The Continuity Method to Deform Cone Angle

Abstract

The continuity method is used to deform the cone angle of a weak conical K\"ahler-Einstein metric with cone singularities along a smooth anti-canonical divisor on a smooth Fano manifold. This leads to an alternative proof of Donaldson's Openness Theorem on deforming cone angle \cite{Don} by combining it with the regularity result of Guenancia-P$\breve{\text{a}}$un \cite{GP} and Chen-Wang \cite{CW}. This continuity method uses relatively less regularity of the metric (only weak conical K\"ahler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a \emph{weak conical K\"ahler-Einstein metric} along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).