The Continuity Method to Deform Cone Angle

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 Donaldson (Essays in Mathematics and Its Applications, 2012) by combining it with the regularity result of Guenancia–P?un (arXiv:1307.6375 2013) and Chen–Wang (arXiv:1405.1201 2014). 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 weak conical Kähler–Einstein metric along a simple normal crossing divisor (pluri-anticanonical) on a smooth Fano manifold (assuming no tangential holomorphic vector fields).