I 6.1. As always in this book a divisor means an integral divisor. A divisor D on X is called big,if 0 g h OX(nD) ≥ c · n for some positive constant c and n , 0. A Q-divisor D is called big, if there is a positive integer m such that mD is a big divisor. For a proof of the following Proposition see Kollar-Mori [1] Proposition 2.61.

Proposition C.2. A nef Q-divisor D on X is big if and only if (Dg)>0.

Seshadri’s Criterion C.3. For a divisor D on X the following statements are equivalent i) D is ample, ii) there is a constant ε>0 such that

(D · C) ≥ ε · m(C)

for all irreducible and reduced curves C on X.

Here m(C) = maxx∈C multxC. For a proof see Hartshorne [2] Theorem I 7.1. The R-vector space NSR(X) := NS(X) ⊗Z R ∗ is of ﬁnite dimension. Let K (X) be the convex cone in NSR(X) generated by classes of ample divisors on X.AnR-divisor D is called ample, if its class is contained in K∗(X). It is well known (see Hartshorne [2]) that the closure of K∗(X) consists of the real divisors on X which are nef. For the proof of the following theorem see Campana-Peternell [1].

Theorem of Campana-Peternell C.4. For a real divisor D on X the following statements are equivalent i) D is ample, ii) (Ds · Y) > 0 for any irreducible subvariety Y ⊂ X of dimension s and all integers s with 0 ≤ s ≤ g.

In particular, if D is nef but not ample, there exists an irreducible subvariety Y ⊂ X, say of dimension s, such that (Ds · Y) = 0.

C.2 The Kodaira Dimension

The Kodaira dimension of a projective variety X is deﬁned as follows: Let ν : X −→ X be a desingularization of X, and ϕ : X P H 0(ωn )∗ n X C.3 Vanishing Theorems 579 the rational map corresponding to the linear system |ωn | whenever h0(ωn )>0. The X X Kodaira dimension of X is −∞ if h0(ωn ) = 0 for all n ≥ 1, κ(X) := X maxn≥1 dim im ϕn otherwise. The variety X is called to be of general type,if

κ(X) = dim X.

Let p : X −→ Y be a proper and surjective morphism of complex projective varieties whose general ﬁbre Z is connected. Then the Kodaira dimensions of X, Y and Z are related by Proposition C.5. κ(X) ≤ κ(Y ) + κ(Z). This is a weak version of the addition formula for the Kodaira dimension. For a proof we refer to Ueno [1] Theorem 6.12.

C.3 Vanishing Theorems

Let X be a smooth projective variety.

Kawamata-Viehweg Vanishing Theorem C.6. LetD be a divisor on X admitting a numerical equivalence decomposition D ≡ D + aiDi such that 1. D is a big and nef Q-divisor, 2. Di is a divisor with simple normal crossings, 3. ai ∈ Q with 0 ≤ ai < 1 for all i. Then j H ωX(D) = 0 for all j>0.

For the proof see Kollar-Mori [1] Theorem 2.64 or Lazarsfeld [3] Theorem 9.1.15. We apply the Kawamata-Viehweg Vanishing Theorem sometimes in the following form, which is an immediate consequence. Corollary C.7. If B is a big and nef Q-divisor on X, whose fractional part has simple normal crossing support, then j H ωX(-B. = 0 for all j>0. Generic Vanishing Theorem C.8. If X has maximal Albanese dimension, then

j 0 H (ωX ⊗ P)= 0 for all j>0 and generic P ∈ Pic (X). Recall that X has maximal Albanese dimension, if the Albanese map α : X → Alb(X) is generically ﬁnite over its image or equivalently if dim α(X) = dim X. For a proof of the Generic Vanishing Theorem see Lazarsfeld [3] Theorem 4.4.3. 580 C. Some Algebraic Geometric Results

C.6 Some Results from Intersection Theory

Let X be a smooth projective variety and p ∈ X a point. For any subscheme V of X passing through p let P(CpV) denote the projective tangent cone of V at p (see Fulton [1] p. 227). It is a closed subscheme of the projectivized tangent space P(TpX) of X at p. In particular the degree of the projective tangent cone is well deﬁned. Let V1,...,Vr be pure dimensional subschemes of X with r dim Vi = (r − 1) dim X (C.1) i=1 such that the intersection V1 ∩···∩Vr is ﬁnite. One says that V1,...,Vr intersect transversally at p,if 2r P(TpVi) =∅. i=1 For the proof of the following result see Fulton [1] Corollary 12.4.

Proposition C.9. Let V1,...Vr be closed subschemes of X satisfying (C.1). Then (V1 · ...· Vr ) ≥ multpV1 · ...· multpVr p∈V1∩···∩Vr with equality if and only if V1,...,Vr intersect transversally. The following proposition is a consequence of the Blow-up Formula (see Fulton [1] Corollary 6.7.1). Proposition C.10. Let V be a subvariety of X and V the proper transform of V in the Blow-up of X at the point p with exceptional divisor E. Then dim V (E · V)= multpV.

C.7 Adjoint Ideals

Let X be a smooth variety and D ⊂ X a reduced effective divisor. The Adjunction formula (see Kollar-Mori [1] Proposition 5.73) gives the following exact sequence

0 −→ ωX −→ ωX(D) −→ ωD −→ 0, (C.2) where ωD denotes the dualizing sheaf of D. The following proposition relates this with a desingularization of D. For this let g : Y → X be an embedded resolution of D, i.e., a desingularization of X such that there is a cartesian diagram / D Y

f g // D X C.7 Adjoint Ideals 581 where the proper transform D of D is a smooth divisor in Y . Obviously we have ∗ ωY (D) = g ωX(D) ⊗ OY (P − N) with effective g-exceptional divisors P and N with no common component. Consider the ideal JD := g∗OY (−N) ⊆ g∗OY = OX. With this notation we have:

Proposition C.11. 1. The embedding OX → OX(D) defined by D induces an exact sequence −→ −→ ⊗ J −→ −→ 0 ωX ωX(D) D f∗ωD 0. (C.3) J = O = 2. D X if and only if f∗ωD ωD. J Note that the sheaf f∗ωD and thus the ideal D is independent of the choice of the resolution (see Lazarsfeld [3] Theorem 9.2.18). The ideal JD is called the adjoint ideal of the divisor D in X. By definition JD is cosupported in the singular locus of D. Recall that by definition a variety V has rational singularities, if there is a resolution f : W → V such that f∗OW = OV (equivalently such that V is normal) and i R f∗OW = 0 for i>0. According to Kollar-Mori [1] Lemma 5.12 V has rational singularities if and only if V is Cohen-Macauley and f∗ωW = ωV . Hence condition 2) in Proposition C.11 means just that JD = OX if and only if D is normal and has only rational singularities. (Note that D is Cohen-Macauley as a reduced effective divisor.) The following proof is due to Ein-Lazarsfeld [1].

Proof (of Proposition C.11). The divisor D being smooth, the adjunction formula −→ −→ −→ −→ gives the exact sequence 0 ωY ωY (D) ωD 0. Applying g∗ we obtain −→ −→ −→ −→ 0 ωX g∗ωY (D) f∗ωD 0 i since g∗ωY = ωX and R g∗ωY = 0 (see Kollar-Mori [1] Theorem 5.10). Moreover g∗OY (P − N) = g∗OY (−N) = JD, implying ∗ g∗ωY (D) = g∗(g ωX(D) ⊗ OY (P − N)) = ωX(D) ⊗ JD.

This completes the proof of a). As for b): The sequences (C.3) and (C.2) yield the following commutative diagramm // / / / 0 ωX ωX(D) ⊗ JD f∗ωD 0 = // / // / 0 ωX ωX(D) ωD 0, which implies the assertion. D. Derived Categories

In Chapter 14 we were naturally lead from the category of coherent sheaves on an abelian variety to some categories derived from it. In this appendix we compile the main deﬁnitions of derived categories and state the results which are applied in Chapter 14. For details we refer to Gelfand-Manin [1] and Hartshorne [3].

D.1 Deﬁnition and First Properties

Let A be an abelian category. In almost all our applications A will be the category of coherent sheaves on a smooth projective variety X. Let K(A) denote the category of complexes over A modulo homotopic equivalence. The objects of K(A) • n n n n+1 are complexes K = (K ,d : K → K )n∈Z and the morphisms of K(A) are the morphisms f • : K• → L• of complexes. Here two morphisms f • and g• : K• → L• are identiﬁed if they differ by a homotopy, i.e., if there is a family of morphisms hn : Kn → Ln−1 such that f n − gn = hn+1dn + dn−1hn for all n ∈ Z:

// / dn / // ··· n−1 Kn n+1 ··· K y yK n yy yy h y n n y yy f −g yy y|y y|y hn+1 // / / / ··· Ln−1 Ln Ln+1 ··· . dn−1 In particular the corresponding homology morphism H n(f •) : H n(K•) → H n(L•) is well deﬁned for all n. A morphism f • : K• → L• in K(A) is called a quasi-isomorphism,ifH n(f •) : H n(K•) → H n(L•) is an isomorphism for all n. The derived category of A is now, roughly speaking, the category with the same objects as K(A), but where all quasi- isomorphisms are replaced by isomorphisms. For the precise deﬁnition we use the following theorem. Theorem D.1. There exists a unique category D(A) and a unique functor q : K(A) → D(A) such that 1. q(f) is an isomorphism for any quasi-homomorphism f ; 2. for any functor p : K(A) → C, mapping any quasi-isomorphism f in K(A) to an isomorphism p(f ) in C, there is a unique functor r : D(A) → C such that p = r ◦ q. 584 D. Derived Categories

For a proof see Gelfand-Manin [1] Chapter III § 2. The category D(A) is called the derived category of A . To understand the philosophy behind it consider the case of the category CohX of coherent sheaves on a variety X. If a coherent sheaf F on X is identiﬁed with the complex ...→ 0 → F → 0 →··· (with F at the 0-th place), a projective resolution of F may be considered as a quasi- isomorphism: // / / / // // ··· P −2 P −1 P 0 0 0 ···

// // // // // // ··· 0 0 F 0 0 ··· .

