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CHAPTER 0 FOUNDATIONAL MATERIAL 1

1.Rudiments of Several Complex Variables 2

Cauchy’s Formula and Applications 2

Several Variables 6

Weierstrass Theorems and Corollaries 7

Analytic Varieties 12

2.Complex Manifolds 14

Complex Manifolds 14

Submanifolds and Subvarieties 18

De Rham and Dolbeault Cohomology 23

Calculus on Complex Manifolds 27

3.Sheaves and Cohomology 34

Origins： The Mittag-Leffler Problem 34

Sheaves 35

Cohomology of Sheaves 38

The de Rham Theorem 43

The Dolbeault Theorem 45

4.Topology of Manifolds 49

Intersection of Cycles 49

Poincare Duality 53

Intersection of Analytic Cycles 60

5.Vector Bundles， Connections， and Curvature 66

Complex and Holomorphic Vector Bundles 66

Metrics， Connections， and Curvature 71

6.Harmonic Theory on Compact Complex Manifolds 80

The Hodge Theorem 80

Proof of the Hodge Theorem Ⅰ： Local Theory 84

Proof of the Hodge Theorem Ⅱ： Global Theory 92

Applications of the Hodge Theorem 100

7.Kahler Manifolds 106

The Kahler Condition 106

The Hodge Identities and the Hodge Decomposition 111

The Lefschetz Decomposition 118

CHAPTER 1 COMPLEX ALGEBRAIC VARIETIES 128

1.Divisors and Line Bundles 129

Divisors 129

Line Bundles 132

Chern Classes of Line Bundles 139

2.Some Vanishing Theorems and Corollaries 148

The Kodaira Vanishing Theorem 148

The Lefschetz Theorem on Hyperplane Sections 156

Theorem B 159

The Lefschetz Theorem on （1，1）-classes 161

3.Algebraic Varieties 164

Analytic and Algebraic Varieties 164

Degree of a Variety 171

Tangent Spaces to Algebraic Varieties 175

4.The Kodaira Embedding Theorem 176

Line Bundles and Maps to Projective Space 176

Blowing Up 182

Proof of the Kodaira Theorem 189

5.Grassmannians 193

Definitions 193

The Cell Decomposition 194

The Schubert Calculus 197

Universal Bundles 207

The Plucker Embedding 209

CHAPTER 2 RIEMANN SURFACES AND ALGEBRAIC CURVES 212

1.Preliminaries 213

Embedding Riemann Surfaces 213

The Riemann-Hurwitz Formula 216

The Genus Formula 219

Cases g=0， 1 222

2.Abel’s Theorem 224

Abel’s Theorem—First Version 224

The First Reciprocity Law and Corollaries 229

Abel’s Theorem—Second Version 232

Jacobi Inversion 235

3.Linear Systems on Curves 240

Reciprocity Law Ⅱ 240

The Riemann-Roch Formula 243

Canonical Curves 246

Special Linear Systems Ⅰ 249

Hyperelliptic Curves and Riemann’s Count 253

Special Linear Systems Ⅱ 259

4.Plucker Formulas 263

Associated Curves 263

Ramification 264

The General Plucker Formulas Ⅰ 268

The General Plucker Formulas Ⅱ 271

Weierstrass Points 273

Plucker Formulas for Plane Curves 277

5.Correspondences 282

Definitions and Formulas 282

Geometry of Space Curves 290

Special Linear Systems Ⅲ 298

6.Complex Tori and Abelian Varieties 300

The Riemann Conditions 300

Line Bundles on Complex Tori 307

Theta-Functions 317

The Group Structure on an Abelian Variety 324

Intrinsic Formulations 326

7.Curves and Their Jacobians 333

Preliminaries 333

Riemann’s Theorem 338

Riemann’s Singularity Theorem 341

Special Linear Systems Ⅳ 349

Torelli’s Theorem 359

CHAPTER 3 FURTHER TECHNIQUES 364

1.Distributions and Currents 366

Definitions； Residue Formulas 366

Smoothing and Regularity 373

Cohomology of Currents 382

2.Applications of Currents to Complex Analysis 385

Currents Associated to Analytic Varieties 385

Intersection Numbers of Analytic Varieties 392

The Levi Extension and Proper Mapping Theorems 395

3.Chern Classes 400

Definitions 400

The Gauss Bonnet Formulas 409

Some Remarks—Not Indispensable—Concerning Chern Classes of Holomorphic Vector Bundles 416

