Bergman metrics and geodesics in the space of K\"ahler metrics on toric varieties

TL;DR

This paper proves convergence of Bergman geodesics to Monge-Ampère geodesics in the space of Kähler metrics on toric varieties, extending previous results to a higher regularity setting.

Contribution

It establishes C^2 convergence of Bergman geodesics to Monge-Ampère geodesics specifically for toric Kähler metrics, advancing understanding of metric approximations.

Findings

Proved C^2 convergence of Bergman geodesics to Monge-Ampère geodesics on toric varieties.

Extended earlier C^0 convergence results to higher regularity in the toric case.

Provided new insights into the approximation of infinite-dimensional geodesics by finite-dimensional metrics.

Abstract

Geodesics on the infinite dimensional symmetric space $\hcal$ of K\"ahler metrics in a fixed K\"ahler class on a projective K\"ahler manifold X are solutions of a homogeneous complex Monge-Amp\`ere equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Amp\`ere geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Amp\`ere geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A \times X)$ in the case of toric K\"ahler metrics, extending our earlier result on $\CP^1$.