Essentially finite vector bundles on varieties with trivial tangent bundle

Abstract

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