Convergence of Bergman geodesics on CP1

Jian Song*, Steve Zelditch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

The space ℋ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X is an infinite dimensional symmetric space whose geodesics ωt are solutions of a homogeneous complex Monge-Ampère equation in A × X, where A ⊂ ℂ is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials φ(t, z) of ωt may be approximated in a weak C0 sense by geodesics φ(t, z) of the finite dimensional symmetric space of Bergman metrics of height N. In this article we prove that φN(t,z) → φ(t, z) in C2([0,1] × X) in the case of toric Kähler metrics on X = CP1.

Original languageEnglish (US)
Pages (from-to)2209-2237
Number of pages29
JournalAnnales de l'Institut Fourier
Volume57
Issue number7
DOIs
StatePublished - 2007

Keywords

  • Bergman metric
  • Bergman-Szegö
  • Kahler potential
  • Kernel
  • Monge-Ampère equation
  • Symplectic potential
  • Toric metric

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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