Complete Moduli in the Presence of Semiabelian Group Action

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 of which is the secondary polytope of Gelfand-Kapranov-Zelevinsky. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of Ag. The main irreducible component of this compactification is described by an ”infinite periodic ” analog of secondary polytope and coincides with the toroidal compactification of