Microscopic stability thresholds and constant scalar curvature K\"{a}hler metrics

TL;DR

This paper establishes a link between microscopic stability thresholds and the existence of unique constant scalar curvature Kähler metrics, extending previous results and providing new sufficient conditions for cone metrics.

Contribution

It proves that certain conditions on microscopic stability thresholds imply the existence of unique constant scalar curvature Kähler metrics, generalizing prior work by Zhang and Fujita-Odaka.

Findings

Conditions on stability thresholds guarantee unique cscK metrics.

Provides sufficient conditions for cscK cone metrics.

Extends stability results beyond K-stability.

Abstract

In this paper, we directly prove that if the limit of microscopic stability thresholds introduced by Berman for a polarized manifold satisfies some condition, then there exists a unique constant scalar curvature K\"{a}hler metric. This is an analogue of K.Zhang's result which is proved by the delta-invariant introduced by Fujita-Odaka. This work is motivated by Berman's result which shows that if a Fano manifold is uniformly Gibbs stable, then there exists a unique K\"{a}hler-Einstein metric, without uniform K-stability. We also give some sufficient conditions of the existence of a constant scalar curvature K\"{a}hler cone metric.