Convergence of Bergman geodesics on CP1
[Convergence des géodésiques de Bergman sur CP1]
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2209-2237.

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 ωt 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 ωt peuvent être approximées dans un sens faible C0 par géodésiques ϕN(t,z) de l’espace symétrique de métriques de Bergman de hauteur N. Le but de cet article est de prouver que ϕN(t,z)ϕ(t,z) dans C2([0,1]×X) dans le cas des métriques toriques sur X=CP1.

The space 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 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 ϕN(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.

DOI : 10.5802/aif.2332
Classification : 53C55
Keywords: Bergman metric, Monge-Ampère equation, Bergman-Szegö kernel, toric metric, Kähler potential, symplectic potential
Mot clés : métrique de Bergman, équation Monge-Ampère, noyau de Bergman-Szegö, métrique torique, potential kählérien, potential symplectique
Song, Jian 1 ; Zelditch, Steve 2

1 Rutgers, The State University of New Jersey Department of Mathematics Hill Center-Busch Campus 110 Frelinghuysen Rd Piscataway, NJ 08854-8019 (USA)
2 Johns Hopkins University Department of Mathematics Baltimore, MD 21218 (USA)
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Song, Jian; Zelditch, Steve. Convergence of Bergman geodesics on $\mathbf{CP}^1$. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2209-2237. doi : 10.5802/aif.2332. https://www.numdam.org/articles/10.5802/aif.2332/

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