The K\"ahler-Ricci flow and optimal degenerations

TL;DR

This paper demonstrates that the K"ahler-Ricci flow on Fano manifolds produces a canonical destabilizing degeneration, links it to a new stability notion, and applies it to resolve conjectures and establish convergence to K"ahler-Ricci solitons.

Contribution

It introduces a new stability concept related to the H-functional, showing the flow produces a canonical degeneration and proving convergence results for K"ahler-Ricci solitons.

Findings

The flow produces a 'most destabilising' degeneration.

A formula for the supremum of Perelman's μ-functional is provided.

Convergence of the flow to K"ahler-Ricci solitons is established.

Abstract

We prove that on Fano manifolds, the K\"ahler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's {\mu}-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a K\"ahler-Ricci soliton, then the K\"ahler-Ricci flow converges to it modulo the action of automorphisms, with any initial metric. This extends work of Tian-Zhu and Tian-Zhang-Zhang-Zhu, where either the manifold was assumed to admit a K\"ahler-Einstein metric, or the initial metric of the flow was assumed to be invariant under a maximal compact group of automorphism.