Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus

Abstract

We compare two rational polyhedral admissible decompositions of the cone of positive definite quadratic forms: the perfect cone decomposition and the 2nd Voronoi decomposition. We determine which cones belong to both the decompositions, thus providing a positive answer to a conjecture of V. Alexeev and A. Brunyate. As an application, we compare the two associated toroidal compactifications of the moduli space of principal polarized abelian varieties: the perfect cone compactification and the 2nd Voronoi compactification.

27 pages, 2 figures, final version, to appear in Mathematische Annalen

References in corpus (2)

• On the tropical Torelli map
• Extending Torelli map to toroidal compactifications of Siegel space

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• The wheel classes in the locally finite homology of ${\mathrm{GL}}_n(\mathbb{Z})$, canonical integrals and zeta values