Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds

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.