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Introduction1

Historical Note1

The Contents of the Book8

Standard Notation12

Ⅰ.Preliminaries13

Topology and Algebra13

1. Notation and Basic Facts13

2. Some Properties of Bilinear Forms15

3. Vector Bundles, Characteristic Classes and the Index Theorem21

Complex Manifolds23

4. Basic Concepts and Facts23

5. Holomorphic Vector Bundles, Serre Duality and Riemann-Roch24

6. Line Bundles and Divisors26

7. Algebraic Dimension and Kodaira Dimension28

General Analytic Geometry30

8. Complex Spaces30

9. The σ-Process34

10. Deformations of Complex Manifolds35

Differential Geometry of Complex Manifolds39

11. De Rham Cohomology39

12. Dolbeault Cohomology41

13. Kanler Manifolds42

14. Weight-1 Hodge Structures48

15. Yaus results on Kahler-Einstein Metrics51

Coverings53

16. Ramification53

17. Cyclic Coverings54

18. Covering Tricks55

Projective-Algebraic Varieties57

19. GAGA Theorems and Proiectivitv Criteria57

20. Tneorems of ertmi and Letscnetz58

Ⅱ.Curves on Surfaces61

Embedded Curves61

1. Some Standard Exact Sequences61

2. The Picard-Group of an Embedded Curve63

3. Riemann-Roch for an Embedded Curve65

4. The Residue Theorem66

5. The Trace Map68

6. Serre Duality on an Embedded Curve70

7. The σ-process75

8. Simple Singularities of Curves78

Intersection Theory81

9. Intersection Multiplicities81

10. Intersection Numbers83

11. The Arithmetical Genus of an Embedded Curve84

12. 1-Connected Divisors85

Ⅲ.Mappings of Surfaces89

Bimeromorphic Geometry89

1. Bimeromorphic Maps89

2. Exceptional Curves90

3. Rational Singularities93

4. Exceptional Curves of the First Kind97

5. Hirzebruch-Jung Singularities99

6. Resoiution or Surface Singuiarities105

7. Singularities of Double Coverings, Simple Singularities of Surfaces107

Fibrations of Surfaces110

8. Generalities on Fibrations110

9. The n-th Root Fibration113

10. Stable Fibrations114

11. Direct Image Sheaves116

12. Relative Duality118

The Period Map of Stable Fibrations121

13. Period Matrices of Stable Curves121

14. Topological Monodromy of Stable Fibrations122

15. Monodromy of the Period Matrix125

16. Extending the Period Map127

17. The Degree of fωx/s129

18. Iitaka's Conjecture C2,1131

Ⅳ.Some General Properties of Surfaces135

1. Meromorphic Maps, Associated to Line Bundles135

2. Hodge Theory on Surfaces137

3. Existence of K？hler Metrics144

4. Deformations of Surfaces154

5. Some Inequalities for Hodge Numbers157

6. Projectivity of Surfaces159

7. The Nef Cone162

8. Surfaces of Algebraic Dimension Zero165

9. Almost-Complex Surfaces without any Complex Structure166

10. Bogomolov's Theorem168

11. Reider's Method174

12. Vanishing Theorems on Surfaces179

Ⅴ.Examples185

Some Classical Examples185

1. The Projective Plane P2185

2. Complete Intersections187

3. Tori of Dimension 2188

Fibre Bundles189

4. Ruled Surfaces189

5. Elliptic Fibre Bundles193

6. Higher Genus Fibre Bundles199

Elliptic Fibrations200

7. Kodaira's Table of Singular Fibres200

8. Stable Fibrations202

9. The Jacobian Fibration204

10. Stable Reduction207

11. Classification211

12. Invariants212

13. Logarithmic Transformations216

Kodaira Fibrations220

14. Kodaira Fibrations220

Finite Quotients223

15. The Godeaux Surface223

16. Kummer Surfaces224

17. Quotients of Products of Curves224

Infinite Quotients225

18. Hopf Surfaces225

19. Inoue Surfaces227

20. Quotients of Bounded Domains in C2230

21. Hilbert Modular Surfaces231

Coverings236

22. Invariants of Double Coverings236

23. An Enriques Surface238

24. Kummer Coverings240

Ⅵ.The Enriques Kodaira Classification243

1. Statement of the Main Result243

2. Characterising Minimal Surfaces whose Canonical Bundle is Nef247

3. The Rationality Theorem and Castelnuovo's Criterion248

4. The Case a（X）＝2252

5. The Case a（X）＝1255

6. The Case a（X）＝0257

7. The Final Step262

8. Deformations263

Ⅶ.Surfaces of General Type269

Preliminaries269

1. Introduction269

2. Some General Theorems271

Two Inequalities273

3. Noether's Inequality273

4. The Inequalityc21≤3c2275

Pluricanonical Maps279

5. The Main Results279

6. Proof of the Main Results281

7. The Exceptional Cases and the 1-Canonical Map286

Surfaces with Given Chern Numbers290

8. The Geography of Chern Numbers291

9. Surfaces on the Noether Lines296

10. Surfaces with q＝pg＝0299

Ⅷ.K3-Surfaces and Enriques Surfaces307

Introduction307

1. Notation307

2. The Results309

K 3-Surfaces310

3. Topological and Analvtical Invariants310

4. Digression on Affne Geometrv over F2314

5. ne Neron-Severi Lattice of Kummer Surfaces316

6. The Torelli Theorem for Kummer Surfaces322

7. The Local Torelli Theorem for K3-Surfaces323

8. A Density Theorem325

9. Behaviour of the K？hler Cone under Deformations327

10. Degenerations of Isomorphisms between K3-Surfaces329

11. The Torelli T neorems for K3-Surfaces332

12. Construction of Moduli Spaces334

13. Digression on Quaternionic Structures336

14. Surjectivity of the Period Map338

Enriques Surfaces339

15. Topological and Analytic Invariants339

16. Divisors on an Enriques Surface Y340

17. Elliptic Pencils342

18. Double Coverings of Quadrics345

19. The Period Map350

20. The Period Domain for Enriques Surfaces352

21. Global Properties of the Period Map354

Special Topics358

22. Proiective K3-surfaces and Mirror Symmetry358

23. Special Curves onK3-Surfaces364

24. An Application to Hyperbolic Geometry369

Ⅸ.Topological and Differentiable Structure of Surfaces375

Topology of Simply Connected Compact Complex Surfaces375

1. Freedman's Results375

2. Representability of Unimodular Forms377

Donaldson Invariants379

3. Introduction379

4. The Donaldson Invariant,a Bird's Eve View380

5. Infiniteiy many Homeomorphic Surfaces which are not Ditteomorphic383

6. FurtherResults obtained by the Donaldson Method390

Seiberg-Witten Invariants391

7. Introduction391

8. Properties of the Invariants393

9.Surfaces Diffeomorpnic to a Rational urface395

Bibliography401

Notation425

Index429