Test configurations, large deviations and geodesic rays on toric varieties

Abstract

This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesic ray of potentials are discontinuous across certain hypersurfaces and are degenerate on certain open sets. A novelty in the analysis is the connection between Bergman metrics, Bergman kernels and the theory of large deviations.

42 pages, no figures

References in corpus (4)

• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
• Convergence of Bergman geodesics on CP^1
• Test configurations and Geodesic rays
• Bernstein polynomials, Bergman kernels and toric Kähler varieties

Cited by in corpus (4)

• Lectures on Stability and Constant Scalar Curvature
• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
• The Dirichlet problem for degenerate complex Monge-Ampere equations
• Regularity of geodesic rays and Monge-Ampere equations