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

Song, Jian; Zelditch, Steve

doi:10.5802/aif.2332

Convergence of Bergman geodesics on

{CP}^{1}

[Convergence des géodésiques de Bergman sur

{CP}^{1}

]

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)

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

Résumé
Abstract

L’espace $ℋ$ des métriques de Kähler dans une classe donnée sur une variété projective kählérienne $X$ est un espace symétrique de dimension infinie dont les géodésiques $ω_{t}$ 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 $ϕ (t, z)$ 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 $ϕ_{N} (t, z) \to ϕ (t, z)$ dans $C^{2} ([0, 1] \times X)$ dans le cas des métriques toriques sur $X = {CP}^{1}$ .

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

MR Zbl

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

Affiliations des auteurs :

Song, Jian ¹ ; Zelditch, Steve ²

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     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},
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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://aif.centre-mersenne.org/articles/10.5802/aif.2332/

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