We study the stability of direct images by Frobenius morphisms. We prove that if the cotangent vector bundle of a nonsingular projective surface $X$ is semistable with respect to a numerically positive polarization divisor satisfying certain conditions, then the direct images of the cotangent vector bundle tensored with line bundles on $X$ by Frobenius morphisms are semistable with respect to the polarization. Hence we see that the de Rham complex of $X$ consists of semistable vector bundles if $X$ has the semistable cotangent vector bundle with respect to the polarization with certain mild conditions.
"Canonical filtrations and stability of direct images by Frobenius morphisms II." Hiroshima Math. J. 38 (2) 243 - 261, July 2008. https://doi.org/10.32917/hmj/1220619460