#### 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
##### Abstract

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space $\mathsc{ℋ}$ of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold $M$ should be well approximated by finite-dimensional submanifolds ${\mathsc{ℬ}}_{k}\subset \mathsc{ℋ}$ of Bergman metrics of height $k$ (Yau, Tian, Donaldson). The Bergman metric spaces are symmetric spaces of type ${G}_{ℂ}∕G$ where $G=U\left({d}_{k}+1\right)$ 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 $\mathsc{ℋ}$, which are solutions of a homogeneous complex Monge–Ampère equation in $A×X$, where $A\subset ℂ$ is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether $\mathsc{ℋ}$-geodesics with fixed endpoints can be approximated by geodesics of ${\mathsc{ℬ}}_{k}$. 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 ${C}^{2}\left(A×X\right)$ 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