Conical Kahler-Einstein metric revisited

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.

44 pages. The paper is reorganized. The methods and applications are highlighted. Some more comments and arguments are added. Normalization of coefficients are made consistent. Some typos are corrected

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Cited by in corpus (25)

• K-stability and Kähler-Einstein metrics
• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Asymptotically log Fano varieties
• Kahler-Einstein metrics and stability
• Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors
• Smooth approximations of the Conical Kahler-Ricci flows
• Bessel Functions, Heat Kernel and the Conical Kähler-Ricci Flow
• The conical Kähler-Ricci flow on Fano manifolds
• On the long time behaviour of the Conical Kähler- Ricci flows
• Kähler-Einstein metrics and volume minimization
• Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds
• Connecting toric manifolds by conical Kahler-Einstein metrics
• Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties
• Unnormalize conical Kähler-Ricci flow
• Existence of Weak Conical Kähler-Einstein Metrics Along Smooth Hypersurfaces
• $C^{2,\alpha }$-estimate for conical Kähler-Ricci flow
• On the regularity problem of complex Monge-Ampere equations with conical singularities
• Properness of log $F$-functionals
• Modified Futaki invariant and equivariant Riemann-Roch formula
• On flops and canonical metrics
• On the Geodesics in the space of Kähler metrics with prescribed singularities
• $C^{2,\alpha }$-estimate for Monge-Ampere equations with Hölder-continuous right hand side
• Twisted and conical Kähler-Ricci soliton on Fano manifolds
• Numerical Solutions of Kahler-Einstein metrics on $P^2$ with conical singularities along a conic curve