Ochiai Recent Topics in Differential and Analytic Geometry

1. Auflage 2014
ISBN: 978-1-4832-1468-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 18, 460 Seiten, Web PDF

Reihe: Advanced Studies in Pure Mathematics

Advanced Studies in Pure Mathematics, Volume 18-I: Recent Topics in Differential and Analytic Geometry presents the developments in the field of analytical and differential geometry. This book provides some generalities about bounded symmetric domains. Organized into two parts encompassing 12 chapters, this volume begins with an overview of harmonic mappings and holomorphic foliations. This text then discusses the global structures of a compact K„hler manifold that is locally decomposable as an isometric product of Ricci-positive, Ricci-negative, and Ricci-flat parts. Other chapters consider the most recognized non-standard examples of compact homogeneous Einstein manifolds constructed via Riemannian submersions. This book discusses as well the natural compactification of the moduli space of polarized Einstein-K„hler orbitfold with a given Hilbert polynomials. The final chapter deals with solving a degenerate Monge-AmpŠre equation by constructing a family of Einstein-K„hler metrics on the smooth part of minimal varieties of general kind. This book is a valuable resource for graduate students and pure mathematicians.

Autoren/Hrsg.

Ochiai, T.

Inhaltsverzeichnis

1;Front Cover;1
2;Recent Topics in Differential and Analytic Geometry;4
3;Copyright Page;5
4;Table of Contents;12
5;Foreword;8
6;Preface to the Present Volume;10
7;CONTENTS OF VOLUME 18-II;14
8;Part I;16
8.1;Section A: Topics in Complex Differential Geometry;18
8.1.1;TABLE OF CONTENTS;22
8.1.2;Lecture I. Harmonic Mappings and Holomorphic Foliations;24
8.1.2.1;(1.1) Some generalities about bounded symmetric domains;24
8.1.2.2;(1.2) Some remarks on quotients of bounded symmetric domains;26
8.1.2.3;(1.3) Local rigidity for compact quotients of bounded symmetric domains;27
8.1.2.4;(1.4) Mostow's Strong Rigidity Theorem;28
8.1.2.5;(1.5) Harmonic mappings into compact manifolds of non-positive curvature;28
8.1.2.6;(1.6) Siu's Strong Rigidity Theorem for Kähler manifolds;30
8.1.2.7;(1.7) Irreducible compact quotients of the polydisc;33
8.1.2.8;(1.8) Holomorphic foliations arising from harmonic maps into irreducible compact quotients of the polydisc;36
8.1.2.9;(1.9) Strong rigidity for quotients of the ball of finite volume;41
8.1.2.10;(1.10) Strong rigidity for irreducible quotients of the polydisc of finite volume;46
8.1.3;Lecture II. Uniformization of Compact Kähler Manifolds of Nonnegative Curvature;51
8.1.3.1;(2.1) Hermitian symmetric manifolds of compact type;51
8.1.3.2;(2.2) Bochner formulas and the maximum principle on tensors;55
8.1.3.3;(2.3) Existence of rational curves and the Hartshorne conjecture;59
8.1.3.4;(2.4) Stable harmonic mappings and the Frankel Conjecture;62
8.1.3.5;(2.5) Evolution of Kähler metric by the parabolic Einstein equation;68
8.1.3.6;(2.6) Compact Kähler-Einstein manifolds of non-negative holomorphic bisectional curvature;74
8.1.3.7;(2.7) Characterization of locally symmetric spaces of rank = 2 by the holonomy group;76
8.1.3.8;(2.8) The space of minimal rational curves on Hermitian symmetric manifolds of compact type;78
