Stability of sheaves of locally closed and exact forms

TL;DR

This paper investigates the stability properties of sheaves of exact and closed forms on smooth projective varieties over fields of positive characteristic, establishing conditions under which these sheaves are stable or semi-stable.

Contribution

It provides new stability criteria for sheaves of exact and closed forms based on the semi-stability of tensor powers of the cotangent bundle.

Findings

Sheaf B^1_X of exact 1-forms is stable under certain semi-stability conditions.

Sheaf B^2_X of exact 2-forms is stable on surfaces with semi-stable cotangent bundle.

Sheaf Z^1_X of closed 1-forms is stable for p>3 and semi-stable for p=3.

Abstract

For any smooth projective variety $X$ of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$ with $\mu(\Omega^1_X)>0$. If ${\rm T}^{\ell}(\Omega^1_X)$ ($0<\ell<n(p-1)$) are semi-stable, then the sheaf $B^1_X$ of exact 1-forms is stable. When $X$ is a surface with $\mu(\Omega^1_X)>0$ and $\Omega^1_X$ is semi-stable, the sheaf $B^2_X$ of exact 2-forms is also stable. Moreover, under the same condition, the sheaf $Z^1_X$ of closed 1-forms is stable when $p>3$, and $Z^1_X$ is semi-stable when $p=3$.