Essentially finite vector bundles on varieties with trivial tangent bundle

Abstract

Abstract. Let X be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle T X is trivial. Let F X : X −→ X be the absolute Frobenius morphism of X. We prove that for any n ≥ 1, the n-fold composition F n X is a torsor over X for a finite group-scheme that depends on n. For any vector bundle E −→ X, we show that the direct image (F n X ) * E is essentially finite (respectively, F -trivial) if and only if E is essentially finite (respectively, F -trivial).