Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics

Abstract

The seminar was made possible through the support of the Stiftung Volkswagenwerk.

Einstein metrics on spheres

“…The second step is to use the continuity method developed by [Aub82], [Siu88], [Siu87], [Tia87] to construct Kähler-Einstein metrics on orbifolds. With minor modifications, the method of [Nad90], [DK01] arrives at a sufficient condition, involving the integrability of inverses of polynomials on Y (a).…”

“…Let (X, ∆) be a compact orbifold of dimension n such that K −1 X orb is ample. The continuity method for finding a Kähler-Einstein metric on (X, ∆) was developed by [Aub82], [Siu88], [Siu87], [Tia87], [Nad90], [DK01].…”

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

“…It is well-known, as seen, for example, in the exposition of Siu [17], that for Theorem 2 we have an a priori estimate…”

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

“…In order to do this, we need to have a priori C k estimates for all k. The usual bootstrapping argument for the Monge-Ampère equation allows us to deduce those estimates from the C 2,α ones for some α ∈]0, 1[. The crucial fact here is that we have at our disposal the following local result, taken from [17] (see also [27], [6, Theorem 5.1]), which gives interior estimates. It is a consequence of Evans-Krylov's theory: In our case, we choose some point p outside the support of the divisor ∆, and consider two coordinate open sets Ω ⊂ Ω containing p, but not intersecting Supp(∆).…”