Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -

TL;DR

This paper introduces a new compactification of the moduli space of principally polarized abelian varieties by attaching flat tori, explicitly describing Gromov-Hausdorff limits and their behavior in holomorphic families.

Contribution

It explicitly constructs a tropical geometric compactification of $A_g$ and characterizes Gromov-Hausdorff limits, extending previous work on curves to abelian varieties.

Findings

Explicit description of the compactification of $A_g$.

Determination of Gromov-Hausdorff limits for abelian varieties.

Identification of special subsets of the boundary in holomorphic limits.

Abstract

We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and show that they form special non-trivial subsets of the whole boundary. We also do it for algebraic curves case and observe a crucial difference with the case of abelian varieties.