LitNote-Alexeev-2004-Complete moduli in the presence of semiabelian group action-Ale04

- CiteKey: “Ale04” - Type: report - Title: “Complete moduli in the presence of semiabelian group action,” - Author: “Alexeev, Valery;” - Publisher: “arXiv,” - Year: 2004 - Keywords: “14K10”; “Mathematics - Algebraic Geometry”

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.

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