Homogeneous projective bundles over abelian varieties

TL;DR

This paper characterizes homogeneous projective bundles over abelian varieties using projective representations, extending Mukai's work on semi-homogeneous vector bundles and providing insights into the Brauer group.

Contribution

It describes the structure of homogeneous projective bundles via projective representations and relates them to Heisenberg groups, expanding understanding of the Brauer group of abelian varieties.

Findings

Homogeneous bundles correspond to projective representations of commutative algebraic groups.

Irreducible bundles are associated with Heisenberg groups and their standard representations.

Provides a geometric perspective on the Brauer group of abelian varieties.

Abstract

We consider those projective bundles (or Brauer-Severi varieties) over an abelian variety that are homogeneous, i.e., invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative algebraic groups; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semi-homogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties.