An $\mathcal{O}$-acyclic variety of even index

Abstract

We give the first examples of $\mathcal{O}$-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over $\mathbb{P}^{1}$ such that any multi-section has even degree over the base $\mathbb{P}^{1}$ and show moreover that we can find such a family defined over $\mathbb{Q}$. This answers affirmatively a question of Colliot-Th\'el\`ene and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel-Jacobi maps.