Compact Complex Surfaces

438 Pages·2004·38.21 MB·English

by  Wolf P. Barth

Ergebnisse der Mathematik Volume 4
und ihrer Grenzgebiete
3. Folge
A Series of Modern Surveys
in Mathematics
Editorial Board
S. Feferman, Stanford M. Gromov, Bures-sur-Yvette
J. Jost, Leipzig J. Kollâr, Princeton
H.W. Lenstra, Jr., Berkeley P.-L. Lions, Paris
M. Rapoport, Koln J. Tits, Paris D.B. Zagier, Bonn
G.M. Ziegler, Berlin
Managing Editor R. Remmert, Miinster
Springer-V erlag Berlin Heidelberg GmbH
Wolf P. Barth
Klaus Hulek
Chris A. M. Peters
Antonius Van de Ven
Compact
Complex Surfaces
Second Enlarged Edition
Springer
Mathematics Subject Classification (2ooo ):
14B05, 14C3o, 14Eo5, 14E3o, 14F17, 14JI5, 14]17, 14]25, 14]26, 14]27, 14]28,
14]29, 32Go5, 32G13, 32]15, 32Q6o, 53D45, 57R57
ISBN 978-3-540-00832-3 ISBN 978-3-642-57739-0 (eBook)
DOI 10.1007/978-3-642-57739-0
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@ Springer-Verlag Berlin Heidelberg 2004
Softcover reprint of the hardcover 2nd edition 2004
Typeset by the authors using a Springer T]llC macro package.
Printed on acid-free paper 44/3142LK-5 4 3 2 1 o
Preface to the Second Edition
In the 19 years which passed since the first edition was published, several
important developments have taken place in the theory of surfaces. The most
sensational one concerns the differentiable structure of surfaces. Twenty years
ago very little was known about differentiable structures on 4-manifolds, but
in the meantime Donaldson on the one hand and Seiberg and Witten on
the other hand, have found, inspired by gauge theory, totally new invariants.
Strikingly, together with the theory explained in this book these invariants
yield a wealth of new results about the differentiable structure of algebraic
surfaces.
Other developments include the systematic use of nef-divisors (in ac
cordance with the progress made in the classification of higher dimensional
algebraic varieties), a better understanding of Kahler structures on surfaces,
and Reider's new approach to adjoint mappings.
All these developments have been incorporated in the present edition,
though the Donaldson and Seiberg-Witten theory only by way of examples.
Of course we use the opportunity to correct some minor mistakes, which we
ether have discovered ourselves or which were communicated to us by careful
readers to whom we are much obliged.
We gratefully acknowledge the support of various bodies which helped
us prepare this new edition; in particular the following grants and institu
tions: EAGER (European Algebraic Geometry Research Network) and the
DFG (Deutsche Forschungsgemeinschaft) as well as the universities of Essen,
Grenoble, Hannover and Leiden for the hospitality we were offered at various
occasions. Our thanks go to those who have read and commented on parts
of the manuscript: R. Eckert, C. Erdenberger, M. Friedland, A. Gathmann,
M. Lonne, K. Ludwig, John D. McCarthy, M. Schutt, J. Spandaw and H.
Verrill. We are in particular grateful to J.-P. Demailly, L. Bonavero and A.
Teleman for all the advice they offered which helped us to understand some
of the hard analysis needed in various new parts of the book.
Special thanks also to Mme. A. Guttin-Lombard, who efficiently prepared
a major part of the book, and to Mrs. S. Guttner for the careful typing of
several chapters.
Grenoble/Erlangen/Hannover/Leiden, July 2003 W. Barth
K. Hulek
C. Peters
A. Van de Ven
Preface to the First Edition
Par une belle matinee du mois de mai,
une eIegante amazone parcourait, sur
une superbe jument alezane, les allees
fleuries de Bois de Boulogne.
(A. Camus, La Peste)
Early versions of parts of this work date back to the mid-sixties, when the
third author started to write a book on surfaces. But for several reasons, in
particular the appearance of SafareviC's book, he postponed the projects. It
was revived about ten years later, when all three authors were in Leiden. It
is impossible to cover in one book the vast and rapidly developing theory
of surfaces. Choices have to be made, with respect to content as well as
to presentation. We have chosen for a complex-analytic point of view; this
distinguishes our text from most of the existing treatments. Relations with
the case of characteristic p are not discussed.
We hope to have succeeded in writing a readable book; a book that can
be used by non-specialists. The specialist will find very little that is new to
him anyhow.
As to acknowledgements, the authors certainly have to thank the Konink
lijke Shellprijs, awarded to the third author in 1964. The numerous contacts
with colleagues from other countries made possible by that award have had
a very favourable influence on this book. Our thanks are furthermore due
to G. Angermiiller, G. Barthel, G. Fischer, G. van der Geer, N. Hitchin, D.
Husemoller, M. Reid, T. A. Springer, D. Zagier and S. Zucker. Each of them
