Journal of Differential Geometry

Scalar Curvature and Stability of Toric Varieties

S.K. Donaldson

Abstract

We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety.

Article information

Source
J. Differential Geom., Volume 62, Number 2 (2002), 289-349.

Dates
First available in Project Euclid: 27 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090950195

Digital Object Identifier
doi:10.4310/jdg/1090950195

Mathematical Reviews number (MathSciNet)
MR1988506

Zentralblatt MATH identifier
1074.53059

Citation

Donaldson, S.K. Scalar Curvature and Stability of Toric Varieties. J. Differential Geom. 62 (2002), no. 2, 289--349. doi:10.4310/jdg/1090950195. https://projecteuclid.org/euclid.jdg/1090950195


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