Kähler Einstein metrics and the Stability of Projective Varieties

In the mid eighties S.T. Yau conjectured that a Fano manifold would admit a K.E. metric provided the variety is Stable. The precise stability condition at the time was not clear. Over the past few years, mainly through the work of Gang Tian, a precise conjecture-and several theorems-have appeared. This talk will focus on some of these developements: First we will recall Mumfords' Geometric Invariant theory of Chow and Hilbert points, G. Tians' K and CM stability, and finally the relationship of these to the K-Energy map of Mabuchi, the (classical) Futaki invariant, the generalized Futaki invariant of Ding and Tian, and K-stability as it appeared in work of Simon Donaldson.

This is joint work with Gang Tian.