Compact complex surfaces

W. Barth, C. Peters, A. Van de Ven

Contents: Introduction. - Standard Notations. - Preliminaries. - Curves on Surfaces. - Mappings of Surfaces. - Some General Properties of Surfaces. - Examples. - The Enriques-Kodaira Classification. - Surfaces of General Type. - K3-Surfaces and Enriques Surfaces. - Bibliography. - Subject Index.

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  • Historical Note.- References.- The Content of the Book.- Standard Notations.- I. Preliminaries.- Topology and Algebra.- 1. Notations and Basic Facts.- 2. Some Properties of Bilinear forms.- 3. Vector Bundles, Characteristic Classes and the Index Theorem.- Complex Manifolds.- 4. Basic Concepts and Facts.- 5. Holomorphic Vector Bundles, Serre Duality and the Riemann-Roch Theorem.- 6. Line Bundles and Divisors.- 7. Algebraic Dimension and Kodaira Dimension.- General Analytic Geometry.- 8. Complex Spaces.- 9. The ?-Process.- 10. Deformations of Complex Manifolds.- Differential Geometry of Complex Manifolds.- 11. De Rham Cohomology.- 12. Dolbeault Cohomology.- 13. Kahler Manifolds.- 14. Weight-1 Hodge Structures.- 15. Yau's Results on Kahler-Einstein Metrics.- Coverings.- 16. Ramification.- 17. Cyclic Coverings.- 18. Covering Tricks.- Projective-Algebraic Varieties.- 19. GAGA Theorems and Projectivity Criteria.- 20. Theorems of Bertini and Lefschetz.- II. Curves on Surfaces.- Embedded Curves.- 1. Some Standard Exact Sequences.- 2. The Picard-Group of an Embedded Curve.- 3. Riemann-Roch for an Embedded Curve.- 4. The Residue Theorem.- 5. The Trace Map.- 6. Serre Duality on an Embedded Curve.- 7. The ?-Process.- 8. Simple Singularities of Curves.- Intersection Theory.- 9. Intersection Multiplicities.- 10. Intersection Numbers.- 11. The Arithmetical Genus of an Embedded Curve.- 12. 1-Connected Divisors.- III. Mappings of Surfaces.- Bimeromorphic Geometry.- 1. Bimeromorphic Maps.- 2. Exceptional Curves.- 3. Rational Singularities.- 4. Exceptional Curves of the First Kind.- 5. Hirzebruch-Jung Singularities.- 6. Resolution of Surface Singularities.- 7. Singularities of Double Coverings, Simple Singularities of Surfaces.- Fibrations of Surfaces.- 8. Generalities on Fibrations.- 9. The n-th Root Fibration.- 10. Stable Fibrations.- 11. Direct Image Sheaves.- 12. Relative Duality.- The Period Map of Stable Fibrations.- 13. Period Matrices of Stable Curves.- 14. Topological Monodromy of Stable Fibrations.- 15. Monodromy of the Period Matrix.- 16. Extending the Period Map.- 17. The Degree of f* ?X/S.- 18. Iitaka's Conjecture C2, 1.- IV. Some General Properties of Surfaces.- 1. Meromorphic Maps Associated to Line Bundles.- 2. Hodge Theory on Surfaces.- 3. Deformations of Surfaces.- 4. Some Inequahties for Hodge Numbers.- 5. Projectivity of Surfaces.- 6. Surfaces of Algebraic Dimension Zero.- 7. Almost-Complex Surfaces without any Complex Structure.- 8. The Vanishing Theorems of Ramanujam and Mumford.- V. Examples.- Some Classical Examples.- 1. The Projective Plane ?2.- 2. Complete Intersections.- 3. Tori of Dimension 2.- Fibre Bundles.- 4. Ruled Surfaces.- 5. Elliptic Fibre Bundles.- 6. Higher Genus Fibre Bundles.- Elliptic Fibrations.- 7. Kodaira's Table of Singular Fibres.- 8. Stable Fibrations.- 9. The Jacobian Fibration.- 10. Stable Reduction.- 11. Classification.- 12. Invariants.- 13. Logarithmic Transformations.- Kodaira Fibrations.- 14. Kodaira Fibrations.- Finite Quotients.- 15. The Godeaux Surface.- 16. Kummer Surfaces.- 17. Quotients of Products of Curves.- Infinite Quotients.- 18. Hopf Surfaces.- 19. Inoue Surfaces.- 20. Quotients of Bounded Domains in C2.- 21. Hilbert Modular Surfaces.- Double Coverings.- 22. Invariants.- 23. An Enriques Surface.- VI. The Enriques-Kodaira Classification.- 1. Statement of the Main Result.- 2. The Castelnuovo Criterion.- 3. The Case a(X) = 2.- 4. The Case a(X) = 1.- 5. The Case a (X) = 0.- 6. The Final Step.- 7. Deformations.- VII. Surfaces of General Type.- Preliminaries.- 1. Introduction.- 2. Some General Theorems.- Two Inequalities.- 3. Noether's Inequality.- 4. The Inequality c12 ? 3c2.- Pluricanonical Maps.- 5. The Main Results.- 6. Connectedness Properties of Pluricanonical Divisors.- 7. Proof of the Main Results.- 8. The Exceptional Cases and the 1-canonical Map.- Surfaces with Given Chern Numbers.- 9. The Geography of Chern Numbers.- 10. Surfaces on the Noether Lines.- 11. Surfaces with q = pg = 0.- VIII. K3-Surfaces and Enriques Surfaces.- 1. Notations.- 2. The Results.- K3-Surfaces.- 3. Topological and Analytical Invariants.- 4. Digression on Affine Geometry over ?2.- 5. The Picard Lattice of Kummer Surfaces.- 6. The Torelli Theorem for Kummer Surfaces.- 7. The Local Torelli Theorem for K3-Surfaces.- 8. A Density Theorem.- 9. Behaviour of the Kahler Cone Under Deformations.- 10. Degenerations of Isomorphisms Between Kahler K3-Surfaces.- 11. The Torelli Theorems for Kahler K3-Surfaces.- 12. Construction of Moduli Spaces.- 13. Digression on Quaternionic Structures.- 14. Surjectivity of the Period Map
  • Every K3-Surface is Kahlerian.- Enriques Surfaces.- 15. Topological and Analytic Invariants.- 16. Divisors on an Enriques Surface Y.- 17. Elliptic Pencils.- 18. Double Coverings of Quadrics.- 19. The Period Map.- 20. The Period Domain for Enriques Surfaces.- 21. Global Properties of the Period Map.- Notations.

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書名 Compact complex surfaces
著作者等 Barth, Wolf
Peters, C.
Ven, A. van de
Peters C. (Universite de Grenoble)
Van De Ven A.
Barth W.
シリーズ名 Ergebnisse der Mathematik und ihrer Grenzgebiete
出版元 Springer-Verlag
刊行年月 1984
版表示 Softcover reprint of the original 1st ed. 1984
ページ数 x, 304 p.
大きさ 25 cm
ISBN 0387121722
NCID BA0017553X
※クリックでCiNii Booksを表示
言語 英語
出版国 ドイツ

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