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

TL;DR

This paper develops a systematic geometric approach to studying Vinberg theta-representations using free resolutions and degeneracy loci, connecting these to moduli spaces of Abelian varieties.

Contribution

It introduces a novel method combining free resolutions with degeneracy loci to analyze Vinberg theta-representations and their relation to Abelian varieties.

Findings

Demonstrates the effectiveness of free resolutions in studying degeneracy loci

Connects geometric representation theory to moduli spaces of Abelian varieties

Provides examples illustrating the approach

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.