Uniformization of singular varieties

Abstract: By proving Calabi's conjecture, Yau proved that the first and second Chern classes of a compact complex manifold with ample canonical bundle encode the symmetries of the Kahler-Einstein metric via a simple inequality; the so-called Miyaoka-Yau inequality. In the case of equality, such symmetries lead to the uniformization by the ball. In the classification theory of complex spaces, one looks at a class of varieties far bigger than manifolds with ample canonical bundle. There are referred to as the minimal models of general type. These varieties are not necessarily smooth and their existence has been one of the most important recent breakthroughs in the classification theory. Singularity of such spaces pose a significant difficulty to a successful application of the analytic tools. In this talk I will briefly explain how Hodge theoretic methods can be used to remedy this problem. The talk is based on a joint work with Greb, Kebekus and Peternell.