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 This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any otherway, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science+ Business Media springeronline.com @ 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