Introduction 1
Historical Note 1
The Contents of the Book 8
Standard Notation 12
Ⅰ.Preliminaries 13
Topology and Algebra 13
1. Notation and Basic Facts 13
2. Some Properties of Bilinear Forms 15
3. Vector Bundles, Characteristic Classes and the Index Theorem 21
Complex Manifolds 23
4. Basic Concepts and Facts 23
5. Holomorphic Vector Bundles, Serre Duality and Riemann-Roch 24
6. Line Bundles and Divisors 26
7. Algebraic Dimension and Kodaira Dimension 28
General Analytic Geometry 30
8. Complex Spaces 30
9. The σ-Process 34
10. Deformations of Complex Manifolds 35
Differential Geometry of Complex Manifolds 39
11. De Rham Cohomology 39
12. Dolbeault Cohomology 41
13. Kanler Manifolds 42
14. Weight-1 Hodge Structures 48
15. Yaus results on Kahler-Einstein Metrics 51
Coverings 53
16. Ramification 53
17. Cyclic Coverings 54
18. Covering Tricks 55
Projective-Algebraic Varieties 57
19. GAGA Theorems and Proiectivitv Criteria 57
20. Tneorems of ertmi and Letscnetz 58
Ⅱ.Curves on Surfaces 61
Embedded Curves 61
1. Some Standard Exact Sequences 61
2. The Picard-Group of an Embedded Curve 63
3. Riemann-Roch for an Embedded Curve 65
4. The Residue Theorem 66
5. The Trace Map 68
6. Serre Duality on an Embedded Curve 70
7. The σ-process 75
8. Simple Singularities of Curves 78
Intersection Theory 81
9. Intersection Multiplicities 81
10. Intersection Numbers 83
11. The Arithmetical Genus of an Embedded Curve 84
12. 1-Connected Divisors 85
Ⅲ.Mappings of Surfaces 89
Bimeromorphic Geometry 89
1. Bimeromorphic Maps 89
2. Exceptional Curves 90
3. Rational Singularities 93
4. Exceptional Curves of the First Kind 97
5. Hirzebruch-Jung Singularities 99
6. Resoiution or Surface Singuiarities 105
7. Singularities of Double Coverings, Simple Singularities of Surfaces 107
Fibrations of Surfaces 110
8. Generalities on Fibrations 110
9. The n-th Root Fibration 113
10. Stable Fibrations 114
11. Direct Image Sheaves 116
12. Relative Duality 118
The Period Map of Stable Fibrations 121
13. Period Matrices of Stable Curves 121
14. Topological Monodromy of Stable Fibrations 122
15. Monodromy of the Period Matrix 125
16. Extending the Period Map 127
17. The Degree of fωx/s 129
18. Iitaka's Conjecture C2,1 131
Ⅳ.Some General Properties of Surfaces 135
1. Meromorphic Maps, Associated to Line Bundles 135
2. Hodge Theory on Surfaces 137
3. Existence of K？hler Metrics 144
4. Deformations of Surfaces 154
5. Some Inequalities for Hodge Numbers 157
6. Projectivity of Surfaces 159
7. The Nef Cone 162
8. Surfaces of Algebraic Dimension Zero 165
9. Almost-Complex Surfaces without any Complex Structure 166
10. Bogomolov's Theorem 168
11. Reider's Method 174
12. Vanishing Theorems on Surfaces 179
Ⅴ.Examples 185
Some Classical Examples 185
1. The Projective Plane P2 185
2. Complete Intersections 187
3. Tori of Dimension 2 188
Fibre Bundles 189
4. Ruled Surfaces 189
5. Elliptic Fibre Bundles 193
6. Higher Genus Fibre Bundles 199
Elliptic Fibrations 200
7. Kodaira's Table of Singular Fibres 200
8. Stable Fibrations 202
9. The Jacobian Fibration 204
10. Stable Reduction 207
11. Classification 211
12. Invariants 212
13. Logarithmic Transformations 216
Kodaira Fibrations 220
14. Kodaira Fibrations 220
Finite Quotients 223
15. The Godeaux Surface 223
16. Kummer Surfaces 224
17. Quotients of Products of Curves 224
Infinite Quotients 225
18. Hopf Surfaces 225
19. Inoue Surfaces 227
20. Quotients of Bounded Domains in C2 230
21. Hilbert Modular Surfaces 231
Coverings 236
22. Invariants of Double Coverings 236
23. An Enriques Surface 238
24. Kummer Coverings 240
Ⅵ.The Enriques Kodaira Classification 243
1. Statement of the Main Result 243
2. Characterising Minimal Surfaces whose Canonical Bundle is Nef 247
3. The Rationality Theorem and Castelnuovo's Criterion 248
4. The Case a（X）＝2 252
5. The Case a（X）＝1 255
6. The Case a（X）＝0 257
7. The Final Step 262
8. Deformations 263
Ⅶ.Surfaces of General Type 269
Preliminaries 269
1. Introduction 269
2. Some General Theorems 271
Two Inequalities 273
3. Noether's Inequality 273
4. The Inequalityc21≤3c2 275
Pluricanonical Maps 279
5. The Main Results 279
6. Proof of the Main Results 281
7. The Exceptional Cases and the 1-Canonical Map 286
Surfaces with Given Chern Numbers 290
8. The Geography of Chern Numbers 291
9. Surfaces on the Noether Lines 296
10. Surfaces with q＝pg＝0 299
Ⅷ.K3-Surfaces and Enriques Surfaces 307
Introduction 307
1. Notation 307
2. The Results 309
K 3-Surfaces 310
3. Topological and Analvtical Invariants 310
4. Digression on Affne Geometrv over F2 314
5. ne Neron-Severi Lattice of Kummer Surfaces 316
6. The Torelli Theorem for Kummer Surfaces 322
7. The Local Torelli Theorem for K3-Surfaces 323
8. A Density Theorem 325
9. Behaviour of the K？hler Cone under Deformations 327
10. Degenerations of Isomorphisms between K3-Surfaces 329
11. The Torelli T neorems for K3-Surfaces 332
12. Construction of Moduli Spaces 334
13. Digression on Quaternionic Structures 336
14. Surjectivity of the Period Map 338
Enriques Surfaces 339
15. Topological and Analytic Invariants 339
16. Divisors on an Enriques Surface Y 340
17. Elliptic Pencils 342
18. Double Coverings of Quadrics 345
19. The Period Map 350
20. The Period Domain for Enriques Surfaces 352
21. Global Properties of the Period Map 354
Special Topics 358
22. Proiective K3-surfaces and Mirror Symmetry 358
23. Special Curves onK3-Surfaces 364
24. An Application to Hyperbolic Geometry 369
Ⅸ.Topological and Differentiable Structure of Surfaces 375
Topology of Simply Connected Compact Complex Surfaces 375
1. Freedman's Results 375
2. Representability of Unimodular Forms 377
Donaldson Invariants 379
3. Introduction 379
4. The Donaldson Invariant,a Bird's Eve View 380
5. Infiniteiy many Homeomorphic Surfaces which are not Ditteomorphic 383
6. FurtherResults obtained by the Donaldson Method 390
Seiberg-Witten Invariants 391
7. Introduction 391
8. Properties of the Invariants 393
9.Surfaces Diffeomorpnic to a Rational urface 395
Bibliography 401
Notation 425
Index 429