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1 Categories,Products,Projective and Inductive Limits1

1.1 The Notion of a Category and Examples1

1.2 Functors3

1.3 Products,Projective Limits and Direct Limits in a Category4

1.3.1 The Projective Limit4

1.3.2 The Yoneda Lemma6

1.3.3 Examples6

1.3.4 Representable Functors8

1.3.5 Direct Limits9

1.4 Exercises10

2 Basic Concepts of Homological Algebra11

2.1 The Category ModΓ of Γ-modules11

2.2 More Functors13

2.2.1 Invariants,Coinvariants and Exactness13

2.2.2 The First Cohomology Group15

2.2.3 Some Notation16

2.2.4 Exercises17

2.3 The Derived Functors19

2.3.1 The Simple Principle20

2.3.2 Functoriality22

2.3.3 Other Resolutions24

2.3.4 Injective Resolutions of Short Exact Sequences24

A Fundamental Remark26

The Cohomology and the Long Exact Sequence27

The Homology of Groups27

2.4 The Functors Ext and Tor28

2.4.1 The Functor Ext28

2.4.2 The Derived Functor for the Tensor Product30

2.4.3 Exercise32

3 Sheaves35

3.1 Presheaves and Sheaves35

3.1.1 What is a Presheaf？35

3.1.2 A Remark about Products and Presheaf36

3.1.3 What is a Sheaf？36

3.1.4 Examples38

3.2 Manifolds as Locally Ringed Spaces39

3.2.1 What Are Manifolds？39

3.2.2 Examples and Exercise41

3.3 Stalks and Sheafification45

3.3.1 Stalks45

3.3.2 The Process of Sheafification of a Presheaf46

3.4 The Functors f＊ and f＊47

3.4.1 The Adjunction Formula48

3.4.2 Extensions and Restrictions49

3.5 Constructions of Sheaves49

4 Cohomology of Sheaves51

4.1 Examples51

4.1.1 Sheaves on Riemann surfaces51

4.1.2 Cohomology of the Circle54

4.2 The Derived Functor55

4.2.1 Injective Sheaves and Derived Functors55

4.2.2 A Direct Definition of H156

4.3 Fiber Bundles and Non Abelian H159

4.3.1 Fibrations59

Fibre Bundle59

Vector Bundles60

4.3.2 Non-Abelian H161

4.3.3 The Reduction of the Structure Group62

Orientation62

Local Systems63

Isomorphism Classes of Local Systems64

Principal G-bundels64

4.4 Fundamental Properties of the Cohomology of Sheaves65

4.4.1 Introduction65

4.4.2 The Derived Functor to f＊66

4.4.3 Functorial Properties of the Cohomology68

4.4.4 Paracompact Spaces69

4.4.5 Applications75

Cohomology of Spheres75

Orientations76

Compact Oriented Surfaces77

4.5 ？ech Cohomology of Sheaves77

4.5.1 The ？ech-Complex77

4.5.2 The ？ech Resolution of a Sheaf81

4.6 Spectral Sequences83

4.6.1 Introduction83

4.6.2 The Vertical Filtration88

4.6.3 The Horizontal Filtration94

Two Special Cases95

Applications of Spectral Sequences96

4.6.4 The Derived Category98

The Composition Rule101

Exact Triangles102

4.6.5 The Spectral Sequence of a Fibration103

Sphere Bundles an Euler Characteristic104

4.6.6 ？ech Complexes and the Spectral Sequence105

A Criterion for Degeneration107

An Application to Product Spaces109

4.6.7 The Cup Product111

4.6.8 Example:Cup Product for the Comology of Tori115

A Connection to the Cohomology of Groups116

4.6.9 An Excursion into Homotopy Theory117

4.7 Cohomology with Compact Supports120

4.7.1 The Definition120

4.7.2 An Example for Cohomology with Compact Supports121

The Cohomology with Compact Supports for Open Balls121

Formulae for Cup Products123

4.7.3 The Fundamental Class125

