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For a quasi-projective smooth geometrically integral variety over a numberfield $k$, we prove that the iterated descent obstruction is equivalent to thedescent obstruction. This generalizes a result of Skorobogatov, and thisanswers an open question of Poonen. The key idea is the notion of invariantBrauer subgroup and the notion of invariant \'etale Brauer-Manin obstructionfor a $k$-variety equipped with an action of a connected linear algebraicgroup.

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