Kähler–Einstein Metrics and Stability

Abstract

This is a note to announce a solution of the well-known question: which Fano manifolds admit Kahler–Einstein metrics? The idea that the appropriate condition should be in terms of “algebro-geometric stability” was proposed about 20 years ago by Yau [21, 22] (partly by analogy with the “Kobayashi–Hitchin correspondence” in the case of holomorphic bundles). For more on the history of this problem please see our forthcoming full paper. Over the years various different notions of stability have been discussed in the literature, both in the Fano/Kahler–Einstein case and in the more general situation of constant scalar curvature Kahler metrics on polarized manifolds. But the condition we end up using is essentially as proposed by Tian [20] and which we now recall (see also [14] for the equivalence with a priori stronger definitions).