Preface Introduction Chapter 1.Kahler Geometry §1.1.Complex manifolds §1.2.Almost complex structures §1.3.Hermitian and Kahler metrics §1.4.Covariant derivatives and curvature §1.5.Vector bundles §1.6.Connections and curvature of line bundles §1.7.Line bundles and projective embeddings Chapter 2.Analytic Preliminaries §2.1.Harmonic functions on Rn §2.2.Elliptic differential operators §2.3.Schauder estimates §2.4.The Laplace operator on Kahler manifolds Chapter 3.Kaihler-Einstein Metrics §3.1.The strategy §3.2.The CO-and C2-estimates §3.3.The C3-and higher-order estimates §3.4.The case cl(M)=0 §3.5.The case Cl(M)＞0 §3.6.Futher reading Chapter 4.Extremal Metrics §4.1.The Calabi functional §4.2.Holomorphic vector fields and the Futaki invariant §4.3.The Mabuchi functional and geodesics §4.4.Extremal metrics on a ruled surface §4.5.Toric manifolds Chapter 5.Moment Maps and Geometric Invariant Theory §5.1.Moment maps §5.2.Geometric invariant theory (GIT) §5.3.The Hilbert-Mumford criterion §5.4.The Kempf-Ness theorem §5.5.Relative stability Chapter 6.K-stability §6.1.The scalar curvature as a moment map §6.2.The Hilbert polynomial and flat limits §6.3.Test-configurations and K-stability §6.4.Automorphisms and relative K-stability §6.5.Relative K-stability of a ruled surface §6.6.Filtrations §6.7.Toric varieties Chapter 7.The Bergman Kernel §7.1.The Bergman kernel §7.2.Proof of the asymptotic expansion §7.3.The equivariant Bergman kernel §7.4.The algebraic and geometric Futaki invariants §7.5.Lower bounds on the Calabi functional §7.6.The partial C0-estimate Chapter 8.CscK Metrics on Blow-ups
§8.1.The basic strategy §8.2.Analysis in weighted spaces §8.3.Solving the non-linear equation when n＞2 §8.4.The case when n=2 §8.5.The case when M admits holomorphic vector fields §8.6.K-stability of cscK manifolds Bibliography Index
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