Bergman metrics and geodesics in the space of K\"{a}hler metrics on principally polarized Abelian varieties

TL;DR

This paper demonstrates the existence of smooth and asymptotic approximations of geodesics in the space of Kähler metrics on principally polarized Abelian varieties using Bergman metrics, extending previous results to a broader class.

Contribution

It establishes $C^{ abla}$-smooth and complete asymptotic expansions for Bergman geodesics on principally polarized Abelian varieties, generalizing prior work on toric varieties.

Findings

Existence of $C^{ abla}$-smooth approximations of geodesics.

Complete asymptotic expansion for Bergman geodesics.

Generalization of harmonic map approximations to Abelian varieties.

Abstract

It's well-known in \kahler geometry that the infinite dimensional symmetric space $\hcal$ of smooth \kahler metrics in a fixed \kahler class on a polarized \kahler manifold is well approximated by finite dimensional submanifolds $\bcal_k \subset \hcal$ of Bergman metrics of height $k$. Then it's natural to ask whether geodesics in $\hcal$ can be approximated by Bergman geodesics in $\bcal_k$. For any polarized \kahler manifold, the approximation is in the $C^0$ topology. While Song-Zelditch proved the $C^2$ convergence for the torus-invariant metrics over toric varieties. In this article, we show that some $C^{\infty}$ approximation exists as well as a complete asymptotic expansion for principally polarized Abelian varieties. We also get a $C^\infty$ complete asymptotic expansion for harmonic maps into $\bcal_k$ which generalizes the work of Rubinstein-Zelditch on toric varieties.