Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Abstract

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, F_X : X → X the absolute Frobenius morphism on X. Let E be a rational GL_n(k)-bundle on X, and ρ : GL_n(k) → GL_m(k) a rational GL_n(k)-representation of degree at most d such that ρ maps the radical RGL_n(k)) of GL_n(k) into the radical R(GL_m(k)) of GL_m(k). We show that if $F_X^{N*}(E)$ is semistable for some integer $N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)$, then the induced rational GL_m(k)-bundle E(GL_m(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image $F_{X*}(\rho*(\Omega_X^1))$, where $\rho*(\Omega_X^1)$ is the vector bundle obtained from $\Omega_X^1$ via the rational representation ρ.