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

Abstract

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space $\mathcal{\mathscr{H}}$ of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold $M$ should be well approximated by finite-dimensional submanifolds ${\mathcal{ℬ}}_k\subset \mathcal{\mathscr{H}}$ 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 $\mathcal{\mathscr{H}}$, 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 $\mathcal{\mathscr{H}}$-geodesics with fixed endpoints can be approximated by geodesics of ${\mathcal{ℬ}}_k$. 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 $C^2\left(A\times X\right)$ convergence in the case of toric Kähler metrics, extending our earlier result on $ℂ{\mathbb{P}}^1$.