Principal bundles, quasi-abelian varieties and structure of algebraic groups

TL;DR

This paper classifies principal bundles over anti-affine schemes with affine, commutative groups, leading to a comprehensive classification of quasi-abelian varieties and simplifying the understanding of smooth algebraic groups.

Contribution

It provides a classification of quasi-abelian varieties over a field by analyzing principal bundles over anti-affine schemes with affine, commutative structural groups.

Findings

Classification of principal bundles over anti-affine schemes

Reduction of smooth group scheme classification to quasi-abelian varieties

New insights into the structure of algebraic groups

Abstract

We classify principal bundles over anti-affine schemes with affine and commutative structural group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes with no non constant global functions). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes.