By Gabor Szekelyhidi
A uncomplicated challenge in differential geometry is to discover canonical metrics on manifolds. the simplest identified instance of this is often the classical uniformization theorem for Riemann surfaces. Extremal metrics have been brought through Calabi as an try out at discovering a higher-dimensional generalization of this consequence, within the environment of Kahler geometry. This e-book supplies an advent to the research of extremal Kahler metrics and particularly to the conjectural photo concerning the lifestyles of extremal metrics on projective manifolds to the steadiness of the underlying manifold within the feel of algebraic geometry. The ebook addresses a number of the simple principles on either the analytic and the algebraic facets of this photograph. an outline is given of a lot of the required historical past fabric, corresponding to easy Kahler geometry, second maps, and geometric invariant idea. past the fundamental definitions and homes of extremal metrics, numerous highlights of the idea are mentioned at a degree available to graduate scholars: Yau's theorem at the life of Kahler-Einstein metrics, the Bergman kernel growth because of Tian, Donaldson's decrease certain for the Calabi power, and Arezzo-Pacard's life theorem for consistent scalar curvature Kahler metrics on blow-ups.
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Extra info for An Introduction to Extremal Kahler Metrics
The section v1 of T 0•1M is obtained by taking the tensor product of these four sections and performing various contractions between pairwise dual spaces. vdV) of the contraction of the volume form with v, so we can use Stokes's theorem). Using the product rule, we have 'Vzv 1 = ('Vmk1)h('Vks)s + l 1('Vrh)('Vks)s + l 1h('Vr'Vks)s + l 1h('Vks)('Vrs), where each time the covariant derivative of the appropriate bundle is used. By the defining properties of the Levi-Civita and Chern connections, we have \lg= 0 and 'Vh = 0, so 'Vzv 1 = l 1h('Vz'Vks)s + gk1h('Vks)('Vrs).
1. Let M be a compact Kahler manifold with c1(M) < 0. Then there is a unique Kahler metric w E -27rc1(M) such that Ric(w) = -w. 31), so using this theorem, it is possible to construct many examples of Einstein manifolds. at. Finally we briefly discuss the case c1(M) > 0, which has only recently been solved. The algebrogeometric obstructions that appear in this and the more general case of extremal metrics will be our subject of study in the remainder of the book. 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 .
4) maximal at p, this means that -F(p). p:::;;; -F(p):::;;; suplFI. p. p. One more useful notation is to write tr9 g' = gikg;k and tr91g = g'ikgjk· We will also write A' for the Laplacian with respect to the metric g'. The key calculation is the following. 7. 3k , tr g/ 9 where R;k is the Ricci curvature of g'. 2. The C 0 - and C 2 -estimates 41 Proof. We will compute in normal coordinates for the metric g around a point p E M. In addition we can assume that g' is diagonal at p since any Hermitian matrix can be diagonalized by a unitary transformation.
An Introduction to Extremal Kahler Metrics by Gabor Szekelyhidi