Calabi-Yau threefolds over finite fields and torsion in cohomologies

TL;DR

This paper explores Calabi-Yau threefolds over finite fields, providing counterexamples to existing conjectures, computing p-adic cohomologies, and revealing new phenomena in positive characteristic algebraic geometry.

Contribution

It presents counterexamples to conjectures, computes p-adic cohomologies of specific Calabi-Yau threefolds, and investigates Hodge number invariance in positive characteristic.

Findings

Counterexample to Joshi's conjecture on lifting Calabi-Yau threefolds.

Computed p-adic cohomologies of Cynk-van Straten and Hirokado threefolds.

Hodge numbers are not invariants in positive characteristic.

Abstract

We study various examples of Calabi-Yau threefolds over finite fields. In particular, we provide a counterexample to a conjecture of K. Joshi on lifting Calabi-Yau threefolds to characteristic zero. We also compute the p-adic cohomologies of some Calabi-Yau threefolds constructed by Cynk-van Straten which have remarkable arithmetic properties, as well as those of the Hirokado threefold. These examples and computations answer some outstanding questions of B. Bhatt, T. Ekedahl, van der Geer-Katsura and Patakfalvi-Zdanowicz, and shed new light on the Beauville-Bogomolov decomposition in positive characteristic. Our tools include p-adic Hodge theory as well as classical algebraic topology. We also give potential examples showing that Hodge numbers of threefolds in positive characteristic are not derived invariants, contrary to the case of characteristic zero.