Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

Abstract

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.

arXiv comment41 pages, uses tabmac.sty, Dedicated to David Eisenbud on the occasion of his 65th birthday; v2: fixed some typos and added references

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Cited by in corpus (11)

• Tropicalization of classical moduli spaces
• The Universal Kummer Threefold
• Invariant theory of ${\bigwedge }^3(9)$ and genus 2 curves
• Alternating trilinear forms on a 9-dimensional space and degenerations of (3,3)-polarized Abelian surfaces
• The geometry of the Coble cubic and orbital degeneracy loci
• Gradings of Lie algebras, magical spin geometries and matrix factorizations
• Vector bundles on genus 2 curves and trivectors
• The Coble Quadric
• Free resolutions of orbit closures for the representations associated to gradings on Lie algebras of type $E_6$, $F_4$ and $G_2$
• Topics on the geometry of homogeneous spaces
• Smooth Surfaces in Smooth Fourfolds with Vanishing First Chern Class