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On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold

Part of: Global differential geometry Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  12 October 2012

Chi Li
Chi Li*
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544, USA (email: [email protected])
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Abstract

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This work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

Keywords

Kähler–Einstein metriccontinuity methodtoric Fano manifoldconic singularity

MSC classification

Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55: Hermitian and Kählerian manifolds 58E11: Critical metrics
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1985 - 2003
Copyright
Copyright © Foundation Compositio Mathematica 2012

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