Conical Kähler-Einstein Metrics Revisited

In this paper we introduce the "interpolation-degeneration" strategy to study Kähler-Einstein metrics on a smooth Fano manifold with cone singularities along a smooth divisor that is proportional to the anti-canonical divisor. By "interpolation" we show the angles in (0, 2π] that admit a conical Kähler-Einstein metric form a connected interval, and by "degeneration" we determine the boundary of the interval in some important cases. As a first application, we show that there exists a Kähler-Einstein metric on P² with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in (π/2, 2π]. When the angle is 2π/3 this proves the existence of a Sasaki-Einstein metric on the link of a three dimensional A ₂ singularity, and thus answers a question posed by Gauntlett-Martelli-Sparks-Yau. As a second application we prove a version of Donaldson's conjecture about conical Kähler-Einstein metrics in the toric case using Song-Wang's recent existence result of toric invariant conical Kähler-Einstein metrics.