Bergman Metrics and Geodesics in the Space of Kähler Metrics on Toric Varieties

Geodesics on the infinite dimensional symmetric space H 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×A, where A ⊂ C is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces GC/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 convergence of Bergman to Monge-Ampère geodesics on a general Kähler manifold. In this article we prove convergence in C2(A×X) in the case of toric Kähler metrics, extending our earlier result on CP.