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Pré-Publication, Document De Travail Année : 2020

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

Résumé

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.

Dates et versions

hal-03017757 , version 1 (21-11-2020)

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John Christian Ottem, Fumiaki Suzuki, Olivier Wittenberg. An $\mathcal{O}$-acyclic variety of even index. 2020. ⟨hal-03017757⟩
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