Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds

TL;DR

This paper establishes a criterion linking the coercivity of the Mabuchi functional to Tian's alpha invariant on Fano manifolds, extending Tian's results and providing new examples of K"ahler classes with coercive Mabuchi functionals.

Contribution

It generalizes Tian's criterion for the coercivity of the Mabuchi functional to all K"ahler classes on Fano manifolds and proves the alpha invariant's continuity on the K"ahler cone.

Findings

Established a criterion for coercivity in terms of alpha invariant.

Proved the alpha invariant is continuous on the K"ahler cone.

Identified new K"ahler classes with coercive Mabuchi functional on degree one del Pezzo surfaces.

Abstract

We give a criterion for the coercivity of the Mabuchi functional for general K\"ahler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the existence of a K\"ahler-Einstein metric. We also prove the alpha invariant is a continuous function on the K\"ahler cone. As an application, we provide new K\"ahler classes on a general degree one del Pezzo surface for which the Mabuchi functional is coercive.