The Continuity Method to Deform Cone Angle

TL;DR

This paper introduces a continuity method for deforming cone angles in weak conical Kähler-Einstein metrics, providing an alternative proof of Donaldson's Openness Theorem and extending to more general divisors with less regularity requirements.

Contribution

It presents a novel continuity approach that reduces regularity assumptions and simplifies the deformation process of cone angles in Kähler-Einstein metrics, extending previous results.

Findings

Provides an alternative proof of Donaldson's Openness Theorem.

Generalizes the deformation method to weak conical Kähler-Einstein metrics on simple normal crossing divisors.

Reduces regularity requirements for the metrics involved.

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).