L’espace des métriques de Kähler dans une classe donnée sur une variété projective kählérienne est un espace symétrique de dimension infinie dont les géodésiques sont des solutions d’une équation Monge-Ampère complexe homogène sur , ou . Phong-Sturm ont prouvé que les géodésiques Monge-Ampère des potentiels kählériens de peuvent être approximées dans un sens faible par géodésiques de l’espace symétrique de métriques de Bergman de hauteur . Le but de cet article est de prouver que dans dans le cas des métriques toriques sur .
The space of Kähler metrics in a fixed Kähler class on a projective Kähler manifold is an infinite dimensional symmetric space whose geodesics are solutions of a homogeneous complex Monge-Ampère equation in , where is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials of may be approximated in a weak sense by geodesics of the finite dimensional symmetric space of Bergman metrics of height . In this article we prove that in in the case of toric Kähler metrics on .
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
@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}, pages = {2209--2237}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2332}, zbl = {1144.53089}, mrnumber = {2394541}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2332/} }
TY - JOUR AU - Song, Jian AU - Zelditch, Steve TI - Convergence of Bergman geodesics on $\mathbf{CP}^1$ JO - Annales de l'Institut Fourier PY - 2007 SP - 2209 EP - 2237 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2332/ DO - 10.5802/aif.2332 LA - en ID - AIF_2007__57_7_2209_0 ER -
%0 Journal Article %A Song, Jian %A Zelditch, Steve %T Convergence of Bergman geodesics on $\mathbf{CP}^1$ %J Annales de l'Institut Fourier %D 2007 %P 2209-2237 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2332/ %R 10.5802/aif.2332 %G en %F AIF_2007__57_7_2209_0
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. http://www.numdam.org/articles/10.5802/aif.2332/
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