Compact Complex Surfaces

ISBN-13: 9783642967566 / Angielski / Miękka / 2012 / 304 str.

W. Barth; C. Peters; A. Van De Ven

Opis

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.

Spis treści

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. Kähler Manifolds.- 14. Weight-1 Hodge Structures.- 15. Yau’s Results on Kähler-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 Kähler Cone Under Deformations.- 10. Degenerations of Isomorphisms Between Kähler K3-Surfaces.- 11. The Torelli Theorems for Kähler K3-Surfaces.- 12. Construction of Moduli Spaces.- 13. Digression on Quaternionic Structures.- 14. Surjectivity of the Period Map; Every K3-Surface is Kählerian.- 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.