Convergence of Bergman geodesics on $\mathbf{CP}^1$

[Convergence des géodésiques de Bergman sur $\textbf{CP}^1$]

Song, Jian ; Zelditch, Steve

Annales de l'Institut Fourier, Tome 57 (2007), p. 2209-2237 / Harvested from Numdam

Résumé

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

Abstract

The space $\mathcal{H}$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold $X$ is an infinite dimensional symmetric space whose geodesics $\omega _t$ are solutions of a homogeneous complex Monge-Ampère equation in $A \times X$, where $A \subset \mathbb{C}$ is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials $\varphi (t, z)$ of $\omega _t$ may be approximated in a weak $C^0$ sense by geodesics $\varphi _N(t, z) $ of the finite dimensional symmetric space of Bergman metrics of height $N$. In this article we prove that $\varphi _N(t, z) \rightarrow \varphi (t,z)$ in $C^2([0,1] \times X)$ in the case of toric Kähler metrics on $X = \mathbf{CP}^1$.

Détail

Publié le : 2007-01-01

MR 2394541 | Zbl 1144.53089

DOI : https://doi.org/10.5802/aif.2332
Classification:  53C55
Mots clés: métrique de Bergman, équation Monge-Ampère, noyau de Bergman-Szegö, métrique torique, potential kählérien, potential symplectique

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