Compactification of the moduli of polarized abelian varieties and mirror symmetry

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.