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An Introduction to Extremal Kahler Metrics
发布日期:2015-12-17  浏览

An Introduction to Extremal Kahler Metrics

[Book Description]

A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result, in the setting of Kahler geometry. This book gives an introduction to the study of extremal Kahler metrics and in particular to the conjectural picture relating the existence of extremal metrics on projective manifolds to the stability of the underlying manifold in the sense of algebraic geometry. The book addresses some of the basic ideas on both the analytic and the algebraic sides of this picture. An overview is given of much of the necessary background material, such as basic Kahler geometry, moment maps, and geometric invariant theory. Beyond the basic definitions and properties of extremal metrics, several highlights of the theory are discussed at a level accessible to graduate students: Yau's theorem on the existence of Kahler-Einstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and Arezzo-Pacard's existence theorem for constant scalar curvature Kahler metrics on blow-ups.

[Table of Contents]
Preface                                            xi
Introduction                                       xiii
    Chapter 1 Kahler Geometry                      1   (22)
      § 1.1 Complex manifolds                 1   (3)
      § 1.2 Almost complex structures         4   (2)
      § 1.3 Hermitian and Kahler metrics      6   (4)
      § 1.4 Covariant derivatives and         10  (3)
      curvature
      § 1.5 Vector bundles                    13  (3)
      § 1.6 Connections and curvature of      16  (3)
      line bundles
      § 1.7 Line bundles and projective       19  (4)
      embeddings
    Chapter 2 Analytic Preliminaries               23  (12)
      § 2.1 Harmonic functions on Rn          23  (2)
      § 2.2 Elliptic differential operators   25  (1)
      § 2.3 Schauder estimates                26  (6)
      § 2.4 The Laplace operator on Kahler    32  (3)
      manifolds
    Chapter 3 Kahler-Einstein Metrics              35  (22)
      § 3.1 The strategy                      36  (4)
      § 3.2 The C0- and C2-estimates          40  (4)
      § 3.3 The C3- and higher-order          44  (3)
      estimates
      § 3.4 The case c1{M) = 0                47  (4)
      § 3.5 The case c1{M) > 0             51  (2)
      § 3.6 Futher reading                    53  (4)
    Chapter 4 Extremal Metrics                     57  (28)
      § 4.1 The Calabi functional             57  (5)
      § 4.2 Holomorphic vector fields and     62  (5)
      the Futaki invariant
      § 4.3 The Mabuchi functional and        67  (4)
      geodesies
      § 4.4 Extremal metrics on a ruled       71  (5)
      surface
      § 4.5 Toric manifolds                   76  (9)
    Chapter 5 Moment Maps and Geometric            85  (20)
    Invariant Theory
      § 5.1 Moment maps                       85  (5)
      § 5.2 Geometric invariant theory        90  (4)
      (GIT)
      § 5.3 The Hilbert-Mumford criterion     94  (3)
      § 5.4 The Kempf-Ness theorem            97  (4)
      § 5.5 Relative stability                101 (4)
    Chapter 6 K-stability                          105 (24)
      § 6.1 The scalar curvature as a         105 (4)
      moment map
      § 6.2 The Hilbert polynomial and        109 (2)
      flat limits
      § 6.3 Test-configurations and           111 (4)
      K-stability
      § 6.4 Automorphisms and relative        115 (1)
      K-stability
      § 6.5 Relative K-stability of a         116 (4)
      ruled surface
      § 6.6 Filtrations                       120 (4)
      § 6.7 Toric varieties                   124 (5)
    Chapter 7 The Bergman Kernel                   129 (24)
      § 7.1 The Bergman kernel                129 (3)
      § 7.2 Proof of the asymptotic           132 (6)
      expansion
      § 7.3 The equivariant Bergman kernel    138 (2)
      § 7.4 The algebraic and geometric       140 (2)
      Futaki invariants
      § 7.5 Lower bounds on the Calabi        142 (8)
      functional
      § 7.6 The partial C°-estimate       150 (3)
    Chapter 8 CscK Metrics on Blow-ups             153 (32)
      § 8.1 The basic strategy                153 (6)
      § 8.2 Analysis in weighted spaces       159 (13)
      § 8.3 Solving the non-linear            172 (2)
      equation when n > 2
      § 8.4 The case when n = 2               174 (4)
      § 8.5 The case when M admits            178 (3)
      holomorphic vector fields
      § 8.6 K-stability of cscK manifolds     181 (4)
Bibliography                                       185 (6)
Index                                              191
 

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