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Book 1：Varieties in Projective Space 3

1 Basic Notions 3

1 Algebraic Curves in the Plane 3

1.1 Plane Curves 3

1.2 Rational Curves 6

1.3 Relation with Field Theory 9

1.4 Rational Maps 11

1.5 Singular and Nonsingular Points 13

1.6 The Projective Plane 17

1.7 Exercises to Section 1 22

2 Closed Subsets of Affine Space 23

2.1 Definition of Closed Subsets 23

2.2 Regular Functions on a Closed Subset 25

2.3 Regular Maps 27

2.4 Exercises to Section 2 32

3 Rational Functions 34

3.1 Irreducible Algebraic Subsets 34

3.2 Rational Functions 36

3.3 Rational Maps 37

3.4 Exercises to Section 3 40

4 Quasiprojective Varieties 41

4.1 Closed Subsets of Projective Space 41

4.2 Regular Functions 46

4.3 Rational Functions 50

4.4 Examples of Regular Maps 52

4.5 Exercises to Section 4 53

5 Products and Maps of Quasiprojective Varieties 54

5.1 Products 54

5.2 The Image of a Projective Variety is Closed 57

5.3 Finite Maps 60

5.4 Noether Normalisation 65

5.5 Exercises to Section 5 65

6 Dimension 66

6.1 Definition of Dimension 66

6.2 Dimension of Intersection with a Hypersurface 69

6.3 The Theorem on the Dimension of Fibres 75

6.4 Lines on Surfaces 77

6.5 Exercises to Section 6 80

2 Local Properties 83

1 Singular and Nonsingular Points 83

1.1 The Local Ring of a Point 83

1.2 The Tangent Space 85

1.3 Intrinsic Nature of the Tangent Space 86

1.4 Singular Points 92

1.5 The Tangent Cone 94

1.6 Exercises to Section 1 95

2 Power Series Expansions 98

2.1 Local Parameters at a Point 98

2.2 Power Series Expansions 100

2.3 Varieties over the Reals and the Complexes 104

2.4 Exercises to Section 2 106

3 Properties of Nonsingular Points 106

3.1 Codimension 1 Subvarieties 106

3.2 Nonsingular Subvarieties 110

3.3 Exercises to Section 3 111

4 The Structure of Birational Maps 113

4.1 Blowup in Proiective Space 113

4.2 Local Blowup 115

4.3 Behaviour of a Subvariety Under a Blowup 117

4.4 Exceptional Subvarieties 119

4.5 Isomorphism and Birational Equivalence 120

4.6 Exercises to Section 4 123

5 Normal Varieties 124

5.1 Normal Varieties 124

5.2 Normalisation of an Affine Variety 128

5.3 Normalisation of a Curve 130

5.4 Projective Embedding of Nonsingular Varieties 134

5.5 Exercises to Section 5 136

6 Singularities of a Map 137

6.1 Irreducibility 137

6.2 Nonsingularity 139

6.3 Ramification 140

6.4 Examples 143

6.5 Exercises to Section 6 146

3 Divisors and Differential Forms 147

1 Divisors 147

1.1 The Divisor of a Function 147

1.2 Locally Principal Divisors 151

1.3 Moving the Support of a Divisor away from a Point 153

1.4 Divisors and Rational Maps 155

1.5 The Linear System of a Divisor 156

1.6 Pencil of Conics over P1 159

1.7 Exercises to Section 1 161

2 Divisors on Curves 163

2.1 The Degree ofa Divisor on a Curve 163

2.2 Bézout's Theorem on a Curve 167

2.3 The Dimension of a Divisor 168

2.4 Exercises to Section 2 169

3 The Plane Cubic 170

3.1 The Class Group 170

3.2 The Group Law 173

3.3 Maps 177

3.4 Applications 179

3.5 Algebraically Nonclosed Field 181

3.6 Exercises to Section 3 183

4 Algebraic Groups 184

4.1 Algebraic Groups 184

4.2 Quotient Groups and Chevalley's Theorem 185

4.3 Abelian Varieties 186

4.4 The Picard Variety 188

4.5 Exercises to Section 4 189

5 Differential Forms 190

5.1 Regular Differential 1-Forms 190

