Conical Kahler-Einstein metric revisited

TL;DR

This paper introduces an interpolation-degeneration strategy to analyze conical Kähler-Einstein metrics on Fano manifolds, determining existence intervals for cone angles and applying results to specific geometric cases including Sasaki-Einstein metrics.

Contribution

It develops a novel interpolation-degeneration approach to study cone angles for Kähler-Einstein metrics and applies it to solve problems in Fano manifolds and toric geometry.

Findings

Interval of cone angles admitting Kähler-Einstein metrics established

Existence of metrics on P^2 with cone singularities along conics characterized

Confirmed a version of Donaldson's conjecture in the toric case

Abstract

In this paper we introduce the "interpolation-degneration" strategy to study Kahler-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 $(0, 2\pi]$ that admit a conical Kahler-Einstein metric form an interval; and by "degeneration" we figure out the boundary of the interval. As a first application, we show that there exists a Kahler-Einstein metric on $P^2$ with cone singularity along a smooth conic (degree 2) curve if and only if the angle is in $(\pi/2, 2\pi]$. When the angle is $2\pi/3$ this proves the existence of a Sasaki-Einstein metric on the link of a three dimensional $A_2$ singularity, and thus answers a problem posed by Gauntlett-Martelli-Sparks-Yau. As a second application we prove a version of Donaldson's conjecture about conical Kahler-Einstein metrics in the toric case using Song-Wang's recent existence result of toric invariant conical Kahler-Einstein metrics.