Complete moduli in the presence of semiabelian group action

TL;DR

This paper constructs and describes the structure of moduli spaces for projective varieties with semiabelian group actions and ample divisors, linking them to polytopal configurations and compactifications of abelian varieties.

Contribution

It establishes the existence and detailed structure of moduli spaces for varieties with semiabelian actions, connecting them to secondary polytopes and toroidal compactifications.

Findings

Connected components are proper.

Components are described by polytopal configurations.

Provides a compactification of the moduli space of abelian varieties.

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 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.