Ochiai Kähler Metric and Moduli Spaces

Advanced Studies in Pure Mathematics, Vol. 18.2
1. Auflage 2013
ISBN: 978-1-4832-1467-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 472 Seiten, Web PDF

K„hler Metric and Moduli Spaces, Volume 18-II covers survey notes from the expository lectures given during the seminars in the academic year of 1987 for graduate students and mature mathematicians who were not experts on the topics considered during the sessions about partial differential equations. The book discusses basic facts on Einstein metrics in complex geometry; Einstein-K„hler metrics with positive or non-positive Ricci curvature; Yang-Mills connections; and Einstein-Hermitian metrics. The text then describes the tangent sheaves of minimal varieties; Ricci-Flat K„hler metrics on affine algebraic manifolds; and degenerations of K„hler-Einstein. The moduli of Einstein metrics on a K3 surface and degeneration of Type I and the uniformization of complex surfaces are also considered. Mathematicians and graduate students taking differential and analytic geometry will find the book useful.

Autoren/Hrsg.

Ochiai, T.

Inhaltsverzeichnis

1;Front Cover;1
2;Kähler Metric and Moduli Spaces;4
3;Copyright Page;5
4;Table of Contents;12
5;Foreword;8
6;Preface to the Present Volume;10
7;Chapter 1. Einstein Metrics in Complex Geometry: An Introduction;16
7.1;1. Einstein-Kähler metrics and Calabi's Conjecture;16
7.2;2. Einstein-Kähler metrics for compact Riemann surfaces;17
7.3;3. Birational moduli of algebraic surfaces of general type;18
7.4;4. Uniformization by Miyaoka-Van de Ven-Yau's inequality;19
7.5;5. Einstein-Hermitian metrics and stable vector bundles;22
7.6;References;23
8;Chapter 2. Einstein-Kähler Metrics with Positive Ricci Curvature;26
8.1;Introduction;26
8.2;1. Matsushima's obstruction and Kobayashi's semistability;28
8.3;2. Futaki's obstruction F;34
8.4;3. Symplectic geometry and the character F;42
8.5;4. Chern-Simons invariants and the group lifting of F;48
8.6;5. The uniqueness theorem;54
8.7;6. Existence of Einstein-Kähler metrics I;65
8.8;7. Existence of Einstein-Kähler metrics II;80
8.9;Appendix;83
8.10;References;94
9;Chapter 3. On Tangent Sheaves of Minimal Varieties;100
9.1;1. Terminologies;102
9.2;2. Construction of degenerate Einstein-Kähler metrics;104
9.3;3. Stability for tangent sheaves of minimal varieties;109
9.4;4. An inequality between Chern numbers;111
9.5;5. A criterion for stability;114
9.6;References;116
10;Chapter 4. Einstein Kähler Metrics of Negative Ricci Curvature on Open Kahler Manifolds;120
10.1;Introduction;120
10.2;1. Calabi's construction;122
10.3;2. V-manifolds;123
10.4;3. Schauder estimates;134
10.5;4. Monge-Ampère equations on V-manifolds;135
10.6;5. On quasi-projective manifolds;140
10.7;6. Miyaoka-Yau inequality;146
10.8;References;148
11;Chapter 5. Ricci-Flat Kähler Metrics on Affine Algebraic Manifolds and Degenerations of Kähler-Einstein K3 Surfaces;152
11.1;Abstract;152
11.2;0. Introduction;152
11.3;1. Some Kähler geometry;157
11.4;2. Ricci-flat Kähler metrics on affine algebraic manifolds;170
11.5;References;239
12;Chapter 6. Compact Ricci-Fiat Kähler Manifolds;244
12.1;1. Bogomolov decomposition;244
12.2;2. Deformation;247
12.3;3. Symplectic manifolds;250
12.4;4. Period map and Weil-Peterson metric;260
12.5;5. Examples;265
12.6;References;269
13;Chapter 7. Moduli of Einstein Metrics on a K3 Surface and Degeneration of Type I;272
13.1;Abstract;272
13.2;0. Preliminaries;272
13.3;1. The moduli of Kähler-Einstein K3 surfaces;280
13.4;2. Ricci-flat K3 surfaces with concentrated curvature;306
13.5;References;322
14;Chapter 8. Uniformization of Complex Surfaces;328
14.1;Abstract;328
14.2;0. Introduction;328
14.3;1. Kähler-Einstein metrics and uniformization;334
14.4;2. Holomorphic G-structures and uniformization;342
14.5;3. Numerical characterization of ball quotients for normal surfaces with branch loci;350
14.6;4. GHCS and uniformization of complex surfaces;373
14.7;5. Appendix;394
14.8;References;404
15;Yang-Mills Connections and Einstein - Hermitian Metrics;410
15.1;Introduction;411
15.2;Chapter 1. Notation and Preliminaries;413
15.2.1;1.1. Yang-Mills functional and Yang-Mills connections;413
15.2.2;1.2. Self-dual connections;416
15.2.3;1.3. Einstein-Hermitian connections;418
15.2.4;1.4. Moment maps;421
15.3;Chapter 2. Local Goometries of Moduli Spaces;426
15.3.1;2.1. Smooth structure on moduli space;426
15.3.2;2.2. Complex Kähler structure on moduli spaces;430
15.3.3;2.3. Hyperkähler structure of the moduli space;433
15.4;Chapter 3. Twistor Theory and Yang-Mills Connection;434
15.4.1;3.1. Self-dual connections on self-dual manifolds;434
15.4.2;3.2. B2-connections on quaternionic Kähler manifolds;437
15.5;Chapter 4. Analysis for Yang-Mills Equations;442
15.5.1;4.1. A priori estimates for Yang-Mills connections;442
15.5.2;4.2. Removable singularities theorem;445
15.5.3;4.3. Examples of convergence of Yang-Mills connections;447
15.6;Chapter 5. Existence Theorems for Yang-Mills Connections;448
15.6.1;5.1. Anti-self-dual connections on 4-manifolds;448
15.6.2;5.2. Einstein-Hermitian connections on stable bundles;453
15.7;Chapter 6. Vector Bundles over Moduli Spaces;461
15.7.1;6.1. Universal connections;461
15.7.2;6.2. Determinant line bundles;464
16;References;469