Hence in the derived category D(CohX) a coherent sheaf on X is isomorphic to all its projective resolutions. The category D(A) can be described as follows: Objects of D(A) are the objects of • n n • • K(A), i.e., the complexes K = (K ,d )n∈Z. A morphism K → L in D(A) is an equivalence class of diagrams (s, f ) in K(A) of the form

• Z ?? s ??f ? K• L• with a morphism f and a quasi-isomorphism s. A diagram (s,f) is equivalent to (s, f ) if and only if the two diagrams can be completed to a diagram

• W ?? t ??g ? Z• TTT i Z• TiTiTii ?? s iiii TTTT ??f ii TTT ? iii s f TT • ti T* K L• with a morphism g and a quasi-isomorphism t. The composition of morphisms (s, f ) : K• → L• and (t, g) : L• → M• is given by a diagram (sr, gh) :

• W ?? r ??h ? • • U ?? V ?? s ??f t ??g ? ? K• L• M• with a morphism h and a quasi-isomorphism r. Of course one has to show that the diagram (r, h) exists and (sr, gh) is uniquely determined up to equivalence. Consider the full subcategories K+(A) and K−(A) of K(A) consisting of com- • i plexes K with K = 0 for i ≤ i0, respectively i ≥ i0 for some integer i0, as well as D.1 Deﬁnition and First Properties 585

Kb(A) = K+(A) ∩ K−(A). For example injective resolutions (respectively projec- + tive resolutions) of a coherent sheaf on a variety X belong to K (CohX) (respectively − K (CohX)). Quasi-isomorphisms in these categories are defined as in K(A). Using this one defines the derived categories D+(A) of left bounded complexes, D−(A) of right bounded complexes, and Db(A) of bounded complexes in the same way as D(A). In the sequel we use the notation D∗(A) to indicate that we work in either of the categories D(A), D+(A), D−(A) or Db(A). Similarly we use the notation K∗(A). In abelian categories one has the notion of a short exact sequence, which is fundamental in homological algebra. Now it turns out that derived categories are not abelian categories in general. So one needs a replacement for the short exact sequences. These are the distinguished triangles which we introduce next. For this we need some auxiliary notions. • = n n For any complex K (K ,dK )n∈Z and any integer m define the shifted complex •[ ] •[ ] n := m+n n := − m m+n ∈ Z K m by (K m ) K and morphisms dK[m] ( 1) dK for all n . For a morphism of complexes f : K• → L• define f [m]:K•[m]→L•[m] by (f [m])n := f m+n. Now let f : K• → L• be a morphism of complexes. The cone • of f is the complex Cf with n n n dK[1] 0 C = K[1] ⊕ L and dC = . f f f [1] dL It fits into the following exact sequence

//• ι2 / • p1 / • // 0 L Cf K [1] 0 , (D.1) ∗ where ι2 and p1 denote the natural maps. A triangle in the category K (A) (respectively D∗(A)) is a diagram of the form

• α / • β / • γ / • K L M K [1]. It is often written in the form of a triangle, which explains the name. A morphism of triangles is a commutative diagram

• α / • β / • γ / • K L M K [1]

f g h f [1] • α / • β / • γ / • K L M K [1]. A triangle is called distinguished if it is isomorphic in D∗(A) to the diagram

• f / • ι2 / • p1 / • K L Cf K [1], (D.2) derived from the exact sequence (D.1). It is easy to see that the composition of any two consecutive morphisms in (D.2) is equal to 0 in the category K(A), i.e. up to homotopy equivalence. The main property of distinguished triangles is contained in the following theorem. 586 D. Derived Categories

α β γ Theorem D.2. For any distinguished triangle K• → L• → M• → K•[1] the following sequence is exact

• γ [−1] • α • β ···→H 0(M [−1]) → H 0(K ) → H 0(L ) →

β • γ • α[1] • → H 0(M ) → H 0(K [1]) → H 0(L [1]) →··· where α[n]:=H 0(α[n]) and similarly β[n] and γ [n] for all n ∈ Z. For a proof we refer to Gelfand-Manin [1] Chapter III § 3. Note that by deﬁnition of K•[n] we have H 0(K•[n]) = H n(K•) and similarly for H 0(L•[n]) and H 0(M•[n]). Hence the exact sequence of Theorem D.2 is just the usual long exact sequence

//− • // • // • // • // • // • // ··· H 1(M ) H 0(K ) H 0(L ) H 0(M ) H 1(K ) H 1(L ) ··· The category D(A) is additive. To see this, just note that one can show that any two morphisms f, g : K• → L• in D(A) can be represented by diagrams

• s • ϕ • • s • ψ • K ← M → L and K ← M → L with the same quasi-isomorphism s. Then f +g is defined by the class of the diagram s ϕ+ψ K• ← M• → L•. Any object F ∈ A will be identified with the complex consisting of F at place 0 and 0 elsewhere. Then F [n] is the complex with F at place −n and 0 elsewhere. These complexes can be considered as objects in any of the categories D∗(A). Using this one defines for any F and G ∈ A and n ∈ Z: n Ext A(F, G) := HomD∗(A)(F, G[n]). (D.3) It is easy to see that this definition does not depend on the choice of ∗. The n ∗ Ext A(F, G)’s are abelian groups, since D (A) is an additive category. Moreover, n ◦ ◦ Ext A determines a bifunctor A × A → Ab, where A denotes the opposite category and Ab the category of abelian groups. Let us recall the classical definition of the Ext-groups. An object F ∈ A is called projective (respectively injective) if the functor G → Hom(F,G)(respectively the functor G → Hom(G,F)) is exact. A projective (respectively injective) resolution of F ∈ A is an exact sequence ···→P −2 → P −1 → P 0 → F → 0 (respectively 0 → F → I 0 → I 1 → I 2 → ···) with all P ν projective (respectively all I ν injective). Suppose now A admits sufficiently many projective (respectively injective) objects, i.e., any F ∈ A admits a projective (respectively injective) resolution. n In the first case the groups ExtA(F, G) are defined in the following way: Choose a projective resolution P • → F → 0. Then 0 → Hom(F,G) → Hom(P•,G)is a n complex, and ExtA(F, G) is defined to be its n-th cohomology group. Similarly, in • n the second case choose an injective resolution 0 → G → I and define ExtA(F, G) to be the n-th cohomology group of the complex 0 → Hom(F,G) → Hom(F,I•). This definition does not depend on the choice of the projective or injective resolution. D.2 Derived Functors 587

Theorem D.3. Suppose A admits sufﬁciently many projective or injective objects. Then n ≥ n ExtA(F, G) if n 0, Ext A(F, G) = 0 if n<0.

Note that this isomorphism is functorial in F and G. For the proof see Gelfand-Manin [1] Chapter III § 6.

D.2 Derived Functors

Let F : A → B be an additive functor of abelian categories. In general, F will not take quasi-isomorphisms into quasi-isomorphisms. Hence it does not extend directly to a functor D∗(A) → D∗(B). Under some additional assumptions however such an extension RF (respectively LF ) can be defined. It is called the right derived functor (respectively left derived functor) of F . For simplicity we define these functors only for the cases we need. For the general definition we refer to Gelfand-Manin [1] and Hartshorne [3].

For a variety X denote by ShX the category of OX-modules on X.IfY is a second variety and F : ShX → ShY a left exact functor, the right derived functor RF : + + • + D (ShX) → D (ShY ) can be deﬁned as follows: Let K ∈ D (ShX) be a complex • of OX-modules. An injective resolution of K is by deﬁnition a quasi-isomorphism

K• → I •,

• + where I ∈ D (ShX) is a complex of injective OX-modules. Note that injective resolutions always exist. Deﬁne

• • + RF(K ) := F(I ) ∈ D (ShX), where F(I•) is the complex

F(dn−1) F(dn) F(I•) : ···→F(In−1) → F(In) → F(In+1) →··· .

• • + For a morphism f : K → L in D (ShX) the morphism

RF (f ) : RF(K•) → RF(L•) is defined in an obvious way. These definitions do not depend on the choice of the injective resolutions. We use this definition only for the following left exact functors:

1. the functor f∗ : ShX → ShY , where f : X → Y is a morphism of abelian varieties. ∗ 2. the functor S : ShX → Sh,S(F) := p2∗(P ⊗ p F), with an abelian variety X 1 X, the Poincare´ bundle P on X ×X and the projections pi of X ×X (see Section 14.7). 588 D. Derived Categories

F : S → S F G := ∗F ⊗ ∗G 3. the functor hX hX, μ∗(π1 π2 ), where X is an abelian variety, F ∈ ShX, and μ, πi : X × X → X are addition respectively projection maps (see Section 14.3). The relation of RF to the (classical) derived functors RiF (see Hartshorne [1]) is as follows: Consider any F ∈ ShX in the usual way as a complex supported in degree 0. Then RiF(F) = Hi RF(F) is the i-th cohomology sheaf of the complex RF(F).

