# Complete moduli in the presence of semiabelian group action

### Abstract

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 satisyfing 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 compactificatin 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 comapctification of $A_g$ for the second Voronoi decomposition.

Valery Alexeev