An introduction to extremal Kähler metrics

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 Kähler geometry. This book gives an introduction to the study of extremal Kähler 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 Kähler 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 Kähler-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 Kähler metrics on blow-ups. 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. -- Provided by Publisher In revised lecture notes he used for a graduate topics course he taught in the spring of 2012, Szkelyhidi introduces ideas surrounding recent developments in extreme Khler metrics. The main ideas from Khler geometry and analysis should be accessible with just a knowledge of Riemannian geometry, and graduate-level analysis, he says, but the sections on geometric invariant theory and K-stability would be difficult to follow without a background in complex algebraic geometry. Annotation 2014 Ringgold, Inc., Portland, OR (protoview.com) Cover Title page Contents Preface Introduction Kähler geometry Analytic preliminaries Kähler-Einstein metrics Extremal metrics Moment maps and geometric invariant theory K-stability The Bergman kernel CscK metrics on blow-ups Bibliography Index Back Cover