Existence of Einstein metrics on Fano manifolds

This is largely an expository paper and dedicated to my friend J. Cheeger for his 65th birthday. The purpose of this paper is to discuss some of my works on the existence of Kähler-Einstein metrics on Fano manifolds and some related topics. I will describe a program I have been following for the last twenty years. It includes some of my results and speculations which were scattered in my previous publications or mentioned in my lectures. I also take this opportunity to clarify and make them more accessible. In the course of doing so, I will also discuss some recent advances and problems which arise from studying the existence problem. A Fano manifold is a compact Kähler manifold with positive first Chern class. It has been one of main problems in Kähler geometry to study if a Fano manifold admits a Kähler-Einstein metrics since the Aubin-Yau theorem on Kähler-Einstein metrics with negative scalar curvature and the Calabi-Yau theorem on Ricci-flat Kähler metrics in 70’s. This problem is much more difficult because there are new obstructions to the existence. The classical one was given by Matsushima: If a Fano manifold M admits a Kähler-Einstein metric, then its Lie algebra of holomorphic vector fields must be reductive. In the early 80’s, A. Futaki introduced a new invariant, now referred to the Futaki invariant, whose vanishing is a necessary condition for M to have a Kähler-Einstein metric. Since late 80’s, inspired by my works on Kähler-Einstein metrics on complex surfaces [Ti89], I have been developing methods of relating certain geometric stability of underlying manifolds to Kähler-Einstein metrics. In [Ti97], I introduced the K-stability for any Fano manifold and proved that a Fano manifold with trivial holomorphic vector fields and which admits a Kähler-Einstein metric is K-stable. An algebraic version of the K-stability was given by Donaldson in [Do02]. It was conjectured that the existence of Kähler-Einstein metrics on M is equivalent to the asymptotic K-stability. As said at the beginning, this is not intended to be a complete survey on Kähler-Einstein metrics with positive scalar curvature. Unfortunately, there are