Counterexample to Néron-Ogg-Shafarevich criterion for Calabi-Yau threefolds

Abstract

For any prime $p>5$ we construct a Calabi-Yau threefold $X$ defined over a finite extension $K$ of ${\mathbb{Q}}_p$ such that every model of $X$ over $\mathrm{Spec}{\mathcal{O}}_K$ has singular special fiber, yet the Galois action on the $\ell$-adic cohomology group $H_{\acute{e}t}^3(X,{\mathbb{Q}}_{\ell })$ is unramified for $\ell \ne p$ and crystalline for $\ell =p$. This provides a counterexample to the analogue of the Néron-Ogg-Shafarevich criterion for three-dimensional Calabi-Yau manifolds.