Complex Abelian Varieties

Preface to the Second Edition
Contents
Introduction
Notation
1. Complex Tori
	1.1 Complex Tori
	1.2 Homomorphisms
	1.3 Cohomology of Complex Tori
	1.4 The Hodge Decomposition
	1.5 Exercises and Further Results
2. Line Bundles on Complex Tori
	2.1 Line Bundles on Complex Tori
	2.2 The Appell-Humbert Theorem
	2.3 Canonical Factors
	2.4 The Dual Complex Torus
	2.5 The Poincar´e Bundle
	2.6 Exercises and Further Results
3. Cohomology of Line Bundles
	3.1 Characteristics
	3.2 Theta Functions
	3.3 The Positive Semidefinite Case
	3.4 The Vanishing Theorem
	3.5 Cohomology of Line Bundles
	3.6 The Riemann-Roch Theorem
	3.7 Exercises and Further Results
4. Abelian Varieties
	4.1 Polarized Abelian Varieties
	4.2 The Riemann Relations
	4.3 The Decomposition Theorem
	4.4 The Gauss Map
	4.5 Projective Embeddings
	4.6 Symmetric Line Bundles
	4.7 Symmetric Divisors
	4.8 Kummer Varieties
	4.9 Morphisms into Abelian Varieties
	4.10 The Pontryagin Product
	4.11 Homological Versus Numerical Equivalence
	4.12 Exercises and Further Results
5. Endomorphisms of Abelian Varieties
	5.1 The Rosati Involution
	5.2 Polarizations
	5.3 Norm-Endomorphisms and Symmetric Idempotents
	5.4 Endomorphisms Associated to Cycles
	5.5 The Endomorphism Algebra of a Simple Abelian Variety
	5.6 Exercises and Further Results
6. Theta and Heisenberg Groups
	6.1 Theta Groups
	6.2 Theta Groups under Homomorphisms
	6.3 The Commutator Map
	6.4 The Canonical Representation of the Theta Group
	6.5 The Isogeny Theorem
	6.6 Heisenberg Groups and Theta Structures
	6.7 The Schrödinger Representation
	6.8 The Isogeny Theorem for Finite Theta Functions
	6.9 Symmetric Theta Structures
	6.10 Exercises and Further Results
7. Equations for Abelian Varieties
	7.1 The Multiplication Formula
	7.2 Surjectivity of the Multiplication Map
	7.3 Projective Normality
	7.4 The Ideal of an Abelian Variety in P_N
	7.5 Riemann’s Theta Relations
	7.6 Cubic Theta Relations
	7.7 Exercises and Further Results
8. Moduli
	8.1 The Siegel Upper Half Space
	8.2 The Analytic Moduli Space
	8.3 Level Structures
	8.3.1 Level D-Structure
	8.3.2 Generalized Level n-Structure
	8.3.3 Decomposition of the Lattice
	8.4 The Theta Transformation Formula, Preliminary Version
	8.5 Classical Theta Functions
	8.6 The Theta Transformation Formula, Final Version
	8.7 The Universal Family
	8.8 The Action of the Symplectic Group
	8.9 Orthogonal Level D-Structures
	8.10 The Embedding of A_D(D)0 into Projective Space
	8.11 Exercises and Further Results
9. Moduli Spaces of Abelian Varieties with Endomorphism Structure
	9.1 Abelian Varieties with Endomorphism Structure
	9.2 Abelian Varieties with Real Multiplication
	9.3 Some Notation
	9.4 Families of Abelian Varieties with Totally Indefinite Quaternion Multiplication
	9.5 Families of Abelian Varieties with Totally Definite Quaternion Multiplication
	9.6 Families of Abelian Varieties with Complex Multiplication
	9.7 Group Actions on H_{r, s} and H_m
	9.8 Shimura Varieties
	9.9 The Endomorphism Algebra of a General Member
	9.10 Exercises and Further Results
10. Abelian Surfaces
	10.1 Preliminaries
	10.2 The 16_6-Configuration of the Kummer Surface
	10.3 An Equation for the Kummer Surface
	10.4 Reider’s Theorem
	10.5 Polarizations of Type (1, 4)
	10.6 Products of Elliptic Curves
	10.7 The Coble Hypersurface of a Principally Polarized Abelian Surface
