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

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

2007-07-20

arxiv.org/abs/0707.3082v3

Analysis & PDE 3 (2) (2010), 295-358

Mathematics

Complex Variables

A substantial revision. More detail is given on estimates in Section 6, and a precise rate of convergence is given. The introd

Scientific paper

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 $\CP^1$.

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