Recall the following deﬁnition: Let G : ShY → ShZ be a left exact functor for • + n some varieties Y and Z. A complex K ∈ K (ShY ) is called G-acyclic,ifK is G-acyclic for every n ∈ Z, i.e., if RiG(Kn) = 0 for all i>0 and n ∈ Z. In Chapter 14 we need the following results.

Theorem D.4. Let F : ShX → ShY and G : ShY → ShZ be left exact functors • • + such that the complex F(I ) is G-acyclic for every injective complex I ∈ D (ShX). Then there is a natural isomorphism of functors

R(G ◦ F) RG ◦ RF

+ + from D (ShX) to D (ShZ). See Gelfand-Manin [1] Theorem 3.7.1. In terms of spectral sequences this theorem implies

Theorem D.5. Let the hypothesis be as in Theorem D.4. Then for every sheaf F ∈ ShX there is a spectral sequence p,q = p q F ⇒ p+q = p+q ◦ F E2 R G R F( ) E R (G F)( ), which is functorial in F.

See Gelfand-Manin [1] Theorem 3.7.7. In Chapter 14 we work in fact with much smaller categories: For an abelian variety X we denote by D(X) the derived category of the category CohX of coherent sheaves on X and by Db(X) the full subcategory of bounded complexes. A complex in Db(X) in general does not admit an injective resolution in Db(X). But every K• ∈ Db(X) + can be considered as an element of D (ShX). Hence for any left exact functor • + F : ShX → ShY , RF(K ) is deﬁned as an element of D (ShY ).

Proposition D.6. Let F : ShX → ShY denote any of the functors in 1., 2., or 3. above. Then for any K• ∈ Db(X) we have RF(K•) ∈ Db(Y ).

Proof. If F = f∗ for a morphism f of abelian varieties, this follows immediately from Hartshorne [3], Chapter II, Proposition 2.2. = : S → S Let F S hX hX as in 2.. Then S is the composition of the exact functor F → P ⊗ ∗F p1 with the left exact functor p2∗: D.3 The Grothendieck-Riemann-Roch Theorem 589

· = ◦ P ⊗ ∗ · S( ) p2∗ p1( ) . · = ◦ P ⊗ ∗ · P ⊗ ∗ · Hence RS( ) Rp2∗ R p1( ) by Theorem D.4. But p1( ) being P ⊗ ∗ Db ⊆ Db × exact implies R( p1) (X) (X X) and by what we have said above b b Rp2∗ D (X × X) ⊆ D (X). The same proof works for F.

Hence the derived functors Rf∗, RS, and RF of the functors of 1., 2., and 3. are well defined on the category Db(X). In fact, using a result of Spaltenstein [1] it is not difficult to define them on the bigger category D(X) of unbounded complexes, but we do not need this fact.

Projection Formula D.7. Let f : X → Y be a ﬂat projective morphism of varieties and G a locally free sheaf on Y . Then there is a natural functorial isomorphism • ⊗ G • ⊗ ∗G Rf∗(K ) OY Rf∗(K OX f ) for every K• ∈ Db(X).

See Hartshorne [3] Proposition II 5.6.

Flat Base Change D.8. Let f : X → Y be a morphism of abelian varieties. Let g : Y → Y be a ﬂat morphism and X = Y ×Y X with projections p1 and p2. Then there is a functorial isomorphism ∗ • ∗ • g Rf∗(F ) Rp1∗p2F for any F • ∈ Db(X).

See Hartshorne [3] Proposition II 5.12.

D.3 The Grothendieck-Riemann-Roch Theorem

Let X be an abelian variety of dimension g. The Grothendieck group K(X)is deﬁned to be the free abelian group generated by all isomorphism classes [F] of coherent sheaves F on X modulo the relations [F]=[G]+[H] for each exact sequence 0 → G → F → H → 0 of coherent sheaves on X. One can compute K(X) only using locally free sheaves, since X is nonsingular. Consider the canonical map • k : Db(X) → K(X), L → (−1)iLi. i 590 D. Derived Categories

Let f : X → Y be a homomorphism of abelian varieties. For any coherent sheaf F on X deﬁne i i f!(F) := (−1) R f∗F in K(Y). i≥0 This induces a homomorphism of groups f! : K(X) → K(Y) ﬁtting into the following commutative diagram

Rf∗ / Db(X) Db(Y )

k k f! K(X) /K(Y). • • Let Ch (X) and Ch (X)Q be the Chow rings as in Chapter 16. For any x ∈ K(X)the ∈ i = i i-th Chern class ci(x) Ch (X) is well-deﬁned. If ct (x) i≥0 ci(x)t denotes = g + the Chern polynomial of x and one writes formally ct (x) i=1(1 ait), then the Chern character is the homomorphism of groups

g g ν • a ch : K(X) → Ch (X), ch(x) := eai = i . ν! i=1 i=1 ν≥0 Grothendick-Riemann-Roch Theorem D.9. For any homomorphism of abelian varieties f : X → Y the following diagram commutes

Rf∗ / Db(X) Db(Y )

ch ch f∗ Ch•(X) /Ch•(Y ).

The proof is a combination of the above diagramm and the usual Grothendieck-Rie- mann-Roch Theorem (see for example Fulton [1] Section 15.2) using the fact that for homomorphisms of abelian varieties the relative tangent bundle is trivial and thus its Todd class is 1. In Sections 14.3 and 14.6 we need Relative Duality in the following form (see Kleimann [1]): Relative Duality D.10. Let f : X → Y be a smooth morphism of smooth projective varieties, n := dim X −dim Y , F a coherent sheaf on X ﬂat over Y and G a coherent − sheaf on Y . Suppose for some i ≤ n the sheaf Rn if∗F commutes with base change, then there is a canonical isomorphism E i F ⊗ ∗G H n−i F G xtf ( ,ωf f ) omOY (R f∗ , ). = ⊗ ∗ −1 E i · H Here ωf ωX f ωY denotes the relative canonical sheaf, and xtf ( , ) ◦ H · H denotes the i-th derived functor of the composed functor f∗ omOX ( , ) for any coherent sheaf H on X. E. Moduli Spaces of Sheaves

In Chapter 14 we need some properties of moduli spaces of sheaves, which we compile here. Let SchC denote the category of Noetherian schemes over C. A contravariant functor F : SchC −→ { sets} is representable, if there exists a scheme M ∈ SchC and an isomorphism of functors

σ : Hom(· ,M)→ F(· ).

In this case the object ξ = σ(M)(idM ) ∈ F(M) is called the universal object for the functor F . The isomorphism σ is induced by pulling back the universal object:

σ(M)(f ) = F (f )(ξ) for all f : M → M.Iff is representable, the scheme M is uniquely determined (up to isomorphisms) and called the ﬁne moduli space for the functor F . Many functors do not admit a ﬁne moduli space. Therefore one introduces a slightly weaker notion of moduli space. A coarse moduli space for a contravariant functor F : SchC →{sets} is a scheme M ∈ SchC together with a morphism of functors

τ : F(· ) → Hom(· ,M) satisfying (i) τ(Spec C) : F(Spec C) → Hom(Spec C,M)is bijective; (ii) given a scheme M ∈ SchC and a morphism τ : F(· ) → Hom(· ,M ), there is a unique morphism f : M → M such that the following diagram commutes

F(· ) r MM rr MM τ rr MMτ rr MMM rxrr M& Hom(· ,f ) Hom(· ,M) /Hom(· ,M)

A coarse moduli space is unique (up to isomorphisms), if it exists. In order to obtain reasonable spaces as moduli spaces of sheaves we need the following two definitions. Let X be a projective variety over C and L an ample line bundle on X. For any coherent sheaf G on X the function PG : Z → Z defined by := G ⊗ n = − p p G ⊗ n PG(n) χ( L ) p≥0( 1) h ( L ) 592 E. Moduli Spaces of Sheaves is a polynomial, called the Hilbert polynomial of G with respect to L. It is well known that it is of the form = dim X n+ν PG(n) ν=0 αν ν (E.1) with αν ∈ Z. A coherent sheaf E on X is called stable (with respect to the ample line bundle L on X), if it is torsion free and satisfies χ(F⊗Ln) χ(E⊗Ln) , rk F < rk E for n 0 F = E = dim X n+ν for every proper subsheaf 0of . For a given polynomial P (n) ν=0 αν ν MP : S →{ } as in (E.1), consider the functor L chC sets defined by % & E × ∗E MP := coherent sheaves on X S, flat over S, s.t. ιs is ∼ L (S) L P ∗F = P s S / stable w.r.t. and ιs for all closed points of E ∼ E E E ⊗ ∗ where if and only if pSM for some line bundle M on S and ιs : X X ×{s} → X × S denotes the canonical embedding. The following theorem is due to Maruyama [1]. MP P Theorem E.1. The functor L admits a quasi-projective coarse moduli space ML . MP In some cases even a fine moduli space for L exists (see Maruyama [1] II Corollary 6.11.1): = MP Proposition E.2. If gcd(α0,...,αdim X) 1, the functor L is representable.

