Vol. 3, No. 3, 2010

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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

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.

Kähler metrics, Bergman kernels, toric varieties, Monge–Ampère
Mathematical Subject Classification 2000
Primary: 14M25, 35P20, 35S30, 53C22, 53C55
Received: 2 March 2009
Revised: 14 November 2009
Accepted: 14 December 2009
Published: 21 July 2010
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