Topology of Algebraic Varieties

The Torelli map associates to a genus g curve its Jacobian - a $g$-dimensional principally polarized abelian variety. It turns out, by the works of Mumford and Namikawa in the 1970s (resp. Alexeev and Brunyate in 2010s), that the Torelli map extends to a morphism from the Deligne-Mumford moduli of stable curves to the Voronoi (resp. perfect cone) toroidal compactification of the moduli of abelian varieties. The Prym map associates to an etale double cover of a genus g curve its Prym - a principally polarized $(g-1)$-dimensional abelian variety. The indeterminacy locus of the extension of this map to a map to the Voronoi toroidal compactification was studied by Alexeev, Birkenhake, and Hulek in 2000s, and in this talk we discuss the extension of the Prym map to a map to the perfect cone toroidal compactification, a unified approach to all the results above, and applications to the period map for cubic threefolds.