By Gabor Szekelyhidi

A easy challenge in differential geometry is to discover canonical metrics on manifolds. the easiest recognized instance of this is often the classical uniformization theorem for Riemann surfaces. Extremal metrics have been brought via Calabi as an try out at discovering a higher-dimensional generalization of this outcome, within the environment of Kahler geometry. This ebook provides an creation to the research of extremal Kahler metrics and particularly to the conjectural photograph concerning the lifestyles of extremal metrics on projective manifolds to the steadiness of the underlying manifold within the feel of algebraic geometry. The publication addresses a few of the uncomplicated rules on either the analytic and the algebraic aspects of this photograph. an summary is given of a lot of the mandatory history fabric, corresponding to uncomplicated Kahler geometry, second maps, and geometric invariant concept. past the fundamental definitions and homes of extremal metrics, numerous highlights of the idea are mentioned at a degree obtainable 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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**Example text**

8, the c 0-norm off is controlled by the L 1-norm off together with the C°'-norm of Lf. This can be done by using a blow-up argument similar to what we have used above (a good exercise for the reader), although the more standard way is to use similar estimates in Sobolev spaces, together with the Sobolev embedding theorem. 3) for harmonic functions. An important point which does not follow from these arguments is that we do not need to know a priori that f E Ck+ 2•°'. 3) holds. For this, one needs to work harder; see Gilbarg-Trudinger [58, Chapter 6).

It follows that 81

E. -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.