1 Categories,Products,Projective and Inductive Limits 1
1.1 The Notion of a Category and Examples 1
1.2 Functors 3
1.3 Products,Projective Limits and Direct Limits in a Category 4
1.3.1 The Projective Limit 4
1.3.2 The Yoneda Lemma 6
1.3.3 Examples 6
1.3.4 Representable Functors 8
1.3.5 Direct Limits 9
1.4 Exercises 10
2 Basic Concepts of Homological Algebra 11
2.1 The Category ModΓ of Γ-modules 11
2.2 More Functors 13
2.2.1 Invariants,Coinvariants and Exactness 13
2.2.2 The First Cohomology Group 15
2.2.3 Some Notation 16
2.2.4 Exercises 17
2.3 The Derived Functors 19
2.3.1 The Simple Principle 20
2.3.2 Functoriality 22
2.3.3 Other Resolutions 24
2.3.4 Injective Resolutions of Short Exact Sequences 24
A Fundamental Remark 26
The Cohomology and the Long Exact Sequence 27
The Homology of Groups 27
2.4 The Functors Ext and Tor 28
2.4.1 The Functor Ext 28
2.4.2 The Derived Functor for the Tensor Product 30
2.4.3 Exercise 32
3 Sheaves 35
3.1 Presheaves and Sheaves 35
3.1.1 What is a Presheaf？ 35
3.1.2 A Remark about Products and Presheaf 36
3.1.3 What is a Sheaf？ 36
3.1.4 Examples 38
3.2 Manifolds as Locally Ringed Spaces 39
3.2.1 What Are Manifolds？ 39
3.2.2 Examples and Exercise 41
3.3 Stalks and Sheafification 45
3.3.1 Stalks 45
3.3.2 The Process of Sheafification of a Presheaf 46
3.4 The Functors f＊ and f＊ 47
3.4.1 The Adjunction Formula 48
3.4.2 Extensions and Restrictions 49
3.5 Constructions of Sheaves 49
4 Cohomology of Sheaves 51
4.1 Examples 51
4.1.1 Sheaves on Riemann surfaces 51
4.1.2 Cohomology of the Circle 54
4.2 The Derived Functor 55
4.2.1 Injective Sheaves and Derived Functors 55
4.2.2 A Direct Definition of H1 56
4.3 Fiber Bundles and Non Abelian H1 59
4.3.1 Fibrations 59
Fibre Bundle 59
Vector Bundles 60
4.3.2 Non-Abelian H1 61
4.3.3 The Reduction of the Structure Group 62
Orientation 62
Local Systems 63
Isomorphism Classes of Local Systems 64
Principal G-bundels 64
4.4 Fundamental Properties of the Cohomology of Sheaves 65
4.4.1 Introduction 65
4.4.2 The Derived Functor to f＊ 66
4.4.3 Functorial Properties of the Cohomology 68
4.4.4 Paracompact Spaces 69
4.4.5 Applications 75
Cohomology of Spheres 75
Orientations 76
Compact Oriented Surfaces 77
4.5 ？ech Cohomology of Sheaves 77
4.5.1 The ？ech-Complex 77
4.5.2 The ？ech Resolution of a Sheaf 81
4.6 Spectral Sequences 83
4.6.1 Introduction 83
4.6.2 The Vertical Filtration 88
4.6.3 The Horizontal Filtration 94
Two Special Cases 95
Applications of Spectral Sequences 96
4.6.4 The Derived Category 98
The Composition Rule 101
Exact Triangles 102
4.6.5 The Spectral Sequence of a Fibration 103
Sphere Bundles an Euler Characteristic 104
4.6.6 ？ech Complexes and the Spectral Sequence 105
A Criterion for Degeneration 107
An Application to Product Spaces 109
4.6.7 The Cup Product 111
4.6.8 Example:Cup Product for the Comology of Tori 115
A Connection to the Cohomology of Groups 116
4.6.9 An Excursion into Homotopy Theory 117
4.7 Cohomology with Compact Supports 120
4.7.1 The Definition 120
4.7.2 An Example for Cohomology with Compact Supports 121
The Cohomology with Compact Supports for Open Balls 121
Formulae for Cup Products 123
4.7.3 The Fundamental Class 125
4.8 Cohomology of Manifolds 126
