Communications in Mathematical Physics

, Volume 331, Issue 3, pp 927–973 | Cite as

Conical Kähler–Einstein Metrics Revisited

  • Chi LiEmail author
  • Song Sun


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 \({\mathbb{P}^2}\) 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.


Cone Angle Exceptional Divisor Einstein Metrics Bergman Kernel Reeb Vector 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Mathematics DepartmentStony Brook UniversityStony BrookUSA
  3. 3.Department of MathematicsImperial CollegeLondonUK

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