Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Song, Jian; Zelditch, Steve

doi:10.2140/apde.2010.3.295

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Jian Song and Steve Zelditch

Vol. 3 (2010), No. 3, 295–358

DOI: 10.2140/apde.2010.3.295

Abstract

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space $ℋ$ of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold $M$ should be well approximated by finite-dimensional submanifolds $ℬ_{k} \subset ℋ$ of Bergman metrics of height $k$ (Yau, Tian, Donaldson). The Bergman metric spaces are symmetric spaces of type $G_{ℂ} ∕ G$ where $G = U (d_{k} + 1)$ for certain $d_{k}$ . This article establishes some basic estimates for Bergman approximations for geometric families of toric Kähler manifolds.

The approximation results are applied to the endpoint problem for geodesics of $ℋ$ , which are solutions of a homogeneous complex Monge–Ampère equation in $A \times X$ , where $A \subset ℂ$ is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether $ℋ$ -geodesics with fixed endpoints can be approximated by geodesics of $ℬ_{k}$ . Phong and Sturm proved weak $C^{0}$ -convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has $C^{2} (A \times X)$ convergence in the case of toric Kähler metrics, extending our earlier result on $ℂ ℙ^{1}$ .

Keywords

Kähler metrics, Bergman kernels, toric varieties, Monge–Ampère

Mathematical Subject Classification 2000

Primary: 14M25, 35P20, 35S30, 53C22, 53C55

Milestones

Received: 2 March 2009

Revised: 14 November 2009

Accepted: 14 December 2009

Published: 21 July 2010

Authors

Jian Song
Rutgers University Department of Mathematics New Brunswick, NJ 08854 United States

Steve Zelditch
Johns Hopkins University Department of Mathematics 3400 N. Charles Street Baltimore, MD 21218 United States	Northwestern University Department of Mathematics 2033 Sheridan Road Evanston, IL 60208-2370 United States

Vol. 3, No. 3, 2010