K\"ahler-Ricci flow, K\"ahler-Einstein metric, and K-stability

TL;DR

This paper establishes the existence of Kähler-Einstein metrics on K-stable Fano manifolds by analyzing the asymptotic behavior of Kähler-Ricci flows, linking geometric stability to flow convergence.

Contribution

It provides a new proof of Kähler-Einstein metric existence on K-stable Fano manifolds using flow compactness and algebro-geometric methods.

Findings

Existence of Kähler-Einstein metrics on K-stable Fano manifolds.

Connection between K-stability and flow asymptotics.

Application to Calabi flow and K-stability under geometric bounds.

Abstract

We prove the existence of Kahler-Einstein metric on a K-stable Fano manifold using the recent compactness result on Kahler-Ricci flows. The key ingredient is an algebro-geometric description of the asymptotic behavior of Kahler-Ricci flow on Fano manifolds. This is in turn based on a general finite dimensional discussion, which is interesting in its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kahler manifold to K-stability assuming bounds on geometry.