Invariant Brauer group of an abelian variety

TL;DR

This paper investigates the invariant Brauer group associated with abelian varieties, providing explicit bounds, examples with non-trivial groups, and exploring its behavior over various fields and characteristics.

Contribution

It introduces the invariant Brauer group for abelian varieties, offers explicit bounds over complex numbers, and constructs examples with non-trivial groups in all dimensions.

Findings

Invariant Brauer group is an elementary abelian 2-group over complex numbers.

Many abelian varieties have trivial invariant Brauer group.

Constructed examples of abelian varieties with invariant Brauer group of order 2.

Abstract

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper bound on the rank. We exhibit many cases in which the invariant Brauer group is zero, and construct complex abelian varieties in every dimension starting with 2, both simple and non-simple, with invariant Brauer group of order 2. We also address the situation in finite characteristic and over non-closed fields.