4.Fixed-Point and Residue Formulas 419

The Lefschetz Fixed-Point Formula 419

The Holomorphic Lefschetz Fixed-Point Formula 422

The Bott Residue Formula 426

The General Hirzebruch-Riemann-Roch Formula 435

5.Spectral Sequences and Applications 438

Spectral Sequences of Filtered and Bigraded Complexes 438

Hypercohomology 445

Differentials of the Second Kind 454

The Leray Spectral Sequence 462

CHAPTER 4 SURFACES 469

1.Preliminaries 470

Intersection Numbers，the Adjunction Formula，and Riemann-Roch 470

Blowing Up and Down 473

The Quadric Surface 478

The Cubic Surface 480

2.Rational Maps 489

Rational and Birational Maps 489

Curves on an Algebraic Surface 498

The Structure of Birational Maps Between Surfaces 510

3.Rational Surfaces Ⅰ 513

Noether’s Lemma 513

Rational Ruled Surfaces 514

The General Rational Surface 520

Surfaces of Minimal Degree 522

Curves of Maximal Genus 527

Steiner Constructions 528

The Enriques-Petri Theorem 533

4.Rational Surfaces Ⅱ 536

The Castelnuovo-Enriques Theorem 536

The Enriques Surface 541

Cubic Surfaces Revisited 545

The Intersection of Two Quadrics in P4 550

5.Some Irrational Surfaces 552

The Albanese Map 552

Irrational Ruled Surfaces 553

A Brief Introduction to Elliptic Surfaces 564

Kodaira Number and the Classification Theorem Ⅰ 572

The Classification Theorem Ⅱ 582

K-3 Surfaces 590

Enriques Surfaces 594

6.Noether’s Formula 600

Noether’s Formula for Smooth Hypersurfaces 600

Blowing Up Submanifolds 602

Ordinary Singularities of Surfaces 611

Noether’s Formula for General Surfaces 618

Some Examples 628

Isolated Singularities of Surfaces 636

CHAPTER 5 RESIDUES 647

1.Elementary Properties of Residues 649

Definition and Cohomological Interpretation 649

The Global Residue Theorem 655

The Transformation Law and Local Duality 656

2.Applications of Residues 662

Intersection Numbers 662

Finite Holomorphic Mappings 667

Applications to Plane Projective Geometry 670

3.Rudiments of Commutative and Homological Algebra with Applications 678

Commutative Algebra 678

Homological Algebra 682

The Koszul Complex and Applications 687

A Brief Tour Through Coherent Sheaves 695

4.Global Duality 705

Global Ext 705

Explanation of the General Global Duality Theorem 707

Global Ext and Vector Fields with Isolated Zeros 708

Global Duality and Superabundance of Points on a Surface 712

Extensions of Modules 722

Points on a Surface and Rank-Two Vector Bundles 726

Residues and Vector Bundles 729

CHAPTER 6 THE QUADRIC LINE COMPLEX 733

1.Preliminaries： Quadrics 734

Rank of a Quadric 734

Linear Spaces on Quadrics 735

Linear Systems of Quadrics 741

Lines on Linear Systems of Quadrics 746

The Problem of Five Conics 749

2.The Quadric Line Complex： Introduction 756

Geometry of the Grassmannian G（2，4） 756

Line Complexes 759

The Quadric Line Complex and Associated Kummer Surface Ⅰ 762

Singular Lines of the Quadric Line Complex 767

Two Configurations 773

3.Lines on the Quadric Line Complex 778

The Variety of Lines on the Quadric Line Complex 778

Curves on the Variety of Lines 780

Two Configurations Revisited 784

The Group Law 787

4.The Quadric Line Complex： Reprise 791

The Quadric Line Complex and Associated Kummer Surface Ⅱ 791

Rationality of the Quadric Line Complex 796

INDEX 805

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