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

Abstract

Geodesics on the infinite dimensional symmetric space $\hcal$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X are solutions of a homogeneous complex Monge-Ampère 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-Ampère 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-Ampère geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A \times X)$ in the case of toric Kähler metrics, extending our earlier result on $\CP^1$.

A substantial revision. More detail is given on estimates in Section 6, and a precise rate of convergence is given. The introduction is updated to take into account subsequent work by the authors, partly in collaboration with Y. Rubinstein