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

Extended revision of the LATTER half (Abelian varieties case, $A_g$ case) of the original version 1 of arXiv:1406.7772. Many contents added. Submitted

References in corpus (4)

• Moduli of varieties of general type
• Harmonic tropical morphisms and approximation
• Universal moduli spaces of Riemann surfaces
• On the Singularities of Degenerate Abelian Varieties

Cited by in corpus (1)

• Non-archimedean integrals as limits of complex integrals