# Sous-groupe de Brauer invariant et obstruction de descente it\'er\'ee

## TL;DR

This paper proves that for certain algebraic varieties over number fields, the iterated descent obstruction coincides with the descent obstruction, extending previous results and answering an open question.

## Contribution

It introduces the invariant Brauer subgroup and invariant étale Brauer-Manin obstruction, generalizing Skorobogatov's result and resolving Poonen's open question.

## Key findings

- Iterated descent obstruction equals descent obstruction for smooth geometrically integral varieties.
- Introduces the notion of invariant Brauer subgroup and invariant étale Brauer-Manin obstruction.
- Generalizes previous results by Skorobogatov and answers Poonen's open question.

## Abstract

For a quasi-projective smooth geometrically integral variety over a number field $k$, we prove that the iterated descent obstruction is equivalent to the descent obstruction. This generalizes a result of Skorobogatov, and this answers an open question of Poonen. The key idea is the notion of invariant Brauer subgroup and the notion of invariant \'etale Brauer-Manin obstruction for a $k$-variety equipped with an action of a connected linear algebraic group.