Strong Stability of Cotangent Bundles of Cyclic Covers

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