Some remarks on Hodge symmetry

TL;DR

This paper discusses conditions under which Hodge symmetry holds for smooth, proper schemes over perfect fields of characteristic p, providing a proof for specific cases involving Hodge-Witt schemes and crystalline cohomology.

Contribution

It establishes new criteria ensuring Hodge symmetry for certain classes of algebraic schemes in positive characteristic.

Findings

Hodge symmetry holds if Hodge-Witt scheme has degenerate Hodge de Rham sequence at E_1.

Hodge symmetry is valid when crystalline cohomology is torsion-free.

Provides conditions linking Hodge-Witt properties and crystalline cohomology to Hodge symmetry.

Abstract

I make some remarks on Hodge symmetry, and prove for instance that if $k$ is a perfect field of characteristic $p>0$ and $X/k$ smooth, proper and Hodge-Witt scheme, and Hodge de Rham sequence of $X$ degenerates at $E_1$ and $X$ has torsion-free crystalline cohomology, then Hodge symmetry holds for $X$.