Let $\mathbb R(C)$ be the function field of a smooth, irreducible projective curve over $\mathbb R$. Let $X$ be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism $f$ onto a smooth projective rational variety with a smooth generic fibre over $\mathbb R(C)$. Assume that the cohomological obstruction introduced by Colliot-Th\'el\`ene is the only one to the local-global principle for rational points for the smooth fibres of $f$ over $\mathbb R(C)$-valued points. Then we show that the same holds for $X$, too, by adopting the fibration method similarly to Harpaz--Wittenberg. We also show that the strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran.

中文翻译：

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实函数场上的纤维化方法

令 $\mathbb R(C)$ 是 $\mathbb R$ 上平滑的、不可约的投影曲线的函数域。令 $X$ 是一个光滑的、射影的、几何上不可约的变体，它配备了一个在 $\mathbb R(C)$ 上具有光滑泛型纤维的光滑射影有理变体上的显性态射 $f$。假设 Colliot-Th\'el\`ene 引入的上同调阻塞是 $f$ 超过 $\mathbb R(C)$ 值点的光滑纤维的有理点的局部全局原理的唯一一个。然后我们通过采用与 Harpaz--Wittenberg 类似的纤维化方法证明 $X$ 也是如此。我们还表明，使用 Haran 定理，$n$-fold Massey 乘积的强消失猜想适用于虚拟上同调维度最多 $1$ 的领域。