The Continuity Method to Deform Cone Angle

Abstract

The continuity method is used to deform the cone angle of a weak conical Kähler-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 Kähler-Einstein) and bypasses the difficult Banach space set up; it is also generalized to deform the cone angles of a \emph{weak conical Kähler-Einstein metric} along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).

Theorem 1.3 of version 1 removed due to imprecise statement; English grammar mistakes and typos corrected, missing reference added

References in corpus (3)

• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors
• On the regularity problem of complex Monge-Ampere equations with conical singularities

Cited by in corpus (4)

• Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties
• The conical Kähler-Ricci flow on Fano manifolds
• Stability of the conical Kähler-Ricci flows on Fano manifolds
• Twisted and Singular gravitating vortices