Convergence of Bergman geodesics on CP 1

Jian Song[1]; Steve Zelditch[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 (2007)

  • Volume: 57, Issue: 7, page 2209-2237
  • ISSN: 0373-0956

Abstract

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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 × X , where A ⊂ 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 ) → ϕ ( t , z ) in C 2 ( [ 0 , 1 ] × X ) in the case of toric Kähler metrics on X = CP 1 .

How to cite

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Song, Jian, and Zelditch, Steve. "Convergence of Bergman geodesics on $\mathbf{CP}^1$." Annales de l’institut Fourier 57.7 (2007): 2209-2237. <http://eudml.org/doc/10296>.

@article{Song2007,
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 $X = \mathbf\{CP\}^1$.},
affiliation = {Rutgers, The State University of New Jersey Department of Mathematics Hill Center-Busch Campus 110 Frelinghuysen Rd Piscataway, NJ 08854-8019 (USA); Johns Hopkins University Department of Mathematics Baltimore, MD 21218 (USA)},
author = {Song, Jian, Zelditch, Steve},
journal = {Annales de l’institut Fourier},
keywords = {Bergman metric; Monge-Ampère equation; Bergman-Szegö kernel; toric metric; Kähler potential; symplectic potential},
language = {eng},
number = {7},
pages = {2209-2237},
publisher = {Association des Annales de l’institut Fourier},
title = {Convergence of Bergman geodesics on $\mathbf\{CP\}^1$},
url = {http://eudml.org/doc/10296},
volume = {57},
year = {2007},
}

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
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 7
SP - 2209
EP - 2237
AB - 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 $X = \mathbf{CP}^1$.
LA - eng
KW - Bergman metric; Monge-Ampère equation; Bergman-Szegö kernel; toric metric; Kähler potential; symplectic potential
UR - http://eudml.org/doc/10296
ER -

References

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