Bergman metrics and geodesics in the space of Kähler metrics on principally polarized 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.

To appear in Journal of the Institute of Mathematics of Jussieu

References in corpus (3)

• Bergman metrics and geodesics in the space of Kähler metrics on toric varieties
• Probability measures related to geodesics in the space of Kähler metrics
• Convergence of Bergman geodesics on CP^1