	10.8 Exercises and Further Results
11. Jacobian Varieties
	11.1 Definition of the Jacobian Variety
	11.2 The Theta Divisor
		11.2.1 Theta Characteristics
		11.2.2 The Singularity Locus of θ
	11.3 The Poincaré Bundles for a Curve C
	11.4 The Universal Property
	11.5 Correspondences of Curves
	11.6 Endomorphisms Associated to Curves and Divisors
	11.7 Examples of Jacobians
	11.8 The Criterion of Matsusaka-Ran
	11.9 Trisecants of the Kummer Variety
	11.10 Fay’s Trisecant Identity
	11.11 Albanese and Picard Varieties
	11.12 Exercises and Further Results
12. Prym Varieties
	12.1 Abelian Subvarieties of a Principally Polarized Abelian Variety
	12.2 Prym-Tyurin Varieties
	12.3 Prym Varieties
	12.4 Topological Construction of Prym Varieties
	12.5 The Abel-Prym Map
	12.6 The Theta Divisor of a Prym Variety
	12.7 Recillas’ Theorem
	12.8 Donagi’s Tetragonal Construction
	12.9 Kanev’s Criterion
	12.10 The Schottky-Jung Relations
	12.11 Exercises and Further Results
13. Automorphisms
	13.1 Fixed–Point Formulas
	13.2 The Fixed–Point Set of a Finite Automorphism Group
	13.3 Abelian Varieties of CM-Type
	13.4 Abelian Surfaces with Finite Automorphism Group
	13.5 Poincaré’s Reducibility Theorem with Automorphisms
	13.6 The Group Algebra Decomposition of an Abelian Variety
	13.7 Exercises and Further Results
14. Vector bundles on Abelian Varieties
	14.1 Some Properties of the Poincaré Bundle
	14.2 The Fourier Transform for WIT–Sheaves
	14.3 Some Properties of the Fourier Transform
	14.4 The Dual Polarization
	14.5 Application: Global Generation of Vector Bundles
	14.6 Picard Sheaves
	14.7 The Fourier Transform of a Complex
	14.8 Vector Bundles on Abelian Surfaces
	14.9 Exercises and Further Results
15. Further Results on Line Bundles an the Theta Divisor
	15.1 Very Ample Line Bundles on General Abelian Varieties
	15.2 Syzygies of Line Bundles on Abelian Varieties
	15.3 Seshadri Constants
	15.4 Bounds for Seshadri Constants
	15.5 The Minimal Length of a Period
	15.6 Seshadri Constants of Line Bundles on Abelian Surfaces
	15.7 Subvarieties of Abelian Varieties
	15.8 Singularities of the Theta Divisor
	15.9 Exercises and Further Results
16. Cycles on Abelian varieties
	16.1 Chow Groups
	16.2 Correspondences
	16.3 The Fourier Transform on the Chow Ring
	16.4 The Fourier Transform on the Cohomology Ring
	16.5 A Decomposition of Ch(X)_Q
	16.6 The Künneth Decomposition
	16.7 The Bloch Filtration of Ch_0(X)
	16.8 Exercises and Further Results
17. The Hodge Conjecture for General Abelian and Jacobian Varieties
	17.1 Hodge Structures and Complex Structures
	17.2 Symplectic Complex Structures
	17.3 The Hodge Group of an Abelian Variety
	17.4 The Theorem of Mattuck
	17.5 The Hodge Conjecture for a General Jacobian
	17.6 Exercises and Further Results
A. Algebraic Varieties and Complex Analytic Spaces
B. Line Bundles and Factors of Automorphy
C. Some Algebraic Geometric Results
	C.1 Some Properties of Q-Divisors
	C.2 The Kodaira Dimension
	C.3 Vanishing Theorems
	C.7 Adjoint Ideals
D. Derived Categories
	D.1 Definition and First Properties
	D.2 Derived Functors
	D.3 The Grothendieck-Riemann-Roch Theorem
E. Moduli Spaces of Sheaves
F. Abelian Schemes
	F.1 Abelian Schemes and the Poincaré Bundle
	F.2 Relative Fourier Functor
	F.3 The Relative Jacobian
Bibliography
Glossary of Notation
Index