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

Song, Jian; Zelditch, Steve

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2209-2237.

L’espace $\mathscr{H}$ 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 $A=\{z\in ℂ:e^{-1}<|z|<1\}$ . 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 ${\varphi }_N(t,z)\to \varphi (t,z)$ dans $C^2([0,1]\times X)$ dans le cas des métriques toriques sur $X={\mathbf{CP}}^1$.

The space $\mathscr{H}$ 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 $A\subset ℂ$ 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 ${\varphi }_N(t,z)\to \varphi (t,z)$ in $C^2([0,1]\times X)$ in the case of toric Kähler metrics on $X={\mathbf{CP}}^1$.

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

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