Suppose a coarse moduli space M for a functor F : SchC →{sets} exists. Then in principle all information concerning the structure of M is encoded in the functor F . In particular one can investigate its local properties around a closed point m ∈ M by looking at various infinitesimal deformations of the object in F(Spec C) corresponding to m. This means that properties of the local moduli space imply local properties of the coarse moduli space. In the rest of this appendix we try to clarify these statements. Let A denote the category of local artinian C-algebras with residue field C and A its formal completion, the category of complete local Noetherian C-algebras with residue field C. Let F : A →{sets} be a covariant functor. Then F extends in a canonical way to a functor F : A→{sets} defined by

F(R) := lim F(R/mn) ← for every C-algebra R ∈ A with maximal ideal m. The functor F is called pro- representable if F is representable. This means that there exists a C-algebra R ∈ A and an α ∈ F(R) such that the morphism of functors

Hom(R, · ) → F( · ), E. Moduli Spaces of Sheaves 593 deﬁned by Hom(R,S) → F(S), f → F(f )(α) for all S ∈ A, is an isomorphism. The formal spectrum Spf(R) is called the local moduli space of the functor F. Schlessinger [1] proved the following theorem characterizing pro-representable functors.

Theorem E.3. Let F : A →{sets} be a functor such that F(C) consists of one element. F is pro-representable if and only if the following conditions are satisﬁed:

(H ) for all homomorphisms πi : Ai → A, i = 1, 2,inA with π2 surjective with one-dimensional kernel, the canonical map

α : F(A1 ×A A2) → F(A1) ×F(A) F(A2)

is bijective. (Hf ) dimC F(C[])<∞.

Here A1 ×A A2 denotes the fibre product of A1 and A2 over A consisting of all pairs 2 (a1,a2) with ai ∈ Ai and π1(a1) = π2(a2). Moreover C[]=C[t]/t is the algebra of dual numbers over C. The property (H ) implies that F(C[]) admits a natural C-vector space structure. It is called the tangent space of the functor F. It coincides with the Zariski tangent space of the local moduli space of F. Let E be a coherent sheaf on a variety X over C. Consider the functor DE : A → {sets} of infinitesimal deformations of E defined by E A O an -flat coherent XA -module DE(A) := (E,κ) / ∼ and κ : E → E ⊗A C an isomorphism

where XA := X×CA and (E,κ)∼ (E ,κ ) if and only if there exists an isomorphism ∼ ϕ : E → E such that κ = (ϕ ⊗A C) ◦ κ. Recall that the coherent sheaf E is called = C simple,ifEndOX (E) .

Theorem E.4. For a simple coherent sheaf E on X the functor DE is pro-representable.

The local moduli space ME of the functor DE is also called local moduli space of the sheaf E .

Proof. It sufﬁces to check the conditions of Theorem E.3. Obviously DE(C) ={E} and dimC DE(C[])<∞, the sheaf E being coherent. So let πi : Ai → A, i = 1, 2, be homomorphisms in A with π2 surjective and with one-dimensional kernel. It remains to show that the canonical map : D × → D × D α E(A1 A A2) E(A1) DE (A) E(A2) is bijective. Surjectivity of α: 594 E. Moduli Spaces of Sheaves

Let (Ei,κi) ∈ DE(Ai) for i = 1, 2, such that E ⊗ E ⊗ =: E ( 1,κ1) A1 A ( 2,κ2) A2 A ( ,κ) in DE(A). The ﬁbre product E1 ×E E2 is a coherent OX ⊗ (A1 ×A A2)-module and ∼ the induced map κ : (E1 ×E E2) ⊗ C → E is an isomorphism. In order to show the surjectivity of α it sufﬁces to show that

(E1 ×E E2, κ) ∈ DE(A1 ×A A2).

For this it remains to show that E1 ×E E2 is a flat A1 ×A A2-module. Denote A := A1 ×A A2 and let M be an A-module. Then there is a canonical isomorphism ⊗ E × E = ⊗ E × ⊗ E M A ( 1 E 2) (M1 A1 1) M⊗AE (M2 A2 2) := ⊗ := ⊗ E where Mi M A Ai and M M A A. Since i is Ai-flat, for every injective : → ⊗ homomorphism f M N the homomorphism f A idE1×E E2 is injective as well. According to the definition of flatness this implies that α is surjective. Injectivity of α: E ∈ D D × D Let ( , κ) (A). Its image under α in E(A1) DE (A) E(A2) is E⊗ E⊗ ( A A1,κ1), ( A A2,κ2) = ⊗ with κi κ A idAi . According to the proof of the surjectivity the element (E ⊗ A1) ×E⊗ (E ⊗ A2), κ =: (E, κ) A AA A of DE(A) has the same image under α. So it suffices to show that (E,κ)is isomorphic to (E, κ). By assumption there is an isomorphism θ fitting into the commutative diagramm EH vv HH u1 v HHu2 {vvv H# E⊗ E⊗ A A1 A A2 v1 v2 E⊗ θ /E⊗ A A A A.

According to Lemma E.5 below θ is multiplication by some unit a ∈ A. Let a2 ∈ A2 be a lift of a. Replacing v2 by v2a2 we may assume that v1 ◦ u1 = v2 ◦ u2. Hence there is a unique homomorphism ϕ : E → (E ⊗ A1) ×E⊗ (E ⊗ A2) = E A AA A such that pi ◦ϕ = ui, where pi for i = 1, 2 denote the natural projections. According to Lemma E.6 below ϕ is an isomorphism. Obviously κ corresponds to κ under this isomorphism. This completes the proof of the theorem. E. Moduli Spaces of Sheaves 595

Lemma E.5. Let E be a simple coherent sheaf on X. Forany A ∈ A and E ∈ DE(A) the canonical homomorphism A → EndO (E) is an isomorphism. XA Proof. We apply induction on the length l(A):Ifl(A) = 1 then A = C and the assertion is obvious. So assume l(A) = l>1. There is a principal ideal ηA of A such that ηA C as A-modules. Then l(A/ηA) = l − 1 and thus by induction hypothesis the canonical map A/ηA → End(E ⊗A (A/ηA)) is an isomorphism. So, if ϕ ∈ End(E) then ϕ ⊗A (A/ηA) = (a + ηA) id for some a ∈ A. Consider the endomorphism ψ := ϕ − aid: E → E. Then ψ(E) ⊂ η E := E ⊗A ηA E ⊗A C E. Let m denote the maximal ideal of A. Then ψ annihilates m E = E ⊗A m, since η·m = 0. But E E ⊗A/m E/m E,soψ induces a homomorphism ψ : E → E. There is a b ∈ C with ψ = bid, E being simple. Let b ∈ A be a lifting of b. Then the following diagramm commutes

η / / E @ η E E @@ @@ψ @@ b ψ @

η E /E, so ψ = ηb idE . Hence ϕ = (a + ηb)idE implying that the canonical map A → End(E) is surjective. For the injectivity note that if 0 = c ∈ A then Im(c idE ) = c E = E ⊗A cA = 0, E being ﬂat over A.

Lemma E.6. Let J be a nilpotent ideal in a commutative ring A and ϕ : M → N a homomorphism of A-modules with N ﬂat over A.Ifϕ : M/JM → N/JN is an isomorphism, then ϕ is an isomorphism.

Proof. The proof of this Lemma is straightforward.

Proposition E.7. For any simple coherent sheaf E on X the Zariski tangent space of 1 the local moduli space of the functor DE is canonically isomorphic to Ext (E, E). OX See Huybrechts-Lehn [1] Corollary 4.5.2. A functor F : A →{sets} is called smooth if for every surjective homomorphism A → A in A the induced map F(A) → F(A) is surjective. This deﬁnition is motivated by the following

Proposition E.8. A local moduli space M for a functor F : A →{sets} is smooth if and only if F is smooth.

See Schlessinger [1] Remark 2.10.

In the special case of the functor DE of inﬁnitesimal deformations of a simple coherent sheaf E the following proposition gives a criterion for DE and thus for ME to be smooth. 596 E. Moduli Spaces of Sheaves

Proposition E.9. Fora surjective homomorphism A → A in A with one-dimensional kernel and for each deformation (E,κ)∈ DE(A) there is an element ω(E,κ)∈ 2 Ext (E, E) such that an extension (E ,κ ) ∈ DE(A ) of (E,κ)exists if and only OX if ω(E,κ)= 0.

For a proof see Sernesi [1]. The element ω(E,κ)is called the obstruction to lift of (E,κ)over Spec (A). The set of all possible obstructions forms a subvector space of Ext 2 (E, E), called the OX obstruction space of E. It is easy to see that any stable sheaf is simple. Hence for any stable sheaf E on X the local moduli space for the functor DE exists. The local structure of the coarse P D moduli space space ML at E is related to the local moduli space for the functor E as follows ∈ P D Proposition E.10. For any E ML the local moduli space ME of the functor E is isomorphic to the formal spectrum Spf O P of the completion of the local ring ML ,E P of ML at E. F. Abelian Schemes

In Chapters 14 and 16 we need some results on abelian schemes, which we compile in this appendix. We do not give full details here, since we do not consider abelian schemes as a subject of this book. For details we refer to the standard books Faltings- Chai [1], Bosch-Lutkebohmert-Raynaud¨ [1] and Mumford-Fogarty-Kirwan [1] as well as the articles Mukai [3] and Deninger-Murre [1] on the relative Fourier Functor. For simplicity we do not give the deﬁnitions and results in full generality, but only in the cases we need. In this Appendix the base scheme S is always a Noetherian scheme over C.

