Since then, it has been a challenging problem to study 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. Recently, the problem has been solved by relating existence to K-stability, a geometric stability generalizing substantially the stability from classical geometric invariant theory.
In the first lecture, I will first recall some known facts about Riemann surfaces. Then we give a brief tour on the study of Kähler-Einstein metrics in the last two decades. Next I will explain how those Einstein metrics are related to the study of algebraic group actions and geometric stability.
This lecture is aimed at a general audience.