Complete moduli in the presence of semiabelian group action

1999/05/31 by Valery Alexeev · 2 citations
Mathematics · #math.AG #msc:14K10

paper · pdf

published as Ann. of Math. (2), Vol. 155 (2002), no. 3, 611--708 · 98 pages, published version

arxiv created 2004/09/15 · arxiv updated 2009/11/30

Abstract

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 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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