Minimal abelian varieties of algebras, I

TL;DR

This paper proves that non-affine abelian varieties always contain a nontrivial strongly abelian subvariety, providing foundational insight for future research on minimal abelian varieties.

Contribution

It establishes a key structural property of non-affine abelian varieties, advancing the understanding of their subvariety composition.

Findings

Non-affine abelian varieties have nontrivial strongly abelian subvarieties.

The result sets the stage for studying minimal abelian varieties in subsequent work.

Abstract

We show that any abelian variety that is not affine has a nontrivial strongly abelian subvariety. In later papers in this sequence we apply this result to the study of minimal abelian varieties.