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

TL;DR

This paper constructs examples of smooth projective varieties over complex function fields with even index, providing counterexamples to several conjectures and answering a question by Colliot-Thélène and Voisin.

Contribution

It introduces the first known examples of $ ext{O}$-acyclic varieties with even index, constructed via families of Enriques surfaces over $ ext{P}^1$, over $ ext{Q}$.

Findings

Constructed $ ext{O}$-acyclic varieties with even index.

Provided counterexamples to the Hasse principle and the integral Hodge conjecture.

Answered a question of Colliot-Thélène and Voisin.

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.