Stability of Frobenius direct images over surfaces

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

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