Compactification of moduli spaces and mirror symmetry

Olsson gives modular compactifications of the moduli of toric pairs and the moduli of polarized abelian varieties A [subscript g,δ] in (Ols08). We give alternative constructions of these compactifications by using mirror symmetry. Our constructions are toroidal compactifications. 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. Moreover, we reinterpretate the compactification of A [subscript g,δ] in terms of KSBA stable pairs. We find that there is a canonical set of divisors S(K₂) associated with each cusp. Near the cusp, a polarized semiabelic scheme (X, G, L) is the canonical degeneration given by the compactification if and only if (X, G, Θ) is an object in A P [subscript g,d] for any Θ ∈ S(K₂). The two compactifications presented here are a part of a general program of applying mirror symmetry to the compactification problem of the moduli of Calabi–Yau manifolds. This thesis contains the results in (Zhu14b) and (Zhu14a).