Stability of Frobenius direct images over surfaces

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 $p\geq r(r-1)^2+1$.