Japan. J. Math. 10, 1--41 (2015)

Kähler--Einstein metrics on Fano manifolds

G. Tian

Abstract: This is an expository paper on Kähler metrics of positive scalar curvature. It is for my Takagi Lectures at RIMS in November of 2013. In this paper, I first discuss the Futaki invariants, the K-stability and its relation to the K-energy. Next I will outline my work in 2012 on the existence of Kähler--Einstein metrics on K-stable Fano manifolds. Finally, I will present S. Paul's work on stability of pairs with some modifications of mine.