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 can be interpreted in terms of KSBA stable pairs. We find that any degenerating family of polarized abelic sheme over a local normal base is equipped with a canonical set of divisors S ( K 2 ) S(K_2) . Choosing any divisor Θ \Theta from the set S ( K 2 ) S(K_2) , we get a KSBA stable pair. Then the limit in the moduli space of KSBA pairs A P ¯ g , d \overline {\mathscr {AP}}_{g,d} agrees with the canonical degeneration given by Martin Olsson’s compactification. Moreover, we give an alternative construction of the compactification by using mirror symmetry. We construct a toroidal compactification A ¯ g , δ m \overline {\mathscr {A}}_{g,\delta }^m that is isomorphic to Olsson’s compactification over characteristic zero. The collection of fans needed for a toroidal compactification is obtained from the Mori fans of the minimal models of the mirror families.