Compactification of the moduli of polarized abelian varieties and mirror symmetry

• Published in: arXiv.org
• Main Author: Zhu, Yuecheng
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.06.2016
• Subjects: Collection; Degeneration; Symmetry
• ISSN: 2331-8422

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.