Compactification of the moduli of polarized abelian varieties and mirror symmetry

TL;DR

This paper links Olsson's compactification of polarized abelian varieties to KSBA stable pairs, introduces a mirror symmetry-based construction, and constructs a toroidal compactification using Mori fans from mirror families.

Contribution

It provides a new interpretation of Olsson's compactification via stable pairs and offers an alternative mirror symmetry-based construction method.

Findings

Olsson's compactification can be interpreted through KSBA stable pairs.

A canonical set of divisors associated with each cusp is identified.

A toroidal compactification is constructed using Mori fans from mirror families.

Abstract

We show that Martin Olsson's compactification of moduli space of polarized abelian varieties in \cite{ols08} can be interpreted in terms of KSBA stable pairs. We find that there is a canonical set of divisors $S(K_2)$ associated with each cusp. Near the cusp, a polarized semiabelic scheme $(\mathcal{X}, G,\mathcal{L})$ is the canonical degeneration given by the compactification if and only if $(\mathcal{X},G,\Theta)$ is an object in $\overline{\mathscr{AP}}_{g,d}$ for any $\Theta\in S(K_2)$. Moreover, we give an alternative construction of the compactification by using mirror symmetry. We construct a toroidal compactification $\overline{\mathscr{A}}_{g,\delta}^m$ that is isomorphic to Olsson's compactification over characteristic zero. The data needed for a toroidal compactification is a collection of fans. We obtain the collection of fans from the Mori fans of the minimal models of the mirror families.