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 by way of Calabi as an try out at discovering a higher-dimensional generalization of this end result, within the atmosphere of Kahler geometry. This publication supplies an advent to the examine of extremal Kahler metrics and specifically to the conjectural photograph concerning the life of extremal metrics on projective manifolds to the soundness of the underlying manifold within the experience of algebraic geometry. The e-book addresses a few of the uncomplicated rules on either the analytic and the algebraic facets of this photo. an summary is given of a lot of the required historical past fabric, similar to easy Kahler geometry, second maps, and geometric invariant idea. past the fundamental definitions and houses 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 as a result 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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Extra info for An Introduction to Extremal Kahler Metrics
7. Line bundles and projective embeddings Suppose that L --+ M is a holomorphic line bundle over a complex manifold M. If so, ... 16) U-+ cpk pi--+ [so(p) : · · ·: sk(p)]. 38. A line bundle L over M is very ample if for suitable sections so, ... 16) defines an embedding of M into cpk. A line bundle L is ample if for a suitable integer r > 0 the tensor power Lr is very ample. 39. The bundle 0(1) over cpn is very ample, and the sections Zo, ... 29 define the identity map from cpn to itself. More generally for any projective manifold V c CPn, the restriction of 0(1) to V is a very ample line bundle.
Chapter 3 Kahler-Einstein Metrics Recall that a Riemannian metric is Einstein if its Ricci tensor is proportional to the metric. In this section, we are interested in Kahler metrics which are also Einstein. w, for some ).. E R. : Ric(w) = -w, Ric(w) = O, or Ric(w) = w. As we have seen, the Ricci form of a Kahler metric defines a characteristic class of the manifold, namely c1(M) = 2~ [Ric(w)], which is independent of the Kahler metric won M. It follows that in order to find a Kahler-Einstein metric on M, the class c1(M) must either be a negative, zero, or positive cohomology class.
C1(M) is represented by a Kahler metric. More generally, suppose that M c pn is a smooth complex submanifold of codimension r, defined by the intersection of r hypersurfaces of degrees di, ... , dr. If di+···+ dr > n + 1, then show that c1(M) < 0. 6. Connections and curvature of line bundles The Levi-Civita connection that we used before is a canonical connection on the tangent bundle of a Riemannian manifold. Analogously there is a canonical connection on an arbitrary holomorphic vector bundle equipped with a Hermitian metric, called the Chern connection.
An Introduction to Extremal Kahler Metrics by Gabor Szekelyhidi