F.1 Abelian Schemes and the Poincare´ Bundle

A group scheme p : X → S (abbreviated by X /S) is called an abelian scheme over S if p is smooth and projective with connected geometric fibres. A homomorphism of abelian schemes is a homomorphism in the category of group schemes. For any −1 closed point s of S the fibre Xs := p (s) is a proper connected scheme over C. The set of closed points of Xs has the structure of a compact complex Lie group. So according to Lemma 1.1.1 it is a complex torus. Hence Xs (or rather its underly- ing complex manifold) is a complex abelian variety, since it is projective. Roughly speaking we may consider an abelian scheme p : X → S as a family of abelian varieties parametrized by S. The Rigidity Lemma (see Mumford-Fogarty-Kirwan[1] Proposition 6.1) implies that an abelian scheme is a commutative group scheme. As such it admits a zero section e : S → X .Iff : T → S is a morphism of Noetherian schemes over C, then the pull back pT : XT := X ×S T → T is an abelian scheme := ◦ = −1 over T with zero section eT (e f, idT ) and fibres Xt pT (t). Let X /S be an abelian scheme and L a line bundle on X .Arigidification of L along ∗ the zero section e is an isomorphism e L OS. For any morphism of Noetherian schemes f : T → S define % & isomorphism classes of line bundles PicX (T ) := . /S Lon XT , rigidified along eT

PicX /S( · ) is a contravariant functor from the category of Noetherian schemes into the category of abelian groups, called the Picard functor of X /S. The following theorem is due to Grothendieck [1]. 598 F. Abelian Schemes

Theorem F.1. The functor PicX /S( · ) is represented by a locally Noetherian group scheme PicX /S over S, called the Picard scheme of X /S . For f : T → S as above define isomorphism classes of line bundles L on XT , P 0 := L| icX /S(T ) rigidified along eT , such that Xt is algebraically equivalent to zero for any geometric point t of T. P 0 · P · icX /S( ) is an open subfunctor of icX /S( ). As a consequence of Theorem F.1 P 0 · 0 the functor icX /S( ) is represented by an open subscheme PicX /S of PicX /S. 0 → Theorem F.2. PicX /S S is an abelian scheme. For a proof see Mumford-Fogarty-Kirwan [1] Corollary 6.8. 0 → : X → PicX /S S is called the dual abelian scheme and denoted by p S respectively X/S. Let f : Y → X be a homomorphism of abelian schemes over S. The ∗ : P 0 · → P 0 · morphism of functors f icX /S( ) icY/S( ) defined by pulling back line bundles, induces a homomorphism of abelian schemes f: X → Y, called the dual homomorphism of f . X P 0 · The fact that the abelian scheme /S represents the functor icX /S( ) means that P 0 · · X there is a canonical isomorphism of functors icX /S( ) HomS( , ) in the category of Noetherian S-schemes. In particular, for the abelian S-scheme X we obtain a canonical bijection P 0 X = X X icX /S( ) HomS( , ). The line bundle P = PX /S on XX = X ×S X corresponding to the identity idX is called relative Poincare´ bundle of the abelian scheme X /S. The relative Poincare´ bundle is by definition rigidified along eX : X → X × X , i.e., ∗ P O eX X. Denote by e : S → X the zero section of the dual abelian scheme. Then P is also id×e ∗ rigidified alongeX : X X ×S S −−−→ X ×S X , meaning thateX PX /S OX.If ∗ s : X ×S X → X ×S X denotes the exchange morphism, this means that s PX /S is an element of Pic0 (X ). On the other hand, X → S being an abelian scheme, X/S we have a canonical bijection Pic0 (X ) Hom (X , X). X/S S ∗ Let ι : X → X denote the morphism corresponding to the line bundle s PX /S. The Biduality Theorem (see Bosch-Lutkebohmert-Raynaud¨ [1] Theorem 8.4.5) says that ι : X → X is an isomorphism. Hence, identifying X = X via ι, we obtain P ∗P X/S s X /S. F.2 Relative Fourier Functor 599

F.2 Relative Fourier Functor

Let X /S be an abelian scheme of relative dimension g. Denote by pi the projections of the product X ×S X .AnS-ﬂat coherent sheaf F on X /S is called WIT-sheaf of j P ⊗ ∗F = = index i if R p2∗( X /S p1 ) 0 for all j i. In this case F := i P ⊗ ∗F R p2∗( X /S p1 ) is called the Fourier transform of F.

Proposition F.3. Let F be an S-flat coherent sheaf on X /S. Suppose F|Xs is a WIT-sheaf of index i for all closed points s of S. Then 1. F is a WIT-sheaf of index i, 2. F is flat over S, 3. F|Xs (F|Xs)for all closed points s of S. This follows from a modified version of the base change theorem, see also Mukai [3] Theorem 1.6. As in the absolute case there is an inversion theorem for WIT- sheaves on abelian schemes. For this let ωX /S denote the relative canonical line bundle of X /S. Note ∗ that ωX /S = p N for some line bundle N on S, since it restricts to the trivial line bundle on every fibre.

Inversion Theorem F.4. If F is a WIT-sheaf of index i on an abelian scheme X /S of relative dimension g, then its Fourier transform F is a WIT-sheaf of index g − i on X/S and there is a canonical isomorphism

F −1 ⊗ − ∗ F ωX /S ( 1)X . For the proof see Mukai [3] Theorem 1.1. It is analogous to the proof of the Inversion Theorem 14.2.2. We can use the Inversion Theorem to deduce the following result on local moduli spaces: Suppose E is a simple WIT-sheaf of index i on an abelian variety X. Ac- cording to Theorem E.4 the local moduli space ME for the functor of infinitesimal deformations DE exists. By the Inversion Theorem 14.7.2 the Fourier transform E is simple as well and thus the corresponding local moduli space ME exists. Theorem F.5. Let E be a simple WIT-sheaf on an abelian variety X. There is a canonical isomorphism ME ME of local moduli spaces. Proof. It suffices to show that the Fourier transform gives an isomorphism of func- D → D A tors E E. Let the notation be as in Appendix E. In particular let denote the category of artinian C-algebras with residue field C. Suppose A ∈ A and (E,κ)∈ DE(A). According to Proposition F.3 E is a WIT-sheaf of index i, its Fourier ∼ transform E is an A-flat WIT-sheaf and κ induces an isomorphismκ : E ⊗A C → E. E ∈ D Hence ( , κ) E(A). Now the Inversion Theorem F.4 implies the assertion. 600 F. Abelian Schemes

By abuse of notation we denote the image of P = PX /S ∈ Pic(X ×S X ) = 1 1 Ch (X ×S X ) in Ch (X ×S X )Q by the same letter. The relative Fourier transform of the Chow groups is the homomorphism of groups = : X → X := P · ∗ F FX /S Ch( )Q Ch( )Q, F (α) p2∗(e p1α). The relative Fourier transform satisﬁes the same properties as the Fourier transform in the absolute case (see Section 16.3), some of which we compile next. For the proofs, which are generalizations of the proofs given in Section 16.3, we refer to Deninger-Murre [1].

Inversion Theorem F.6. ◦ = − g − : X → X FX/S FX /S ( 1) ( 1)X /S∗ Ch( )Q Ch( )Q. This is the relative version of the Inversion Theorem 16.3.2. The next proposition is the relative version of Corollary 16.3.3. Proposition F.7. The relative Fourier transform Ch(X )Q → Ch(X )Q is bijective with g FX /S(X ) = (−1) e∗(S) and FX /S(e∗(S)) = X .

An isogeny of abelian schemes f : Y → X over S is a surjective homomorphism of group schemes whose kernel is a ﬁnite group scheme over S. The following proposition is the relative version of Proposition 16.3.4

Proposition F.8. Let f : Y → X be an isogeny of abelian schemes over S, then ∗ a) FX ◦ f∗ = f ◦ FY , ∗ b) FY ◦ f = f∗ ◦ FX .

For any integer n denote by nX /S : X → X the multiplication by n of the abelian scheme X → S. Analogously as in the case of an abelian variety (see Section 16.5) deﬁne eigenspaces p X s := { ∈ p X | ∗ = 2p−s ∈ Z} Ch ( )Q α Ch ( )Q nX /Sα n α for all n . The same proof as for Theorem 16.5.1 yields:

Theorem F.9. Let X → S be an abelian scheme of relative dimension g over an irreducible variety S of dimension d. Then p X = min(2p,p+d) p X s Ch ( )Q s=max(p−g,2p−2g) Ch ( )Q. F.3 The Relative Jacobian 601

F.3 The Relative Jacobian

The notion of Picard functor and Picard scheme generalizes to other schemes over S. We only need the special case of a curve over S.Acurve of genus g over S is a −1 smooth projective morphism p : C → S whose geometric ﬁbres Cs := p (s) are smooth irreducible curves of genus g. For any noetherian scheme f : T → S let as : C := C × → = −1 ∈ above pT T S T T and Ct pT (t) for any t T . Deﬁne {isomorphism classes of line bundles on C } Pic (T ) := T . C/S ∗ { } pT isomorphism classes of line bundles on T

If C/S admits a section σ : S → C, one checks immediately that PicC/S(T ) is canonically isomorphic to the group of isomorphism classes of line bundles on CT σ× idT rigidiﬁed along the section σT : T S×S T → C×S T . PicC/S is a contravariant functor from the category of noetherian S-schemes into the category of abelian groups. Moreover consider the subgroup classes of line bundles [L] in PicC (T ), s.t. P 0 := /S icC/S(T ) for any geometric point t of T the Hilbert polynomial of L|Ct is the constant 1 − g P 0 P icC/S is an open subfunctor of icC/S.

