An Introduction to Extremal Kahler Metrics

Author: Gabor Szekelyhidi Series: Graduate Studies in Mathematics

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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.

Critic Reviews

“This is an important book, in a rapidly-developing area, that brings the specialist or graduate student working on K”

ähler geometry to the frontiers of today research. It is not a self-contained textbook. The pre-requisites in geometric invariant theory, for example, would require some devotion from a potential reader grounded on Riemannian geometry; vice-versa, a reader brought-up in algebraic geometry would have to make an effort to follow the part on analysis or differential geometry. The rewards for these efforts justify everything: the book is well organized, and when it sketches an argument, there are precise pointers to the literature for full details." - MAA Reviews

"Very well written, the book provides a survey of extremal ahler metrics and it promotes to tackle other topics." - Zentralblatt Math