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

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.