Title: Working seminar on Kaehler-Einstein metrics and stability

Venue: Room 09,Quan Zhai, BICMR

Time: Every Tuesday, 10:00am-12:00pm

Abstract：
In 1950s, E. Calabi initiated the study of Kahler-Einstein metrics on compact Kahler manifolds. He proved the uniqueness of Kahler-Einstein metrics in the case of non-positive first Chern class. In 1976, the existence of Kahler-Einstein metrics was proved by Aubin and Yau, independently, in the case of negative first Chern class and by Yau in the case of vanishing first Chern class. It has been a difficult problem to study Kahler-Einstein metrics on Kahler manifolds with positive first Chern class, also called Fano manifolds. The uniquesness was proved by Bando-Mabuchi in 1986. There are holomorphic obstructions to the existence, such as the Futaki invariant constructed by Futaki in 1983.In 1996, using the generalized Futaki invariant by Ding-Tian, Tian introduced the notion of K-stability and proved that if a Fano manifold has no non-zero holomorphic fields and admits a Kahler-Einstein metric, then it is K-stable. The K-stability was reformulated by S. Donaldson in 2000s in a more algebraic way. In last decade, there have been many outstanding works on K-stability and the existence of Kahler-Einstein metrics. In 2012, Tian provided a solution for the YTD conjecture in the case of Fano manifolds, that is, if a Fano manifold is K-stable, then it has a Kahler-Einstein metric. In this seminar, we go through Tian's proof and related works involved in the proof.