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 $\dim X\geq 4$ and Picard number $ρ(X)=1$. 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 $ω_Y$, then $\Omg_Y$ is strongly stable $($resp. strongly semistable$)$ with respect to any polarization.

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