4.8 Cohomology of Manifolds126

4.8.1 Local Systems126

4.8.2 ？ech Resolutions of Local Systems127

4.8.3 ？ech Coresolution of Local Systems129

4.8.4 Poincaré Duality132

4.8.5 The Cohomology in Top Degree and the Homology138

4.8.6 Some Remarks on Singular Homology140

4.8.7 Cohomology with Compact Support and Embeddings141

4.8.8 The Fundamental Class of a Submanifold143

4.8.9 Cup Product and Intersections144

4.8.10 Compact oriented Surfaces146

4.8.11 The Cohomology Ring of IPn（？）147

4.9 The Lefschetz Fixed Point Formula147

4.9.1 The Euler Characteristic of Manifolds149

4.10 The de Rham and the Dolbeault Isomorphism150

4.10.1 The Cohomology of Flat Bundles on Real Manifolds150

The Product Structure on the de Rham Cohomology153

The de Rham Isomorphism and the fundamental class154

4.10.2 Cohomology of Holomorphic Bundles on Complex Manifolds156

The Tangent Bundle156

The Bundle Ω？158

4.10.3 Chern Classes160

The Line Bundles OIPn（？）（k）163

4.11 Hodge Theory164

4.11.1 Hodge Theory on Real Manifolds164

4.11.2 Hodge Theory on Complex Manifolds169

Some Linear Algebra169

K？hler Manifolds and their Cohomology172

The Cohomology of Holomorphic Vector Bundles175

Serre Duality176

4.11.3 Hodge Theory on Tori177

5 Compact Riemann surfaces and Abelian Varieties179

5.1 Compact Riemann Surfaces179

5.1.1 Introduction179

5.1.2 The Hodge Structure on H1（S,？）180

5.1.3 Cohomology of Holomorphic Bundles185

5.1.4 The Theorem of Riemann-Roch191

On the Picard Group191

Exercises192

The Theorem of Riemann-Roch193

5.1.5 The Algebraic Duality Pairing194

5.1.6 Riemann Surfaces of Low Genus196

5.1.7 The Algebraicity of Riemann Surfaces197

From a Riemann Surface to Function Fields197

The reconstruction of S from K202

Connection to Algebraic Geometry209

Elliptic Curves211

5.1.8 Géométrie Analytique et Géométrie Algébrique-GAGA212

5.1.9 Comparison of Two Pairings215

5.1.10 The Jacobian of a Compact Riemann Surface217

5.1.11 The Classical Version of Abel's Theorem218

5.1.12 Riemann Period Relations222

5.2 Line Bundles on Complex Tori223

5.2.1 Construction of Line Bundles223

The Poincaré Bundle229

Universality of N230

5.2.2 Homomorphisms Between Complex Tori232

The Neron Severi group and Hom（A,A？）234

The construction of ？ starting from a line bundle235

5.2.3 The Self Duality of the Jacobian236

5.2.4 Ample Line Bundles and the Algebraicity of the Jacobian237

The Kodaira Embedding Theorem237

The Spaces of Sections239

5.2.5 The Siegel Upper Half Space240

Elliptic curves with level structure243

The end of the excursion251

5.2.6 Riemann-Theta Functions252

5.2.7 Projective embeddings of abelian varieties256

5.2.8 Degeneration of Abelian Varieties259

The Case of Genus 1259

The Algebraic Approach269

5.3 Towards the Algebraic Theory271

5.3.1 Introduction271

The Algebraic Definition of the Neron-Severi Group272

The Algebraic Definition of the Intersection Numbers273

The Study of some Special Neron-Severi groups274

5.3.2 The Structure of End（J）278

The Rosati Involution278

A Trace Formula280

The Fundamental Class［S］of S under the Abel Map284

5.3.3 The Ring of Correspondences285

5.3.4 An Algebraic Substitute for the Cohomology286

Bibliography290

Index293