Stability of Frobenius direct images over surfaces

TL;DR

This paper investigates the stability properties of Frobenius direct images of vector bundles over algebraic surfaces in positive characteristic, establishing conditions under which semistability and stability are preserved.

Contribution

It proves that for surfaces with semistable cotangent bundle and positive slope, the Frobenius pushforward of semistable or stable bundles remains semistable or stable under certain characteristic bounds.

Findings

Frobenius pushforward preserves stability under specified conditions

Stability is maintained for bundles of any rank when characteristic is sufficiently large

Results apply to smooth projective surfaces with semistable cotangent bundle

Abstract

Let $X$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $p> 0$ with $\Omega_{X}^{1}$ semistable and $\mu(\Omega_{X}^{1})>0$. For any semistable (resp. stable) bundle $W$ of rank $r$, we prove that $F_*W$ is semistable (resp. stable) when $p\geq r(r-1)^2+1$.