Introduction to the Tian-Yau-Dondalson Conjecture

The most difficult (and still unresolved) case of the existence problem for Kahler Einstein metrics is when the manifold in question is Fano, that is, when the first Chern class is positive. In the mid eighties S.T. Yau conjectured that the existence of a Kahler Einstein metric on a Fano manifold should be related to the stability (in the sense of D. Mumford's G.I.T.) of its plurianticanonical models. The first substantial step in understanding this suggestion was taken by Ding and Tian in 1992, where they introduced the generalized Futaki invariant. This idea was then more fully developed by Tian in many papers out of which grew his notion of K-stability. It is this condition-or a variant of it proposed by Simon Donaldson -which has attracted much attention in the last several years. In my first talk I will give a (baised) historical account of the ideas leading up to K-stability as formulated by Tian and Donaldson. In my subsequent talks I will explain the shortcomings of this approach and offer what seems to be a completely satisfactory alternative notion of stability which is rooted in representation theory and classical projective geometry.