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.

中文翻译：

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Sous-groupe de Brauer invariant et hashede itérée

对于数域 $k$ 上的准射影光滑几何积分变体，我们证明迭代下降障碍等价于下降障碍。这概括了 Skorobogatov 的结果，这回答了 Poonen 的一个悬而未决的问题。关键思想是不变 Brauer 子群的概念和不变的\'etale Brauer-Manin 阻塞的概念，用于 $k$-品种配备了一个连接的线性代数群的动作。