Test configurations, large deviations and geodesic rays on toric varieties

Details Test configurations, large deviations and geodesic rays on toric varieties Test configurations, large deviations and geodesic rays on toric varieties

2007-12-21

arxiv.org/abs/0712.3599v1

Advances in Mathematics 229 (2012) pp. 2338-2378

Mathematics

Differential Geometry

42 pages, no figures

Scientific paper

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.

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