An introduction to extremal Kähler metrics. (English) Zbl 1298.53002

Székelyhidi, Gábor

Graduate Studies in Mathematics 152. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1047-6/hbk). xvi, 192 p. (2014).

The study of extremal Kähler metrics is an important problem in Kähler geometry. Extremal metrics appeared by Calabi’s works (1980), as an attempt to find canonical Kähler metrics on Kähler manifolds as critical points of an energy functional given by the \(L^2\)-norm of the curvature of a metric. Examples of extremal Kähler metrics are provided by the Kähler-Einstein metrics and by Kähler metrics of constant scalar curvature, (briefly, cscK metrics), but extremal metrics do not always exist. Therefore, the existence of the extremal Kähler metrics was imposed on the projective manifolds, related to the stability of the manifold in the algebraic geometry meaning.
The general aim of the author is to present the main notions about extremal Kähler metrics from the analytic and algebraic point of view. In particular, under the hypothesis that \(M\) is a compact Kähler manifold with a cscK metric in \(c_1(L)\), where \(L \rightarrow M\) is an ample line bundle and \(c_1(L)\) is the first Chern class of \(L\), the author gives a complete proof for the following result: If \(M\) admits a cscK metric in \(c_1(L)\) and \(M\) has a non-trivial holomorphic vector fields then the pair \((M,L)\) is \(K\)-stable.
The structure of the book is the following:
Chapter 1: Kähler geometry
Chapter 2: Analytic preliminaries
Chapter 3: Kähler-Einstein metrics
Chapter 4: Extremal metrics
Chapter 5: Moment maps and geometric invariant theory
Chapter 6: \(K\)-stability
Chapter 7: The Bergman kernel
Chapter 8: CscK metrics on blow-ups
Very well written, the book provides a survey of extremal Kähler metrics and it promotes to tackle other topics, (e.g. Calabi flow, Kähler-Ricci flow, etc.).

Reviewer: Nicoleta Aldea (Brasov)

MSC:

53-02	Research exposition (monographs, survey articles) pertaining to differential geometry
53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C26	Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53B35	Local differential geometry of Hermitian and Kählerian structures
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
32Q20	Kähler-Einstein manifolds
14L24	Geometric invariant theory

Keywords:

extremal Kähler metric; Kähler-Einstein metric; Calabi functional; \(K\)-stability; Yau’s theorem; Monge-Ampère equation; blow-up