Canonical Metrics in Kähler Geometry

1 Introduction to Kähler manifolds.- 1.1 Kähler metrics.- 1.2 Curvature of Kähler metrics.- 2 Extremal Kähler metrics.- 2.1 The space of Kähler metrics.- 2.2 A brief review of Chern classes.- 2.3 Uniformization of Kähler-Einstein manifolds.- 3 Calabi-Futaki invariants.- 3.1 Definition of Calabi-Futaki invariants.- 3.2 Localization formula for Calabi-Futaki invariants.- 4 Scalar curvature as a moment map.- 5 Kähler-Einstein metrics with non-positive scalar curvature.- 5.1 The Calabi-Yau Theorem.- 5.2 Kähler-Einstein metrics for manifolds with c1(M) 0.- 6 Kähler-Einstein metrics with positive scalar curvature.- 6.1 A variational approach.- 6.2 Existence of Kähler-Einstein metrics.- 6.3 Examples.- 7 Applications and generalizations.- 7.1 A manifold without Kähler-Einstein metric.- 7.2 K-energy and metrics of constant scalar curvature.- 7.3 Relation to stability.

"This little monograph offers an essentially self-contained introduction to the theory of canonical Kähler metrics on complex manifolds. ...The author presents some advanced topics which are hard [to] find elsewhere... This graduate course on Kähler-Einstein metrics can be recommended to all those interested in recent developments within complex differential geometry."

--Publicationes Mathematicae

"This monograph includes an essentially self-contained introduction to the theory of canonical Kähler metrics on complex manifolds."

--Zentralblatt Math