Conical Kähler–Einstein Metrics Revisited

• Published in: Communications in mathematical physics Vol. 331; no. 3; pp. 927 - 973
• Main Authors: Li, Chi, Sun, Song
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2014
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Bergman Kernel; Cone Angle; Einstein Metrics; Exceptional Divisor; Reeb Vector
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-014-2123-9

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 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.