has read some part of the manuscript and has made valuable suggestions.
Editor and printer have done an excellent job, and the Springer-Verlag
has been very generous in fulfilling all of our last-minute wishes.
We are also indebted to Mrs. W. M. Van de Ven who not only typed the
better part of the book, but also helped in preparing it for the printer, and
to Mrs. H. Dohrman who carefully typed many pages. Finally the authors
want to thank their wives for all their patience and endurance.
Erlangen/Leiden, February 1984 W. Barth
C. Peters
A. Van de Ven
Table of Contents
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 a-Process ........................................................... 34
10. Deformations of Complex Manifolds ..................................... 35
Differential Geometry of Complex Manifolds ............................. 39
11. De Rham Cohomology .................................................. 39
12. Dolbeault Cohomology .................................................. 41
13. Kahler Manifolds ....................................................... 42
14. Weight-1 Hodge Structures .............................................. 48
15. Yau's 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 Projectivity Criteria .............................. 57
20. Theorems of Bertini and Lefschetz ...................................... 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 Theorern .................................................... 66
x Table of Contents
5. The Trace Map .......................................................... 68
6. Serre Duality on an Embedded Curve .................................... 70
7. The a-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. I-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. Resolution of Surface Singularities ....................................... 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 Degreeoff*wx/s ................................................. 129
18. !itaka'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 Kahler 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 lP'2 .....•.......•......•............•.......•..••.. 185
2. Complete Intersections .................................................. 187
3. Tori of Dimension 2 ..................................................... 188
Fibre Bundles ............................................................. 189
4. Ruled Surfaces .......................................................... 189
Table of Contents XI
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 ((;! .................................. 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 Inequality c~ ::; 3C2 ••••••••••••••••••••••••••••••••••••••••••••••••• 275
Pluricanonical Maps ...................................................... 279
5. The Main Results ....................................................... 279
6. Proof of the Main Results ............................................... 281
7. The Exceptional Cases and the I-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
XII Table of Contents
VIII. K3-Surfaces and Enriques Surfaces ......................... 307
Introduction .............................................................. 307
1. Notation ................................................................ 307
2. The Results ............................................................. 309
K 3-Surfaces ............................................................... 310
3. Topological and Analytical Invariants ................................... 310
4. Digression on Affine Geometry over f2 .................................. 314
5. The 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 Kahler Cone under Deformations ....................... 327
10. Degenerations of Isomorphisms between K3-Surfaces ................... 329
11. The Torelli Theorems for K 3-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. Projective K3-surfaces and Mirror Symmetry .......................... 358
23. Special Curves on K3-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 Eye View ............................. 380
5. Infinitely many Homeomorphic Surfaces which are not Diffeomorphic .... 383
6. Further Results obtained by the Donaldson Method ..................... 390
Seiberg-Witten Invariants ................................................ 391
7. Introduction ............................................................ 391
8. Properties of the Invariants ............................................. 393
9. Surfaces Diffeomorphic to a Rational Surface ............................ 395
Bibliography ............................................................ 401
Notation ................................................................. 425
Index ..................................................................... 429