K-stability for K\"ahler Manifolds

TL;DR

This paper extends the concept of K-stability to K"ahler manifolds and establishes a link between the boundedness of the Mabuchi functional and various stability notions, advancing the understanding of constant scalar curvature K"ahler metrics.

Contribution

It formulates K-stability for K"ahler manifolds and proves one direction of the Yau-Tian-Donaldson conjecture in this broader setting.

Findings

Bounded Mabuchi functional implies K-semistability.

Existence of constant scalar curvature K"ahler metric implies K-semistability.

K-stability holds if the automorphism group is discrete.

Abstract

We formulate a notion of K-stability for K\"ahler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. coercive) implies K-semistability (resp. uniformly K-stable). In particular this shows that the existence of a constant scalar curvature K\"ahler metric implies K-semistability, and K-stability if one assumes the automorphism group is discrete. We also show how Stoppa's argument holds in the K\"ahler case, giving a simpler proof of this K-stability statement.