Theorem F.10. 1. The functor PicC/S is represented by a locally Noetherian commutative group scheme PicC/S over S. 2. The connected component JC/S of PicC/S containing the zero section represents P 0 the subfunctor icC/S. For the proof see Bosch-Lutkebohmert-Raynaud¨ [1] Theorem 9.3.1.

JC/S → S is an abelian scheme called the relative Jacobian of the curve C/S which explains the notation. For a closed point s ∈ S the ﬁbre Js of J → S is the Jacobian of the smooth projective curve Cs. So roughly speaking we may consider J → S as a family of Jacobians over S. Bibliography

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: → tx0 Translation map tx0 X X ...... 10 T0X Tangent space of X at0 ...... 8 v¯ Image π(v) of v ∈ V in X ...... 8 π : V → X Universal covering map of a complex torus ...... 8 Period matrix ...... 9 ρa Analytic representation ...... 10 ρr Rational representation ...... 10 (ker f)0 Component of ker f containing 0 ...... 11 im f Image of the homomorphism f ...... 11 ker f Kernel of the homomorphism f ...... 11 Xn Subgroup of n-division points in X ...... 12 deg f Degree of the homomorphism f ...... 12 nX Multiplication by n on X ...... 12 e(f ) Exponent of the isogeny f ...... 13 IFn(X) Vector space of invariant n-forms on X ...... 15 = HomC(V, C) ...... 16 p X Sheaf of holomorphic p-forms on X ...... 16 = HomC(V, C) ...... 16 ¯ = ¯ ∧ ∧ ¯ dvI dvi1 ... dvip ...... 16 = ∧ ∧ dvI dvi1 ... dvip ...... 16 IFp,q(X) Vector space of invariant (p, q)-forms on X ...... 17 Ap,q C∞ X Sheaf of -(p, q)-forms on X ...... 17 Hp,q Vector space of harmonic forms of type (p, q) ...... 19 C1 Circle group ...... 29 NS(X) Neron´ Severi group of X ...... 29 0 Pic (X) Group of line bundles L with c1(L) = 0 ...... 31 aL Canonical factor for the line bundle L ...... 32 Dual lattice ...... 34 X Dual complex torus ...... 34 → → ∗ ⊗ −1 φL The homomorphism X X, x tx L L ...... 36 K(H) Kernel of φL(H,χ) ...... 37 K(L) Kernel of φL ...... 37 (L) ={v ∈ V | Im H(v,) ⊆ Z} ...... 37 P Poincare´ bundle ...... 38 626 Glossary of Notation

H Pic (X) Line bundles L in Pic(X)with c1(L) = H ...... 40 D = diag(d1,...,dg) ...... 46 L0 = L(H, χ0), the line bundle of characteristic zero ...... 47 χ0(v) Semicharacter e πiE(v1,v2) ...... 47 Pf(E) Pfaffian of E ...... 50 ϑc Theta function with characteristic c ...... 52 c = · −1 c ·+ ϑw¯ aL(w, ) ϑ ( w) ...... 53 K(L)0 Connected component of K(L) containing zero ...... 55 Pfr(E) Reduced Pfaffian ...... 55 A0,q C∞ X (L) Sheaf of -(0,q)-forms with values in L ...... 56 i(L) Index of the line bundle L ...... 61 χ(L) Euler-Poincare´ characteristic of the line bundle L ...... 64 (Lg) Self-intersection number of L ...... 65 ϕL Rational map X → Pn associated to L ...... 71 ∼ Linear equivalence ...... 71 (L1 ·····Lg) Intersection number of the line bundles L1,...,Lg ...... 75 Ds Smooth part of the divisor D ...... 81 a(X) Algebraic dimension ...... 86 H H Pics (X) Symmetric line bundles in Pic (X) ...... 89 (−1)L Normalized isomorphism of L over (−1)X ...... 90 H 0(L)+ (+1)-Eigenspace in H 0(L) ...... 90 H 0(L)− (−1)-Eigenspace in H 0(L) ...... 90 multx(D) Multiplicity of the divisor D at x ...... 93 H e Alternating form on X2 associated with H ...... 94 qL(v)¯ Quadratic form on X2 associated to L ...... 94 − = ∈ | ≡ X2 (D) x X2 multx(D) 1 (mod 2) ...... 96 + = ∈ | ≡ X2 (D) x X2 multx(D) 0 (mod 2) ...... 96 KX Kummer variety ...... 97 Pontryagin product Hp(X, Z) × Hq (X, Z) → Hp+q (X, Z) . . 102 I o Ordered multi-index complementary to I in {1,...,2g} .....104 {V} Homology class of the cycle V ...... 104 ∗ Star-operator ...... 106 −1 : → ψL The homomorphism e(L)φL X X ...... 114 e(L) Exponent of the finite group K(L) ...... 114 a Pf Characteristic polynomial of the analytic representation .....115 r Pf Characteristic polynomial of the rational representation .....115 Na(f ) Analytic norm of f ...... 116 Nr (f ) Rational norm of f ...... 116 Tra(f ) Analytic trace of f ...... 116 Trr (f ) Rational trace of f ...... 116 Lδ Dual polarization ...... 122 e(Y) Exponent of the abelian subvariety Y ...... 122 NY Norm-endomorphism of the abelian subvariety Y ...... 123 Xε Abelian subvariety associated to ε ...... 123 Glossary of Notation 627

εY Symmetric idempotent of the abelian subvariety Y ...... 123 δ(V, W) Endomorphism associated to the cycles V and W ...... 129 G(L) Theta group of the line bundle L ...... 147 G(L) Theta group of the line bundle L ...... 148 [α, w] Element of the theta group G(L) ...... 148 eL Commutator map K(L) × K(L) → C∗ ...... 151 K(D) Zg/DZg ⊕ Zg/DZg ...... 160 H(D) Heisenberg group of type D ...... 160 eD Commutator map K(D)2 → C∗ ...... 160 Sp(D) Symplectic group of type D ...... 162 Ge(L) Extended theta group ...... 170 Ge(L) Extended theta group ...... 170 He(D) Extended Heisenberg group ...... 171 HZ Hermitian form on XZ ...... 211 g XZ The abelian variety C /Z ...... 211 Hg Siegel upper half space ...... 211 2g Z The lattice (Z, D)Z ...... 211 t GD = M ∈ Sp2g(Q) | MD ⊆ D ...... 212 M(Z) = (αZ + β)(γ Z + δ)−1 ...... 212 Sp2g(R) Symplectic group ...... 212 A D Moduli space for polarized abelian varieties of type D ...... 215 D R = ∈ R | 0 D t = 0 D Sp2g( ) R M2g( ) R −D 0 R −D 0 ...... 216 AD(D) Moduli space for abelian varieties of type D with level D-structure ...... 218 D(n) = R ∈ D | R ≡ 12g (mod n) ...... 219 AD(n) Moduli space for polarized abelian varieties of type D with level n-structure ...... 219 t g (S)0 = (s11,...,sgg) ∈ R ...... 220 A D Moduli space for polarized abelian varieties of type D * + with a decomposition ...... 220 c1 ϑ c2 (v, Z) Classical theta function with characteristic ...... 223 2g g jZ The isomorphism R → C ,x→ (Z, 1g)x ...... 229 X = (Cg × H )/ ...... 230 D % g D & ˜ 1g+DaD˜ bD ˜ ˜ GD(D) = ˜ ∈ GD |˜a,b, c,˜ d ∈ Mg(Z) ...... 233 c˜ 1g+dD AD(D) Moduli space for polarized abelian varieties of type D with level D-structure ...... 233 AD(D)0 Moduli space for polarized abelian varieties of type D with orthogonal level D-structure ...... 234 A ⊗ B Kronecker product of matrices ...... 244 H = ∈ C | t =− − t m Z Mm( ) Z Z,1m ZZ > 0 ...... 258 H = ∈ × C | − t r,s %Z M(r s, ) 1s ZZ > 0 ...... 263& 1 0 1 0 U = M ∈ M + (C) | t M r M = r ...... 268 r,s r s 0 −1s 0 −1s 628 Glossary of Notation

% & = = αβ ∈ C | t 1 0 = 1 0 Um,m M −β¯ α¯ M2m( ) M 0 −1 M 0 −1 ....268 pa(C) Arithmetic genus of the curve C ...... 282 r gd linear system of dimension r and degree d on a curve C .....316 J(C) Jacobian variety of the curve C ...... 317 Divn(C) Divisors of degree n on C ...... 319 C(n) n−th symmetric product of C ...... 320 Picn(C) Line bundles of degree n on C ...... 320 TC Tangent bundle of C ...... 320 (n) → αDn Abel Jacobi map C J(C) ...... 320 [Y ] Fundamental cohomology class of the subvariety Y ...... 322 sing Singularity locus of ...... 325 Pn C Poincare´ bundle of degree n on C ...... 328 Nf Divisor norm map ...... 331 γL Homomorphism deﬁned by the correspondence L ...... 333 Corr(C1,C2) Group of equivalence classes of correspondences ...... 334 Aut C Reduced automorphism group of a hyperelliptic curve ...... 340 Alb(M) Albanese Variety of M ...... 353 q(M) Irregularity of the Kahler¨ manifold M ...... 353 Pic0(M) Picard variety of M ...... 355 φ The homomorphism φ ∗ ...... 365 Y ιY −1 ψY The homomorphism e(Y)φY ...... 365 P(V) Projective space of lines in the vector space V ...... 401 L(f ) Holomorphic Lefschetz number of the endomorphism f ....412 F Fourier transform of F ...... 445 F G Pontryagin product of the sheaves F and G ...... 450 E Pn n The Picard sheaf q∗ C ...... 460 F 1 Pn n The Picard sheaf R q∗ C ...... 460 P(G) = P roj (S(G)) ...... 462 S Functor from the category of OX-modules into O the category of X-modules ...... 464 b D (X) Derived category of bounded complexes ...... 464 χ(E, F) = g (−1)ν dim Ext ν (E, F) ...... 470 ν=0 OX Np Property Np ...... 484 multxC Multiplicity of the curve C in the point x ...... 488 ε(L) Seshadri constant of the line bundle L ...... 488 m(C) Maximal multiplicity of points on the curve C ...... 488 m(X, H ) Minimal length of a period ...... 496 $k() ={x ∈ X|multx ≥ k} ...... 515 Zk(X) Group of algebraic cycles of dimension k ...... 522 V · W Intersection product ...... 522 Ch•(X) = g Chp(X) p=0 ...... 523 = g Ch•(X) k=0 Chk(X) ...... 523 V ∼rat W Rational equivalence of cycles V and W ...... 523 V ∼alg W Algebraic equivalence of cycles V and W ...... 524 Glossary of Notation 629

