Direct images of bundles under Frobenius morphism

Abstract

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X 1 be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F * W is bounded by instability of W⊗Tℓ(Ω1 X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F * W being stable whenever W is stable.