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2010 Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
Jian Song, Steve Zelditch
Anal. PDE 3(3): 295-358 (2010). DOI: 10.2140/apde.2010.3.295

Abstract

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold M should be well approximated by finite-dimensional submanifolds k of Bergman metrics of height k (Yau, Tian, Donaldson). The Bergman metric spaces are symmetric spaces of type GG where G=U(dk+1) for certain dk. 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 , which are solutions of a homogeneous complex Monge–Ampère equation in A×X, where A is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether -geodesics with fixed endpoints can be approximated by geodesics of k. Phong and Sturm proved weak C0-convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has C2(A×X) convergence in the case of toric Kähler metrics, extending our earlier result on 1.

Citation

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Jian Song. Steve Zelditch. "Bergman metrics and geodesics in the space of Kähler metrics on toric varieties." Anal. PDE 3 (3) 295 - 358, 2010. https://doi.org/10.2140/apde.2010.3.295

Information

Received: 2 March 2009; Revised: 14 November 2009; Accepted: 14 December 2009; Published: 2010
First available in Project Euclid: 20 December 2017

zbMATH: 1282.35428
MathSciNet: MR2672796
Digital Object Identifier: 10.2140/apde.2010.3.295

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

Rights: Copyright © 2010 Mathematical Sciences Publishers

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