Symmetry and parity in Frobenius action on cohomology

TL;DR

This paper demonstrates symmetry properties of Frobenius action on crystalline cohomology, leading to new insights on Betti numbers and extending classical results to varieties over finite fields.

Contribution

It establishes symmetry of Newton polygons for Frobenius on crystalline cohomology and generalizes Betti number parity results to arbitrary varieties over finite fields.

Findings

Newton polygons of Frobenius are symmetric for proper smooth varieties

Odd-degree Betti numbers are even for proper smooth varieties over any field

Generalization of Betti number parity to varieties over finite fields

Abstract

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincar\'e duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.