The Fibration Method over Real Function Fields

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-Thélène 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.