Complete Moduli in the Presence of Semiabelian Group Action

Abstract

I prove the existence, and describe the structure, of moduli space of pairs (P, Θ) consisting of a projective variety P with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions.Every connected component of this moduli space is proper.A component containing a projective toric variety is described by a configuration of several polytopes, the main one of which is the secondary polytope.On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of A g .The main irreducible component of this compactification is described by an "infinite periodic" analog of the secondary polytope and coincides with the toroidal compactification of A g for the second Voronoi decomposition.