Complex abelian varieties and theta functions

Table of Contents......Page 5
1.1 The Definition of Complex Tori......Page 8
1.2 Hermitian Algebra......Page 9
1.3 The Invertible Sheaves on a Complex Torus......Page 10
1.4 The Structure of Pic(V/L)......Page 12
1.5 Translating Invertible Sheaves......Page 14
2.1 The Sections of Invertible Sheaves (Part I)......Page 16
2.2 The Sections of Invertible Sheaves (Part II)......Page 17
2.3 Abelian Varieties and Divisors......Page 20
2.4 Projective Embeddings of Abelian Varieties......Page 22
3.1 The Cohomology of a Real Torus......Page 25
3.2 A Complex Torus as a Kähler Manifold......Page 26
3.3 The Proof of the Appel-Humbert Theorem......Page 27
3.4 A Vanishing Theorem for the Cohomology of Invertible Sheaves......Page 29
3.5 The Final Determination of the Cohomology of an Invertible Sheaf......Page 31
3.6 Examples......Page 32
4.1 Geometric Background......Page 35
4.2 Representations of the Theta Group......Page 37
4.3 The Hermitian Structure on �䜀愀洀洀愀⠀堀Ⰰ 䰀......Page 39
4.4 The Isogeny Theorem up to a Constant......Page 41
5.1 Canonical Decompositions and Bases......Page 43
5.2 The Theta Function......Page 44
5.3 The Isogeny Theorem Absolutely......Page 45
5.4 The Classical Notation......Page 46
5.5 The Length of the Theta Functions......Page 48
6.1 The Addition Formula......Page 50
6.2 Multiplication......Page 52
6.3 Some Bilinear Relations......Page 54
6.4 General Relations......Page 56
7.1 Complex Structures on a Symplectic Space......Page 59
7.2 Siegel Upper-half Space......Page 62
7.3 Families of Abelian Varieties and Moduli Spaces......Page 66
7.4 Families of Ample Sheaves on a Variable Abelian Variety......Page 67
7.5 Group Actions on the Families of Sheaves......Page 70
8.1 The Definition......Page 73
8.2 The Relationship Between �瀀椀✀开⨀ 一开愀 愀渀搀 䠀 椀渀 琀栀攀 倀爀椀渀挀椀瀀愀氀氀礀 倀漀氀愀爀椀稀攀搀 䌀愀猀......Page 74
8.3 Generators of the Relevant Discrete Groups......Page 76
8.4 The Relationship Between �瀀椀✀开⨀ 一开愀 愀渀搀 䠀 椀猀 䜀攀渀攀爀愀......Page 80
8.5 Projective Embedding of Some Moduli Spaces......Page 81
9.1 Integration......Page 84
9.2 Complete Reducibility of Abelian Varieties......Page 85
9.3 The Characteristic Polynomial of an Endomorphism......Page 86
9.4 The Gauss Mapping......Page 87
10.1 When |D| Has No Fixed Components......Page 89
10.2 Projective Normality of |2D|......Page 90
10.3 The Factorization Theorem......Page 91
10.4 The General Case......Page 92
10.5 Projective Normality of |2D| on X/{�瀀洀 ㄀......Page 94
11.1 Hodge Structure......Page 96
11.2 The Moduli of Polarized Hodge Structure......Page 98
11.3 The Jacobian of a Riemann Surface......Page 99
11.4 Picard and Albanese Varieties for a Kähler Manifold......Page 100
Informal Discussions of Immediate Sources......Page 102
References......Page 103
Subject Index......Page 104