Construction of constant scalar curvature Kähler cone metrics

Abstract

Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformations of Kähler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivization of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian–Einstein metric on this bundle enjoys a regularity property along the divisor on the base.