The fibration method over real function fields

Abstract

AbstractLet $$\mathbb R(C)$$ R ( C ) be the function field of a smooth, irreducible projective curve over $$\mathbb R$$ 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 $$\mathbb R(C)$$ 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 $$\mathbb R(C)$$ R ( C ) -valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.