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It is well known in Kähler geometry that the infinite-dimensional symmetric space 
of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds 
of Bergman metrics of height k. Then it is natural to ask whether geodesics in can be approximated by Bergman geodesics in 
. For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.
