  Let $X$ be a smooth projective variety over an algebraically closed field $k$
of characteristic $p>0$ of $\dim X\geq 4$ and Picard number $\rho(X)=1$.
Suppose that $X$ satisfies $H^i(X,F^{m*}_X(\Omg^j_X)\otimes\Ls^{-1})=0$ for any
ample line bundle $\Ls$ on $X$, and any nonnegative integers $m,i,j$ with
$0\leq i+j<\dim X$, where $F_X:X\rightarrow X$ is the absolute Frobenius
morphism. We prove that by procedures combining taking smooth hypersurfaces of
dimension $\geq 3$ and cyclic covers along smooth divisors, if the resulting
smooth projective variety $Y$ has ample (resp. nef) canonical bundle
$\omega_Y$, then $\Omg_Y$ is strongly stable $($resp. strongly semistable$)$
with respect to any polarization.
