Test configurations, large deviations and geodesic rays on toric varieties

Jian Song*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


This article contains a detailed study in the case of a toric variety of the geodesic rays φ t defined by Phong and Sturm corresponding to test configurations T in the sense of Donaldson. We show that the 'Bergman approximations' φ k(t, z) of Phong and Sturm converge in C 1 to the geodesic ray φ t, and that the geodesic ray itself is C 1,1 and no better. In particular, the Kähler metrics ωt=ω0+i∂∂-φt 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. We construct a sequence of measures μkz on the polytope of the toric variety, show that they satisfy a large deviations principle, and relate the rate function to the geodesic ray.

Original languageEnglish (US)
Pages (from-to)2338-2378
Number of pages41
JournalAdvances in Mathematics
Issue number4
StatePublished - Mar 1 2012


  • Bergman kernels
  • Bergman metrics
  • Kähler metrics
  • Large deviations
  • Test configurations
  • Toric varieties

ASJC Scopus subject areas

  • Mathematics(all)

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