An O -acyclic variety of even index: With an appendix by Olivier Wittenberg

Abstract

We give the first examples of 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 P1 such that any multi-section has even degree over the base P1 and show moreover that we can find such a family defined overQ. This answers affirmatively a question of Colliot-Thélène 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.