By Gabor Szekelyhidi
A uncomplicated challenge in differential geometry is to discover canonical metrics on manifolds. the easiest identified instance of this can be the classical uniformization theorem for Riemann surfaces. Extremal metrics have been brought by means of Calabi as an test at discovering a higher-dimensional generalization of this outcome, within the environment of Kahler geometry. This booklet provides an advent to the research of extremal Kahler metrics and particularly to the conjectural photograph pertaining to the lifestyles of extremal metrics on projective manifolds to the steadiness of the underlying manifold within the feel of algebraic geometry. The e-book addresses the various easy rules on either the analytic and the algebraic facets of this photo. an summary is given of a lot of the mandatory heritage fabric, comparable to uncomplicated Kahler geometry, second maps, and geometric invariant idea. past the fundamental definitions and houses of extremal metrics, a number of highlights of the speculation are mentioned at a degree available to graduate scholars: Yau's theorem at the lifestyles of Kahler-Einstein metrics, the Bergman kernel enlargement because of Tian, Donaldson's reduce sure for the Calabi strength, and Arezzo-Pacard's lifestyles theorem for consistent scalar curvature Kahler metrics on blow-ups.
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Additional resources for An Introduction to Extremal Kahler Metrics
The basic reference for this chapter is Yau [122), but there are many places where this material is explained, for instance Siu [97), Tian [113), or Blocki . 1. 1. First we rewrite the equation in terms of Kahler potentials. Let w0 be any Kahler metric in the class -27rc1(M). 1) Ric(wo) = -wo + AalJF. If w = w0 + AalJcp is another Kahler metric in the same class, then Ric(w) = Ric(wo) - AalJlog w:, Wo so in order to make sure that Ric(w) = -w, we need -AalJcp = AalJF - n AalJlog ~. 1. 2. On a compact Kahler manifold M there exists at most one metric w E -27rc1 (M) such that Ric( w) = -w.
Lkl (a, -of)= gka-;:O;J = g a-;:8kf, while (8* a, f) is just the product (8* a)f. 7) implies that 8* a = -gkl'Vkal, and so -8*8 agrees with our operator D.. Note that by using covariant derivatives we do not have to worry about differentiating the metric when we integrate by parts, and at the same time remember that 8kf = 'V kf. The same idea works for arbitrary (p, q) forms, giving rise to the Hodge Laplacian D. = -8*8- 88* (the term 88* is zero on functions). We can also do the same with 8 instead of 8 and on Kahler manifolds both give rise to the same Laplace operator.
It turns out that if we write out the Taylor expansion of the metric in normal coordinates, then each coefficient will only depend on covariant derivatives of the curvature. In particular if the curvature vanishes in a neighborhood of a point, then in normal coordinates the metric is just given by the Euclidean metric. 21. Verify the following identities for the curvature of a Kahler metric: Ri]kl = RilkJ = Rk]il = RkliJ1 \7pRi]kl = \7iRpJkl· Compare these to the identities satisfied by the curvature tensor of a Riemannian metric, in particular the first and second Bianchi identities.