Instability of Truncated Symmetric Powers of sheaves

TL;DR

This paper establishes bounds on the instability of truncated symmetric powers of sheaves on smooth projective varieties in characteristic p, and explores implications for Frobenius pushforwards and sheaves of differential forms.

Contribution

It provides new upper bounds for the instability of truncated symmetric powers and analyzes their impact on Frobenius pushforwards and stability of differential form sheaves.

Findings

Bound on instability of truncated symmetric powers in terms of known invariants.

Upper bounds for Frobenius direct images of sheaves.

Conditions for slope semi-stability of sheaves of differential forms.

Abstract

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $F_X:X\rightarrow X$ be the absolute Frobenius morphism, and $\E$ a torsion free sheaf on $X$. We give a upper bound of instability of truncated symmetric powers $\mathrm{T}^l(\E)(0\leq l\leq\rk(\E)(p-1))$ in terms of $L_{\max}(\Omg^1_X)$, $\mathrm{I}(\Omg^1_X)$ and $\mathrm{I}(\E)$ (Theorem \ref{InstabTl}). As an application, We obtain a upper bound of Frobenius direct image ${F_X}_*(\E)$ and some sufficient conditions of slope semi-stability of ${F_X}_*(\E)$. In addition, we study the slope (semi)-stability of sheaves of locally exact (closed) forms $B^i_X$ ($Z^i_X$).