Convergence of Bergman geodesics on CP 1
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, p. 2209-2237

The space of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X is an infinite dimensional symmetric space whose geodesics ω t are solutions of a homogeneous complex Monge-Ampère equation in A×X, where A is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials ϕ(t,z) of ω t may be approximated in a weak C 0 sense by geodesics ϕ N (t,z) of the finite dimensional symmetric space of Bergman metrics of height N. In this article we prove that ϕ N (t,z)ϕ(t,z) in C 2 ([0,1]×X) in the case of toric Kähler metrics on X=CP 1 .

L’espace des métriques de Kähler dans une classe donnée sur une variété projective kählérienne X est un espace symétrique de dimension infinie dont les géodésiques ω t sont des solutions d’une équation Monge-Ampère complexe homogène sur A×X, ou A={z:e -1 <|z|<1} . Phong-Sturm ont prouvé que les géodésiques Monge-Ampère des potentiels kählériens ϕ(t,z) de ω t peuvent être approximées dans un sens faible C 0 par géodésiques ϕ N (t,z) de l’espace symétrique de métriques de Bergman de hauteur N. Le but de cet article est de prouver que ϕ N (t,z)ϕ(t,z) dans C 2 ([0,1]×X) dans le cas des métriques toriques sur X=CP 1 .

DOI : https://doi.org/10.5802/aif.2332
Classification:  53C55
Keywords: Bergman metric, Monge-Ampère equation, Bergman-Szegö kernel, toric metric, Kähler potential, symplectic potential
@article{AIF_2007__57_7_2209_0,
     author = {Song, Jian and Zelditch, Steve},
     title = {Convergence of Bergman geodesics on $\mathbf{CP}^1$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     pages = {2209-2237},
     doi = {10.5802/aif.2332},
     mrnumber = {2394541},
     zbl = {1144.53089},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_7_2209_0}
}
Song, Jian; Zelditch, Steve. Convergence of Bergman geodesics on $\mathbf{CP}^1$. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2209-2237. doi : 10.5802/aif.2332. http://www.numdam.org/item/AIF_2007__57_7_2209_0/

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