EVENT DETAILS AND ABSTRACT

Title: Negatively curved Kahler-Einstein metrics and moduli spaces of stable varieties
Speaker: Jacob Sturm
Speaker Info: Rutgers Newark
Brief Description:
Special Note:
Abstract:

We will discuss some estimates for negatively curved Kahler-Einstein metrics on compact manifolds that depend only on their dimension and volume. These are applied to prove that if X_i is a sequence of such manifolds with fixed dimension and volume, that there is a subsequence that converges in the Gromov-Hausdorff topology (already known if n=1 by a theorem of Bers and n=2 by work of Cheeger-Colding). The limit is a finite disjoint union of complete Kahler-Einstein metric spaces without loss of volume. We will also describe an application to M(n,V), which is the compactified moduli space of all Kahler-Einstein varieties of dimension n and volume V. We show that the Weil-Petersson metric, which is a positive closed (1,1) current on the projective variety M(n,V), has continuous local potentials (generalizing Wolpert’s theorem which treats the case n=1). We also show how one can combine these ideas with the technique of Donaldson-Sun to give a new proof of the Deligne-Mumford stable reduction theorem. This is joint work with Jian Song and Xiaowei Wang.