f Graph of the morphism f ...... 526 F = FX Fourier transform on Ch(X)Q ...... 528 • FH Fourier transform on H (X, Q) ...... 532 Sp(V, E) Symplectic group associated to E ...... 552 C0(Sp(V, E)) Set of symplectic complex structures in Sp(V, E) ...... 552 Hg(X) Hodge group of X ...... 555 2p HHodge(X) Vector space of Hodge Classes ...... 556 JD Adjoint ideal of the divisor D ...... 581 CohX Category of coherent sheaves on X ...... 584 D(A) Derived category ...... 584 D(X) Derived category ...... 588 PicX /S Picard functor ...... 597 P 0 icX /S(T ) Picard scheme ...... 598 Index

166-Configuration 311 Algebraic classes 559 CM-type 417 Algebraic dimension 312 G-acyclic complex 588 Analytic representation 10 n-Division points 12 Anti-involution – of the first kind 133 Abel-Jacobi – of the second kind 133 – map 319, 320 – positive 132 – Theorem 319 Appell-Humbert Theorem 32 Abel-Prym map 369 Automorphism group Abelian scheme 597 – finite 421 – dual 598 – maximal in the isogeny class 421 Abelian variety 70 – reduced 340 – G-decomposable 428 Azygetic set of 2-division points 112 – G-simple 428 – dual 70, 122 Bertini’s theorem 79 – of CM-type 558 Bidegree of a correspondence 334 – of Weil-type 565 Big divisor 578 – polarized 70 Bogomolov’s Inequality 296 – simple 126 Action Canonical – proper and discontinuous 215, 568 – basis 46 Addition Formula 203 – polarization of J(C) 317 Addition map 8 – theta divisor 324 Adjoint ideal 581 – theta function 49 Adjunction formula 282 Canonical factor 32 Albanese canonical polarization of Pic0(M) 356 – map 354 Characteristic 47 – maximal ∼ dimension 513, 579 – compatible 159 – torus 353 Characteristic polynomial 115 – variety 332, 358 – reduced 131 Algebra Chern – opposite 134 – character 68, 590 – quaternion 133 – class 296, 590 – totally definite quaternion 134 – first ∼ class 24, 41 – totally indefinite quaternion 134 Chow Algebraic – ring 523 – p-cycle 106, 128 – Theorem of 568 – equivalence of line bundles 88 Classical – complex analytic space 567 – factor of automorphy 49, 50 – cycle 106, 128, 522 – theta function 50 – dimension 86 Cocycle relation 571 – equivalence of cycles 524 Commutator map 151 632 Index

Complementary abelian subvariety 125 – totally positive 120 Complex multiplication 262, 266 Endomorphism structure of a Complex structure 550 polarized abelian variety 245 Complex torus 8 Equivalence Correspondence 333, 525 – algebraic 524 – bidegree of a ˜ 334 – analytic 39 – equivalent ∼ 333 – homological 106, 525 Criterion of Matsusaka-Ran 341 – linear 71 Cubic Equations 203 – numerical 106, 337 Cubic Theta Relations 198, 202 – rational 523 Curve Euler-Poincare´ characteristic 64, 469 – d-gonal 385 Exponent – over S 601 – of an abelian subvariety 122 Cycle – of a Prym-Tyurin variety 368 – map 525 – of an isogeny 13 – algebraic 522 Exponential map 8 – generating an abelian variety 341 – Weil-Hodge cycle 565 Factor of automorphy 24, 571 Fay’s Trisecant Identity 348 Decomposition First Chern class 24, 41 – compatible 157, 159 Formal moduli space 593 – for E 46, 49 Fourier transform 445, 528, 532, 599 – for H 46, 49 – relative 600 – for L 46, 49 Functor – pro-representable 592 – of the lattice 219 Fundamental – Theorem 75 – class 322 Degree – system 112 – of a homomorphism 12 – of a polarization 119 Gopel¨ tetrahedron 289 ϑ Derivative of 84 Gauss map 81 Derived category 584 – generalized ∼ 110 Desargues conﬁguration 205 General type 579 Divisor Genus – big 578 – arithmetic 282 –even 93 Graph of a morphism 526 – nef 489, 577 Green’s operator 19 – odd 93 – symmetric 92 Hamiltonian quaternions 132 Dolbeault cohomology groups 56 Harmonic form 19 Donagi’s Tetragonal Construction 388 – with values in a line bundle 56 Dual Heat equation 226 – abelian scheme 598 Heisenberg group 161 – complex torus 34 – extended 171 – Jacobian Variety 359 – normalizer 177 – lattice 34 Hermitian – polarization 453 – metric 18, 56 Hermitian form 29 Elliptic Hilbert – curve 9 – polynomial 592 – involution 294 Hilbert modular varieties 251 Endomorphism Hodge – primitive ∼ 124 – class 556 – symmetric ∼ 119 – conjecture 559 Index 633

– decomposition 15 Level structure 161, 217 – group 555 – D-structure 217 – structure 22, 550 – n-structure 218 Homological equivalence 525 – generalized level n-structure 218 Homomorphism 10 – orthogonal level D-structure 234 – of polarized abelian varieties 70 Line bundle – algebraic equivalent ∼ 88 Index of a line bundle 61 – ample 85 Intersection number 75 – index of 61 – of cycles 104 – nef 489 Intersection product 522 – nondegenerate 37 Invariant form 15 – normally generated 187 – of type (p, q) 17 – positive definite 49 Inverse Formula 90 – positive semidefinite 54 – for Finite Theta Functions 177 – symmetric 34 Inversion Theorem 445, 465 – totally symmetric 176 Irregularity 353 – very ample 85 Isogenous complex tori 13 Log canonical 513 Isogeny 12 – of type D 240 Maximal quotient abelian variety 110 Isogeny Theorem 157 Minimal length of a period 496 – for finite theta functions 168 Moduli space – hypotheses of the ˜ 159 – coarse 591 Isomorphism – fine 591 – of polarized abelian varieties with – formal 593 endomorphism structure 245 − – local 593 Isomorphism of L over ( 1)X 90 Moduli space of polarized abelian varieties IT-sheaf 444 with endomorphism structure 249, 256, Iwasawa Decomposition 241 261, 266 Moving Lemma 128 Jacobian Multi-index 16 – relative 601 Multiplication by n 168 – variety 317 Multiplication Formula 182 Jets 490 Multiplication map 180 Multiplicity of a curve at a point 488 Kahler¨ – manifold 18 Mumford’s Index Theorem 67 – metric 18 Kunneth¨ decomposition 539 Neron-Severi´ group 29 Kanev’s Criterion 394 – of a complex torus 42 Klein quartic 361 Nakai-Moishezon Criterion 77 Kodaira dimension 508, 579 Nef divior 577 Koszul complex 486 Nondegenerate line bundle 37 Kronecker product of matrices 244 Norm Kummer surface 204, 285 – analytic 116 – self-duality 288 – rational 116 Kummer variety 97, 311 norm map 331 – trisecants 344 Norm-endomorphism 123 Norm-endomorphism Criterion 124 Lefschetz Normalized isomorphism 90 – Decomposition 107 – number 412 Period matrix 9 – theorem 86 – for J(C) 317 Level D-structure 163 Period of X 313 634 Index

