November 16 (Sat), 2013, 16:55--17:55 November 17 (Sun), 2013, 10:00--11:00 Lecture Hall (Room No. 420) Research Institute for Mathematical Sciences Kyoto University, Kyoto, Japan |

Gang Tian

(Princeton University, Beijing International Center for Mathematical Research)

(Princeton University, Beijing International Center for Mathematical Research)

**Abstract**

The study of Kähler-Einstein metrics was initiated by E. Calabi in 50's.
In 70s, Yau and Aubin solved the
existence problem for Kähler-Einstein metrics on compact Kähler
manifolds with vanishing or negative
first Chern class. Since then, it has been a challenging problem to
studying the existence of
Kähler-Einstein metrics on Fano manifolds.
A Fano manifold is a compact Kähler manifold with positive first Chern
class. There are obstructions to the
existence of Kähler-Einstein metrics on Fano manifolds, first by
Matsushima in late 50s, secondly by A. Futaki in
early 80s and also K-stability in 90s. These lectures will concern
Kähler-Einstein metrics and K-stability.

In the first lecture, I will give a brief tour on the study of Kähler-Einstein metrics on Fano manifolds in last two decades and then discuss recent solution for the existence of Kähler-Einstein metrics on Fano manifolds which are K-stable. I will also discuss key tools used in the solution.

In the second lecture, I will focus on the K-stability, its original definition as well as new formulations. I will discuss some recent works on K-stability, particularly, S. Paul's work on stable pairs, which generalize certain fundamental results in the Geometric Invariant Theory, and show how the K-stability can be put in this general setting. Some open problems may be discussed in the end if time permits.