5.2 Algebraic Definition of the Module of Differentials 193

5.3 Differential p-Forms 195

5.4 Rational Difierential Forms 197

5.5 Exercises to Section5 199

6 Examples and Applications of Differential Forms 200

6.1 Behaviour Under Maps 200

6.2 Invariant Differential Forms on a Group 202

6.3 The Canonical Class 204

6.4 Hypersurfaces 206

6.5 Hyperelliptic Curves 209

7 The Riemann-Roch Theorem on Curyes 210

7.1 Statement of the Theorem 210

7.2 Preliminary Form of the Riemann-Roch Theorem 213

7.3 The Residue ofa 1-Form 217

7.4 Linear Algebra in Infinite Dimensional Vector Spaces 219

7.5 The Residue Theorem 224

7.6 The Duality Theorem 225

7.7 Exercises to Sections 6-7 227

8 Higher Dimensional Generalisations 229

4 Intersection Numbers 233

1 Definition and Basic Properties 233

1.1 Definition of Intersection Number 233

1.2 Additivity 236

1.3 Invariance Under Linear Equivalence 238

1.4 The General Definition of Intersection Number 242

1.5 Exercises to Section 1 245

2 Applications of Intersection Numbers 246

2.1 Bézout's Theorem in Projective and Multiprojective Space 246

2.2 Varieties over the Reals 248

2.3 The Genus of aN onsingular Curve on a Surface 251

2.4 The Riemann-Roch Inequality on a Surface 253

2.5 The Nonsingular Cubic Surface 255

2.6 The Ring ofCycle Classes 258

2.7 Exercises to Section 2 259

3 Birational Maps of Surfaces 260

3.1 Blowups of Surfaces 260

3.2 Some Intersection Numbers 261

3.3 Resolution of Indeterminacy 263

3.4 Factorisation as a Chain of Blowups 264

3.5 Remarks and Examples 267

3.6 Exercises to Section3 269

4 Singularities 270

4.1 Singular Points of a Curve 270

4.2 Surface Singularities 273

4.3 Du Val Singularities 274

4.4 Degeneration of Curves 278

4.5 Exercises to Section4 281

Algebraic Appendix 283

1 Linear and Bilinear Algebra 283

2 Polynomials 285

3 Quasilinear Maps 285

4 Invariants 287

5 Fields 288

6 Commutative Rings 289

7 Unique Factorisation 292

8 Integral Elements 293

9 Length of a Module 294

References 297

Index 301

Book 2：Schemes and Varieties 3

5 Schemes 3

1 The Spec of a Ring 5

1.1 Definition of Spec A 5

1.2 Properties of Points of Spec A 7

1.3 The Zariski Topology of Spec A 9

1.4 Irreducibility,Dimension 11

1.5 Exercises to Section 1 14

2 Sheaves 15

2.1 Presheaves 15

2.2 The Structure Presheaf 17

2.3 Sheaves 19

2.4 Stalks of a Sheaf 23

2.5 Exercises to Section 2 24

3 Schemes 25

3.1 Definition of a Scheme 25

3.2 Glueing Schemes 30

3.3 Closed Subschemes 32

3.4 Reduced Schemes and Nilpotents 35

3.5 Finiteness Conditions 36

3.6 Exercises to Section 3 38

4 Products of Schemes 40

4.1 Definition of Product 40

4.2 Group Schemes 42

4.3 Separatedness 43

4.4 Exercises to Section 4 46

6 Varieties 49

1 Definitions and Examples 49

1.1 Definitions 49

1.2 Vector Bundles 53

1.3 Vector Bundles and Sheaves 56

1.4 Divisors and Line Bundles 63

1.5 Exercises to Section 1 67

2 Abstract and Quasiprojective Varieties 68

2.1 Chow's Lemma 68

2.2 Blowup Along a Subvariety 70

2.3 Example of Non-quasiprojective Variety 74

2.4 Criterions for Projectivity 79

2.5 Exercises to Section 2 81

3 Coherent Sheaves 81

3.1 Sheaves of Ox-Modules 81

3.2 Coherent Sheaves 85

3.3 Dévissage of Coherent Sheaves 88

3.4 The Finiteness Theorem 92

3.5 Exercises to Section 3 93

4 Classification of Geometric Obiects and Universal Schemes 94

