Sous-groupe de Brauer invariant pour un groupe alg\'ebrique connexe quelconque

TL;DR

This paper introduces an invariant Brauer subgroup for smooth varieties with connected algebraic group actions and proves its equivalence to the étale Brauer-Manin obstruction, extending previous foundational results.

Contribution

It generalizes the concept of invariant Brauer subgroups and the étale Brauer-Manin obstruction to broader classes of algebraic group actions, not limited to linear groups.

Findings

Invariant Brauer subgroup is introduced for smooth varieties with connected algebraic group actions.

The invariant Brauer-Manin obstruction is shown to be equivalent to the étale Brauer-Manin obstruction.

Extends previous results to non-linear algebraic groups and broader varieties.

Abstract

In this paper, for a smooth variety equiped with an action of a connected algebraic group (not necessary linear), we introduce the notion of invariant Brauer sub-group and the notion of invariant \'etale Brauer-Manin obstruction. Then we prove that this obstruction is equivalent to the \'etale Brauer-Manin obstruction. This extends the main notion and the key result of [C17]. As a corollary, this also extends a result of B. Creutz.