Kähler-Einstein Metrics and Integral Invariants

Akito Futaki (auth.)9780387192505, 0-387-19250-6

These notes present very recent results on compact Kähler-Einstein manifolds of positive scalar curvature. A central role is played here by a Lie algebra character of the complex Lie algebra consisting of all holomorphic vector fields, which can be intrinsically defined on any compact complex manifold and becomes an obstruction to the existence of a Kähler-Einstein metric. Recent results concerning this character are collected here, dealing with its origin, generalizations, sufficiency for the existence of a Kähler-Einstein metric and lifting to a group character. Other related topics such as extremal Kähler metrics studied by Calabi and others and the existence results of Tian and Yau are also reviewed. As the rudiments of Kählerian geometry and Chern-Simons theory are presented in full detail, these notes are accessible to graduate students as well as to specialists of the subject.

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Table of contents :
Introduction….Pages 1-6
Preliminaries….Pages 7-30
Kähler-Einstein metrics and extremal Kähler metrics….Pages 31-45
The character f and its generalization to Kählerian invariants….Pages 46-55
The character f as an obstruction….Pages 56-67
The character f as a classical invariant….Pages 68-86
Lifting f to a group character….Pages 87-98
The character f as a moment map….Pages 99-112
Aubin’s approach and related results….Pages 113-132