8.1.3.9;(2.9) Holonomy-invariance of the space of tangents to minimal rational curves;80
8.1.4;Lecture III. Compactification of Complete Kähler Manifolds of Positive Curvature;84
8.1.4.1;The Frankel Conjecture for open manifolds;84
8.1.4.2;(3.2) Techniques of L2-estimate of . for the embedding problem;86
8.1.4.3;(3.3) Siegel's Theorem for the field of rational functions;90
8.1.4.4;(3.4) L2-estimates for the ideal problem and quasi-surjectivity;94
8.1.4.5;(3.5) Desingularizing the quasi-surjective embedding Fo;96
8.1.4.6;(3.6) Completion to a proper holomorphic embedding;101
8.1.4.7;(3.7) Embedding complete Kähler manifolds of positive Ricci curvature;102
8.1.4.8;(3.8) Characterization of affine-algebraic varieties;106
8.1.5;Lecture IV. Compactification of Complete Kähler-Einstein Manifolds of Finite Volume;110
8.1.5.1;(4.1) Compactification of arithmetic quotients of bounded symmetric domains and generalizations;110
8.1.5.2;(4.2) Siegel's Theorem on pseudoconcave manifolds;112
8.1.5.3;(4.3) Embedding certain pseudoconcave manifolds;115
8.1.5.4;(4.4) Scheme for compactifying certain pseudoconcave manifolds of negative Ricci curvature;117
8.1.5.5;(4.5) Existence theorems on complete Kähler-Einstein metrics on non-compact manifolds;118
8.1.5.6;(4.6) L2-Riemann-Roch inequality of Nadel-Tsuji;123
8.1.5.7;(4.7) A local compactification theorem on bounded domains;128
8.1.5.8;(4.8) Compactifying complete Kähler manifolds of finite volume with pinched strictly negative sectional curvature;132
8.1.5.9;(4.9) Siegel's Theorem and Bézout estimates on Kähler manifolds of finite volume;134
8.1.5.10;(4.10) Estimates of Gauss-Bonnet integrals and a criterion of Zariski-openness using the Kontinuitätssatz;139
8.1.5.11;(4.11) Zariski-openness using plurisubharmonic potentials;143
8.1.5.12;(4.12) Bézout estimates on complete Kähler manifolds of positive Ricci curvature;148
8.1.6;References;149
8.2;Section B: Hausdorff Convergence of Riemannian Manifolds and Its Applications;160
8.2.1;Preface;161
8.2.2;Chapter I. HausdorfF Convergence;163
8.2.2.1;1. Definition and elementary properties;163
8.2.2.2;2. Precompactness theorem;165
8.2.2.3;3. Rigidity theorem;168
8.2.2.4;4. Convergence theorem;173
8.2.2.5;5. Smoothing Riemannian metrics;175
8.2.2.6;6. Pointed and equivariant Hausdorff distances;178
8.2.3;Chapter II. Collapsing Riemannian Manifolds;181
8.2.3.1;7. Pseudo fundamental group;181
8.2.3.2;8. Almost flat manifolds I;186
8.2.3.3;9. Almost flat manifolds II;189
8.2.3.4;10. Examples;195
8.2.3.5;11. A compactification of M(n, D);201
8.2.3.6;12. Fibre bundle theorem;204
8.2.3.7;13. Margulis' Lemma;208
8.2.4;Chapter III. Applications;210
8.2.4.1;14. Finiteness theorems;210
8.2.4.2;15. Pinching Theorems;214
8.2.4.3;16. Aspherical manifolds;223
8.2.4.4;17. Minimal volume;227
8.2.4.5;18. Telescope;237
8.2.4.6;19. T- and F- structures;243
8.2.5;References;249
9;Part II;256
9.1;Chapter 1. Compact Kähler Manifolds with Parallel Ricci Tensor;258
9.1.1;Introduction;258
9.1.2;1. Ricci-positive directions;259
9.1.3;2. Bochner's vanishing theorem;260
9.1.4;3. Ricci-negative directions;261
9.1.5;4. Ricci-flat directions;261
9.1.6;Appendix;267
9.1.7;References;268
9.2;Chapter 2. Eta Invariants and Automorphisms of Compact Complex Manifolds;270
9.2.1;1. Introduction;270
9.2.2;2. Signature operators and eta invariants;272
9.2.3;3. F and eta invariants;277
