Bergman Metrics and Geodesics in The Space Of Kähler Metrics On Toric Varieties

A guiding principle in Kähler geometry is that the infinite-dimensional symmetric space ℋ of Kähler metrics in a fixed Kähler class on a polarized projective Kähler manifold M should be well approximated by finite-dimensional submanifolds ß_k ⊂ 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(d_k+1) 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 H, which are solutions of a homogeneous complex Monge–Ampère equation in A×X, where A ⊂ C is an annulus. Donaldson, Arezzo and Tian, and Phong and Sturm raised the question whether H-geodesics with fixed endpoints can be approximated by geodesics of Bk. Phong and Sturm proved weak C⁰-convergence of Bergman to Monge–Ampère geodesics on a general Kähler manifold. Our approximation results show that one has C².A × X convergence in the case of toric Kähler metrics, extending our earlier result on CP1.