Strong Stability of Cotangent Bundles of Cyclic Covers

TL;DR

This paper proves the strong stability of cotangent bundles for certain cyclic covers of smooth projective varieties in positive characteristic, under specific cohomological and geometric conditions.

Contribution

It introduces a method combining hypersurfaces and cyclic covers to establish strong stability of cotangent bundles in positive characteristic.

Findings

Strong stability of cotangent bundles for cyclic covers with ample canonical bundle.

Conditions under which cotangent bundles are strongly semistable.

Applicability to varieties satisfying specific cohomological vanishing conditions.

Abstract

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.