Complete moduli in the presence of semiabelian group action

I prove the existence, and describe the structure, of moduli space of pairs (P, Theta) 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.