Boundedness of the p-primary torsion of the Brauer group of products of varieties

TL;DR

This paper proves the boundedness of the p-primary torsion in the Brauer group of product varieties over a field, providing new insights into the structure of the Brauer group in various geometric contexts.

Contribution

It establishes the finite exponent of the quotient of the Brauer group of product varieties and offers a direct proof of boundedness for p-primary torsion in abelian varieties.

Findings

The quotient of the Brauer group of a product of varieties has finite exponent.

The p-primary torsion of the Brauer group of an abelian variety is bounded.

The transcendental Brauer group of a Kummer surface has finite exponent, but can be infinite in positive characteristic.

Abstract

We prove that the quotient of the Brauer group of a product of varieties over k by the sum of the images of the Brauer groups of factors has finite exponent. The bulk of the proof concerns p-primary torsion in characteristic p. Our approach gives a more direct proof of the boundedness of the p-primary torsion of the Brauer group of an abelian variety, as recently proved by D'Addezio. We show that the transcendental Brauer group of a Kummer surface over k has finite exponent, but can be infinite when k is an infinite field of positive characteristic. This answers a question of Zarhin and the author.