On the crystalline Artin--Mumford invariant

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $W=W(k)$. We prove that the torsion subgroup of $H^3_{\mathrm{cris}}(X/W)$ is a birational invariant of smooth projective varieties over $k$. The proof combines the action of correspondences on Hodge--Witt cohomology and on the first Nygaard piece of the de Rham--Witt complex.