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.
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