Kahler Metric and Moduli Spaces (eBook)

Advanced Studies in Pure Mathematics, Vol. 18.2

T. Ochiai (Herausgeber)

eBook Download: PDF

2013 | 1. Auflage
472 Seiten
Elsevier Science (Verlag)
978-1-4832-1467-2 (ISBN)

Infos

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.

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Kahler 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-Kahler 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 Kahler metrics on affine algebraic manifolds; and degenerations of Kahler-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.

Inhalt

Front Cover 1
Kähler Metric and Moduli Spaces 4
Copyright Page 5
Table of Contents 12
Foreword 8
Preface to the Present Volume 10
Chapter 1. Einstein Metrics in Complex Geometry: An Introduction 16
1. Einstein-Kähler metrics and Calabi's Conjecture 16
2. Einstein-Kähler metrics for compact Riemann surfaces 17
3. Birational moduli of algebraic surfaces of general type 18
4. Uniformization by Miyaoka-Van de Ven-Yau's inequality 19
5. Einstein-Hermitian metrics and stable vector bundles 22
References 23
Chapter 2. Einstein-Kähler Metrics with Positive Ricci Curvature 26
Introduction 26
1. Matsushima's obstruction and Kobayashi's semistability 28
2. Futaki's obstruction F 34
3. Symplectic geometry and the character F 42
4. Chern-Simons invariants and the group lifting of F 48
5. The uniqueness theorem 54
6. Existence of Einstein-Kähler metrics I 65
7. Existence of Einstein-Kähler metrics II 80
Appendix 83
References 94
Chapter 3. On Tangent Sheaves of Minimal Varieties 100
1. Terminologies 102
2. Construction of degenerate Einstein-Kähler metrics 104
3. Stability for tangent sheaves of minimal varieties 109
4. An inequality between Chern numbers 111
5. A criterion for stability 114
References 116
Chapter 4. Einstein Kähler Metrics of Negative Ricci Curvature on Open Kahler Manifolds 120
Introduction 120
1. Calabi's construction 122
2. V-manifolds 123
3. Schauder estimates 134
4. Monge-Ampère equations on V-manifolds 135
5. On quasi-projective manifolds 140
6. Miyaoka-Yau inequality 146
References 148
Chapter 5. Ricci-Flat Kähler Metrics on Affine Algebraic Manifolds and Degenerations of Kähler-Einstein K3 Surfaces 152
Abstract 152
0. Introduction 152
1. Some Kähler geometry 157
2. Ricci-flat Kähler metrics on affine algebraic manifolds 170
References 239
Chapter 6. Compact Ricci-Fiat Kähler Manifolds 244
1. Bogomolov decomposition 244
2. Deformation 247
3. Symplectic manifolds 250
4. Period map and Weil-Peterson metric 260
5. Examples 265
References 269
Chapter 7. Moduli of Einstein Metrics on a K3 Surface and Degeneration of Type I 272
Abstract 272
0. Preliminaries 272
1. The moduli of Kähler-Einstein K3 surfaces 280
2. Ricci-flat K3 surfaces with concentrated curvature 306
References 322
Chapter 8. Uniformization of Complex Surfaces 328
Abstract 328
0. Introduction 328
1. Kähler-Einstein metrics and uniformization 334
2. Holomorphic G-structures and uniformization 342
3. Numerical characterization of ball quotients for normal surfaces with branch loci 350
4. GHCS and uniformization of complex surfaces 373
5. Appendix 394
References 404
Yang-Mills Connections and Einstein - Hermitian Metrics 410
Introduction 411
Chapter 1. Notation and Preliminaries 413
1.1. Yang-Mills functional and Yang-Mills connections 413
1.2. Self-dual connections 416
1.3. Einstein-Hermitian connections 418
1.4. Moment maps 421
Chapter 2. Local Goometries of Moduli Spaces 426
2.1. Smooth structure on moduli space 426
2.2. Complex Kähler structure on moduli spaces 430
2.3. Hyperkähler structure of the moduli space 433
Chapter 3. Twistor Theory and Yang-Mills Connection 434
3.1. Self-dual connections on self-dual manifolds 434
3.2. B2-connections on quaternionic Kähler manifolds 437
Chapter 4. Analysis for Yang-Mills Equations 442
4.1. A priori estimates for Yang-Mills connections 442
4.2. Removable singularities theorem 445
4.3. Examples of convergence of Yang-Mills connections 447
Chapter 5. Existence Theorems for Yang-Mills Connections 448
5.1. Anti-self-dual connections on 4-manifolds 448
5.2. Einstein-Hermitian connections on stable bundles 453
Chapter 6. Vector Bundles over Moduli Spaces 461
6.1. Universal connections 461
6.2. Determinant line bundles 464
References 469

Produkt-Details

• Erscheint lt. Verlag: 22.10.2013
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 1-4832-1467-2 / 1483214672
• ISBN-13: 978-1-4832-1467-2 / 9781483214672