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 ${\overline{\mathcal{AP}}}_{g,d}$ 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 ${\overline{\mathcal{A}}}_{g,\delta }^m$ 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.

62 pages. This is the final version, accepted by the Journal. The major change is on Section 4.2. We removed a wrong lemma thanks to the referee. Instead we provide a new proof and remove the restriction that the base scheme should be defined over complex numbers. Comments welcome!

References in corpus (1)

• Mirror symmetry for log Calabi-Yau surfaces I