9.2.4;4. Examples;284
9.2.5;References;288
9.3;Chapter 3. Poincaré Bundle and Chern Classes;290
9.3.1;1. Theorems;290
9.3.2;2. The Poincaré bundle;294
9.3.3;3. The Chern character formula;295
9.3.4;4. The compactification of the moduli space;296
9.3.5;5. Einstein-Hermitian bundles and Riemann surfaces;297
9.3.6;6. Hodge structure and the determinant bundle;298
9.3.7;References;299
9.4;Chapter 4. Harmonic Functions with Growth Conditions on a Manifold of Asymptotically Nonnegative Curvature II;302
9.4.1;0. Introduction;302
9.4.2;1. Proof of Theorem A;305
9.4.3;2. Proof of Theorem B;308
9.4.4;3. Some other results;314
9.4.5;References;318
9.5;Chapter 5. Homogeneous Einstein Metrics On Certain Kähler C-Spaces;322
9.5.1;0. Introduction;322
9.5.2;1. Preliminaries;323
9.5.3;2. Kähler C-spaces;325
9.5.4;3. G-invariant Einstein metrics;330
9.5.5;4. G-invariant complex structures;333
9.5.6;References;339
9.6;Chapter 6. An Application of Kähler-Einstein Metrics to Singularities of Plane Curves;340
9.6.1;References;345
9.7;Chapter 7. On Rotationally Symmetric Hamilton's Equation for Kähler-Einstein Metrics;346
9.7.1;0. Introduction;346
9.7.2;1. Hamilton's equation and quasi-Einstein metrics;347
9.7.3;2. Rotationally symmetric equations;348
9.7.4;3. Convergence of the solution;351
9.7.5;4. Convergence of the metric;353
9.7.6;References;356
9.8;Chapter 8. An Algebraic Character associated with the Poisson Brackets;358
9.8.1;0. Introduction;358
9.8.2;1. Notation and conventions;359
9.8.3;2. Poisson brackets for complex manifolds;364
9.8.4;3. Factorization of the character Tc,h;367
9.8.5;4. The moment map;370
9.8.6;5. (C*)r-actions and the theorem of stationary phase;372
9.8.7;6. Ga-actions and the character Tc;374
9.8.8;References;376
9.9;Chapter 9. Compactification of the Moduli Space of Einstein-Kähler Orbifolds;378
9.9.1;Introduction;378
9.9.2;1. Construction of the compactification;379
9.9.3;2. The moduli space of compact Riemann surfaces;383
9.9.4;3. Proof of Theorem 2.6;388
9.9.5;4. Concluding remarks;396
9.9.6;References;402
9.10;Chapter 10. Self-Duality of ALE Ricci-Flat 4-Manifolds and Positive Mass Theorem;404
9.10.1;Introduction;404
9.10.2;1. ALE gravitational instantons;405
9.10.3;2. Hausdorff convergence of Einstein manifolds;407
9.10.4;3. Positive mass conjecture;408
9.10.5;4. Inequalities between characteristic numbers;410
9.10.6;References;414
9.11;Chapter 11. Compactification of Moduli Spaces of Einstein-Hermitian Connections for Null-Correlation Bundles;416
9.11.1;0. Introduction;416
9.11.2;1. Notation, conventions and preliminaries;418
9.11.3;2. Construction of Einstein-Hermitian connections;421
9.11.4;3. Injectivity of the mapping .;425
9.11.5;4. The moduli space of B2-connections on (V, hy);430
9.11.6;5. Compactification of .(S);432
9.11.7;References;434
9.12;Chapter 12. Einstein-Kähler Metrics on Minimal Varieties of General Type and an Inequality between Chern Numbers;436
9.12.1;1. A degenerate Monge-Ampère equation;436
9.12.2;2. A uniform bound for .d;437
9.12.3;3. The second order estimate of .d;439
9.12.4;4. Certain examples;443
9.12.5;5. Convergence of Einstein-Kähler metrics;444
9.12.6;6. An inequality between Chern numbers;451
9.12.7;7. Local decomposition theorem;456
9.12.8;8. Global decomposition theorem;459
9.12.9;References;461