Hodge cohomology of invertible sheaves

TL;DR

This paper proves a conjecture relating the Hodge cohomology of invertible sheaves on smooth projective varieties over fields of positive characteristic, under certain conditions including ordinarity and lifting properties.

Contribution

It establishes the conjecture for the case when the torsion order equals the characteristic, assuming the variety is ordinary, extending previous results in characteristic zero.

Findings

Proves the conjecture in characteristic p for n=p under ordinarity.

Extends understanding of Hodge cohomology behavior for torsion sheaves.

Provides conditions under which the dimension equality holds for invertible sheaves.

Abstract

v2: We improved a little bit according to the referee's wishes. v1: On $X$ projective smooth over a field $k$, Pink and Roessler conjecture that the dimension of the Hodge cohomology of an invertible $n$-torsion sheaf $L$ is the same as the one of its $a$-th power $L^a$ if $a$ is prime to $n$, under the assumptions that $X$ lifts to $W_2(k)$ and $dim X\le p$, if $k$ has characteristic $p>0$. They show this if $k$ has characteristic 0 and if $n$ is prime to $p$ in characteristic $p>0$. We show the conjecture in characteristic $p>0$ if $n=p$ assuming in addition that $X$ is ordinary (in the sense of Bloch-Kato).