Generalized Matsushima's theorem and K\"ahler-Einstein cone metrics

TL;DR

This paper extends Matsushima's theorem to K"ahler-Einstein cone metrics on Fano manifolds with cone singularities, providing new proofs of uniqueness and existence results using the continuity method and conic Ding functional.

Contribution

It generalizes Matsushima's theorem to a broader class of cone singularities and offers alternative proofs for the uniqueness and existence of K"ahler-Einstein cone metrics.

Findings

Matsushima's theorem is proven for K"ahler-Einstein cone metrics with non-proportional divisors.

An alternative proof of the uniqueness of K"ahler-Einstein cone metrics is provided.

An existence theorem for K"ahler-Einstein cone metrics using the conic Ding functional is established.

Abstract

In this paper, we prove Matsushima's theorem for K\"ahler-Einstein metrics on a Fano manifold with cone singularities along a smooth divisor that is not necessarily proportional to the anti-canonical class. We then give an alternative proof of uniqueness of K\"ahler-Einstein cone metrics by the continuity method. Moreover, our method provides an existence theorem of K\"ahler-Einstein cone metrics with respect to conic Ding functional.