4.1 Schemes and Functors 94

4.2 The Hilbert Polynomial 100

4.3 Flat Families 103

4.4 The Hilbert Scheme 107

4.5 Exercises to Section 4 110

Book 3：Complex Algebraic Varieties and Complex Manifolds 110

7 The Topology of Algebraic Varieties 115

1 The Complex Topology 115

1.1 Definitions 115

1.2 Algebraic Varieties as Differentiable Manifolds；Orientation 117

1.3 Homology of Nonsingular Projective Varieties 118

1.4 Exercises to Section 1 121

2 Connectedness 121

2.1 Preliminary Lemmas 121

2.2 The First Proof of the Main Theorem 122

2.3 The Second Proof 124

2.4 Analytic Lemmas 126

2.5 Connectedness of Fibres 127

2.6 Exercises to Section 2 128

3 The Topology of Algebraic Curves 129

3.1 Local Structure of Morphisms 129

3.2 Triangulation of Curves 131

3.3 Topological Classification of Curves 133

3.4 Combinatorial Classification of Surfaces 137

3.5 The Topology of Singularities of Plane Curves 140

3.6 Exercises to Section 3 142

4 Real Algebraic Curves 142

4.1 Complex Conjugation 143

4.2 Proof of Harnack's Theorem 144

4.3 Ovals of Real Curves 146

4.4 Exercises to Section 4 147

8 Complex Manifolds 149

1 Definitions and Examples 149

1.1 Definition 149

1.2 Quotient Spaces 152

1.3 Commutative Algebraic Groups as Quotient Spaces 155

1.4 Examples of Compact Complex Manifolds not Isomorphic to Algebraic Varieties 157

1.5 Complex Spaces 163

1.6 Exercises to Section 1 165

2 Divisors and Meromorphic Functions 166

2.1 Divisors 166

2.2 Meromorphic Functions 169

2.3 The Structure of the Field M（X） 171

2.4 Exercises to Section 2 174

3 Algebraic Varieties and Complex Manifolds 175

3.1 Comparison Theorems 175

3.2 Example of Nonisomorphic Algebraic Varieties that Are Isomorphic as Complex Manifolds 178

3.3 Example of a Nonalgebraic Compact Complex Manifold with Maximal Number of Independent Meromorphic Functions 181

3.4 The Classification of Compact Complex Surfaces 183

3.5 Exercises to Section 3 185

4 K？hler Manifolds 185

4.1 K？hler Metric 186

4.2 Examples 188

4.3 Other Characterisations of K？hler Metrics 190

4.4 Applications of K？hler Metrics 193

4.5 Hodge Theory 196

4.6 Exercises to Section 4 198

9 Uniformisation 201

1 The Universal Cover 201

1.1 The Universal Cover of a Complex Manifold 201

1.2 Universal Covers of Algebraic Curves 203

1.3 Projective Embedding of Quotient Spaces 205

1.4 Exercises to Section 1 206

2 Curves of Parabolic Type 207

2.1 Theta Functions 207

2.2 Projective Embedding 209

2.3 Elliptic Functions，Elliptic Curves and Elliptic Integrals 210

2.4 Exercises to Section 2 213

3 Curves of Hyperbolic Type 213

3.1 Poincaré Series 213

3.2 Projective Embedding 216

3.3 Algebraic Curves and Automorphic Functions 218

3.4 Exercises to Section 3 221

4 Uniformising Higher Dimensional Varieties 221

4.1 Complete Intersections are Simply Connected 221

4.2 Example of Manifold with π1 a Given Finite Group 222

4.3 Remarks 226

4.4 Exercises to Section 4 227

Historical Sketch 229

1 Elliptic Integrals 229

2 Elliptic Functions 231

3 Abelian Integrals 233

4 Riemann Surfaces 235

5 The Inversion of Abelian Integrals 237

6 The Geometry of Algebraic Curves 239

7 Higher Dimensional Geometry 241

8 The Analytic Theory of Complex Manifolds 243

9 Algebraic Varieties over Arbitrary Fields and Schemes 244

References 247

References for the Historical Sketch 250

Index 253