Pfaffian 50 Radical of H 55 – reduced 55 Rational equivalence 523 Picard Rational map 101 – bundle 461 Rational representation 10 – number 42 Rational singularities 581 – sheaves 460 Real multiplication 246 – torus 355 Real Riemann matrix 109, 110 – functor 597 Real torus 21 – number 312 Regularity – scheme 598 – m--regular sheaf 519 – variety 356 Reider’s theorem 293 Poincare´ Relative Duality 590 – Reducibility Theorem 67, 125 Representation – Formula 322 – of the theta group 153 Poincare´ bundle 38 – Schrodinger¨ 164 – cohomology of the 67 Riemann – of degree n for C 328 – theta function 223 – of degree n for C 459 – constant 324 – relative 598 – Equations 198 – universal property of the 38 – Relations 73 Polarization 70 – Singularity Theorem 325 – degree of 119 – Theorem 324 – isomorphic 121 – Theta Relation 197, 206 – canonical ∼ of J(C) 317 Riemann-Roch Theorem – dual 122, 453 – geometric version for curves 326 – exponent of 114 – analytic 64 – induced 70 – for vector bundles 68 – principal 70 – geometric 65 – type 70 Rigidity Lemma 100 Polarized abelian variety Rosati involution 114 – with symplectic basis 211 Rosenhain tetrahedron 288 Pontryagin product 21, 102, 530 – relative 542 Schottky-Jung Relations 406 Pontryagin product Schrodinger¨ representation 164 – of sheaves 450 Seesaw Principle 569 Prime form 348 Self-intersection number 65 Primitive form 107 Semicharacter 29 Proper intersection 522 Semistable Property (Np) 484 – μ-semistable 453 Prym canonical map 380 Seshadri constant 488 Prym differential 379 Shimura variety (analytic) 273 Prym map 409 Siegel upper half space 211 Prym variety Signature 264 – associated to a covering 373 Singular hyperplane 311 – generalized 368 Singular plane 286 Prym-Tyurin variety 368 Singularity – exponent 368 – exceptional 384 – for a curve 368 – stable 384 Stable Quadratic form associated to eH 94 – μ-stable 471 Quaternion multiplication – coherent sheaf 592 – totally definite 257 Star-operator ∗ 106 – totally indefinite 254 Stein factorization 12 Index 635

Symmetric idempotent 123 Theta regularity 519 Symmetric line bundle 34 Theta Relation 196 Symplectic basis 46 Theta Relations Symplectic group 212 – cubic 198, 202 Syzygy-module 205 – Riemann 197, 206 Theta structure 161 Tetragonal construction 388 – compatible 166 Theorem – extended 171 – of de Franchis 407 – symmetric 171 – of Abel-Jacobi 319 Theta Transformation Formula 222, 227 – of Narasimhan and Nori 142 Torelli Theorem for the Prym map 409 – Stone-von Neumann 174 Torelli Problem 313 – Bertini 283 Trace – of Lefschetz 84 – analytic 116 – of Recillas 385 – rational 116 – of Reider 293 – reduced 131 – of Riemann 324 Translation map 10 – of Serre-Rosenlicht 174 Type – of the Cube 42 – of a line bundle 46 – of the Square 33 – of a polarization 70 – of Torelli 321, 360 – Riemann’s Singularity ∼ 325 Universal Property – Seesaw ∼ 568 – of the Poincare´ bundle 38 Theta characteristic 324 – of the Abel-Prym map 369, 378 – even 325 – of the Albanese torus 354 – odd 325 – of the Jacobian 330 Theta Characteristics 325 Theta divisor 317 – canonical 324 Vanishing Theorem 60 – singularity locus 325 Vector bundle Theta function 49 – μ-semi-stable 296 – canonical 49 – μ-semistable 453 – classical 50 – μ-stable 296, 471 – differentiable 57 – unipotent 447 – ﬁnite 165 – homogeneous 468 – Riemann ˜ 223 Theta group 149 Welters’ Criterion 370 – extended 170 WIT-sheaf 445 Theta group of L 147 Theta null value 232 Zarhin’s trick 141 Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

A Selection 223. Bergh/Löfstr¨¨ om:¨ Interpolation Spaces. An Introduction 224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order 225. Schütte:¨ Proof Theory 226. Karoubi: K-Theory. An Introduction 227. Grauert/Remmert: Theorie der Steinschen Räume 228. Segal/Kunze: Integralsand Operators 229. Hasse: Number Theory 230. Klingenberg: Lectures on Closed Geodesics 231. Lang: Elliptic Curves. Diophantine Analysis 232. Gihman/Skorohod:TheTheory of Stochastic Processes III 233. Stroock/Varadhan: Multidimensional Diffusion Processes 234. Aigner: Combinatorial Theory 235.Dynkin/Yushkevich: Controlled Markov Processes 236. Grauert/Remmert: Theory of Stein Spaces 237. Kothe:¨ Topological Vector Spaces II 238. Graham/McGehee: Essays in Commutative Harmonic Analysis 239. Elliott: Probabilistic Number Theory I 240. Elliott: Probabilistic Number Theory II n 241. Rudin: Function Theory in the Unit Ball of C 242. Huppert/Blackburn: Finite GroupsII 243. Huppert/Blackburn: Finite Groups III 244. Kubert/Lang:Modular Units 245. Cornfeld/Fomin/Sinai: Ergodic Theory 246. Naimark/Stern: Theory of Group Representations 247. Suzuki: Group Theory I 248. Suzuki: Group Theory II 249. Chung: Lectures from Markov Processes to Brownian Motion 250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations 251. Chow/Hale: Methods of Bifurcation Theory 252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampè`re Equations 253.Dwork: Lectures on ρ-adic Differential Equations 254. Freitag: Siegelsche Modulfunktionen 255. Lang: Complex Multiplication 256. Hörmander: The Analysis of Linear Partial Differential Operators I 257. Hörmander: The Analysis of Linear Partial Differential Operators II 258. Smoller: Shock Waves and Reaction-Diffusion Equations 259. Duren: Univalent Functions 260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems 261. Bosch/Güntzer/Remmert:¨ Non Archimedian Analysis – A System Approach to Rigid Analytic Geometry 262. Doob: Classical Potential Theory and Its Probabilistic Counterpart 263. Krasnosel’skiˇˇı/Zabreˇˇıko: Geometrical Methods of Nonlinear Analysis 264. Aubin/Cellina: Differential Inclusions 265. Grauert/Remmert: Coherent Analytic Sheaves 266. de Rham: Differentiable Manifolds 267. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I 268. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol.II 269. Schapira: Microdifferential Systems in the Complex Domain 270. Scharlau: Quadratic and Hermitian Forms 271. Ellis: Entropy, LargeDeviations, and Statistical Mechanics 272. Elliott: Arithmetic Functions and Integer Products 273. Nikol’skiˇˇı: Treatise on the Shift Operator 274. Hörmander: The Analysis of Linear Partial Differential Operators III 275. Hörmander: The Analysis of Linear Partial Differential Operators IV 276. Liggett: Interacting Particle Systems 277. Fulton/Lang: Riemann-Roch Algebra 278. Barr/Wells: Toposes, Triples and Theories 279. Bishop/Bridges: Constructive Analysis 280. Neukirch: Class Field Theory 281. Chandrasekharan: Elliptic Functions 282. Lelong/Gruman: Entire Functions of Several Complex Variables 283. Kodaira: Complex Manifolds and Deformation of Complex Structures 284. Finn: Equilibrium Capillary Surfaces 285. Burago/Zalgaller: Geometric Inequalities 286. Andrianaov: Quadratic Forms and Hecke Operators 287. Maskit: Kleinian Groups 288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes 289. Manin: Gauge Field Theory and Complex Geometry 290. Conway/Sloane: Sphere Packings, Lattices and Groups 291. Hahn/O’Meara: The Classical Groups and K-Theory 292. Kashiwara/Schapira: Sheaves on Manifolds 293. Revuz/Yor: Continuous Martingales and Brownian Motion 294. Knus: Quadratic and Hermitian Forms over Rings 295. Dierkes/Hildebrandt/Kü¨ster/Wohlrab:MinimalSurfaces I 296. Dierkes/Hildebrandt/Kü¨ster/Wohlrab:MinimalSurfaces II 297. Pastur/Figotin: Spectra of Random and Almost-Periodic Operators 298. Berline/Getzler/Vergne: Heat Kernelsand Dirac Operators 299. Pommerenke: Boundary Behaviour of Conformal Maps 300. Orlik/Terao: Arrangements of Hyperplanes 301. Loday: Cyclic Homology 302. Lange/Birkenhake: Complex Abelian Varieties 303. DeVore/Lorentz: Constructive Approximation 304. Lorentz/v. Golitschek/Makovoz: Construcitve Approximation. Advanced Problems 305. Hiriart-Urruty/Lemaréchal:´ Convex Analysis and Minimization Algorithms I. Fundamentals 306. Hiriart-Urruty/Lemaréchal:´ Convex Analysis and Minimization Algorithms II. Advanced Theory and Bundle Methods 307. Schwarz: Quantum Field Theory and Topology 308. Schwarz: Topology for Physicists 309. Adem/Milgram: Cohomology of Finite Groups 310. Giaquinta/Hildebrandt: Calculus of Variations I: The Lagrangian Formalism 311. Giaquinta/Hildebrandt: Calculus of Variations II: The Hamiltonian Formalism 312. Chung/Zhao: From Brownian Motion to Schrödinger’s¨ Equation 313. Malliavin: Stochastic Analysis 314. Adams/Hedberg: Function Spaces and Potential Theory 315. Bürgisser/Clausen/Shokrollahi:¨ Algebraic Complexity Theory 316. Saff/Totik: Logarithmic Potentials with External Fields 317. Rockafellar/Wets: Variational Analysis 318. Kobayashi: Hyperbolic Complex Spaces 319. Bridson/Haefliger: Metric Spaces of Non-Positive Curvature 320. Kipnis/Landim: Scaling Limits of Interacting ParticleSystems 321. Grimmett: Percolation 322. Neukirch: Algebraic Number Theory 323. Neukirch/Schmidt/Wingberg:Cohomology of Number Fields 324. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes 325. Dafermos: Hyperbolic Conservation Laws in Continuum Physics 326. Waldschmidt: Diophantine Approximation on Linear Algebraic Groups 327. Martinet: Perfect Lattices in Euclidean Spaces 328. van der Put/Singer: Galois Theory of Linear Differential Equations