Splitting unramified Brauer classes by abelian torsors and the period-index problem

TL;DR

This paper introduces a method using twisted Picard varieties to split unramified Brauer classes on projective varieties and establishes a divisibility relation between the index and period of these classes.

Contribution

It develops a novel approach employing abelian torsors to split Brauer classes and proves a key divisibility property linking index and period.

Findings

Brauer classes can be split using torsors for fixed abelian schemes.

The index of unramified Brauer classes divides a power of their period.

New techniques connect Brauer class splitting with period-index problems.

Abstract

We use twisted relative Picard varieties to split Brauer classes on projective varieties over algebraically closed fields by torsors for a fixed abelian scheme independent of the Brauer class. The construction is also used to prove that the index of an unramified Brauer class divides a fixed power of its period.