4.8.1 Local Systems 126
4.8.2 ？ech Resolutions of Local Systems 127
4.8.3 ？ech Coresolution of Local Systems 129
4.8.4 Poincaré Duality 132
4.8.5 The Cohomology in Top Degree and the Homology 138
4.8.6 Some Remarks on Singular Homology 140
4.8.7 Cohomology with Compact Support and Embeddings 141
4.8.8 The Fundamental Class of a Submanifold 143
4.8.9 Cup Product and Intersections 144
4.8.10 Compact oriented Surfaces 146
4.8.11 The Cohomology Ring of IPn（？） 147
4.9 The Lefschetz Fixed Point Formula 147
4.9.1 The Euler Characteristic of Manifolds 149
4.10 The de Rham and the Dolbeault Isomorphism 150
4.10.1 The Cohomology of Flat Bundles on Real Manifolds 150
The Product Structure on the de Rham Cohomology 153
The de Rham Isomorphism and the fundamental class 154
4.10.2 Cohomology of Holomorphic Bundles on Complex Manifolds 156
The Tangent Bundle 156
The Bundle Ω？ 158
4.10.3 Chern Classes 160
The Line Bundles OIPn（？）（k） 163
4.11 Hodge Theory 164
4.11.1 Hodge Theory on Real Manifolds 164
4.11.2 Hodge Theory on Complex Manifolds 169
Some Linear Algebra 169
K？hler Manifolds and their Cohomology 172
The Cohomology of Holomorphic Vector Bundles 175
Serre Duality 176
4.11.3 Hodge Theory on Tori 177
5 Compact Riemann surfaces and Abelian Varieties 179
5.1 Compact Riemann Surfaces 179
5.1.1 Introduction 179
5.1.2 The Hodge Structure on H1（S,？） 180
5.1.3 Cohomology of Holomorphic Bundles 185
5.1.4 The Theorem of Riemann-Roch 191
On the Picard Group 191
Exercises 192
The Theorem of Riemann-Roch 193
5.1.5 The Algebraic Duality Pairing 194
5.1.6 Riemann Surfaces of Low Genus 196
5.1.7 The Algebraicity of Riemann Surfaces 197
From a Riemann Surface to Function Fields 197
The reconstruction of S from K 202
Connection to Algebraic Geometry 209
Elliptic Curves 211
5.1.8 Géométrie Analytique et Géométrie Algébrique-GAGA 212
5.1.9 Comparison of Two Pairings 215
5.1.10 The Jacobian of a Compact Riemann Surface 217
5.1.11 The Classical Version of Abel's Theorem 218
5.1.12 Riemann Period Relations 222
5.2 Line Bundles on Complex Tori 223
5.2.1 Construction of Line Bundles 223
The Poincaré Bundle 229
Universality of N 230
5.2.2 Homomorphisms Between Complex Tori 232
The Neron Severi group and Hom（A,A？） 234
The construction of ？ starting from a line bundle 235
5.2.3 The Self Duality of the Jacobian 236
5.2.4 Ample Line Bundles and the Algebraicity of the Jacobian 237
The Kodaira Embedding Theorem 237
The Spaces of Sections 239
5.2.5 The Siegel Upper Half Space 240
Elliptic curves with level structure 243
The end of the excursion 251
5.2.6 Riemann-Theta Functions 252
5.2.7 Projective embeddings of abelian varieties 256
5.2.8 Degeneration of Abelian Varieties 259
The Case of Genus 1 259
The Algebraic Approach 269
5.3 Towards the Algebraic Theory 271
5.3.1 Introduction 271
The Algebraic Definition of the Neron-Severi Group 272
The Algebraic Definition of the Intersection Numbers 273
The Study of some Special Neron-Severi groups 274
5.3.2 The Structure of End（J） 278
The Rosati Involution 278
A Trace Formula 280
The Fundamental Class［S］of S under the Abel Map 284
5.3.3 The Ring of Correspondences 285
5.3.4 An Algebraic Substitute for the Cohomology 286
Bibliography 290
Index 293