The 13th Takagi Lectures
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

Kähler--Einstein Metrics on Fano Manifolds
Gang Tian
(Princeton University, Beijing International Center for Mathematical Research)


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.