On the F-fundamental group scheme

Abstract

Let X be a smooth projective variety defined over a perfect field k of positive characteristic, and let F_X be the absolute Frobenius morphism of X. For any vector bundle E→X, and any polynomial g with non-negative integer coefficients, define the vector bundle g^- (E) using the powers of F_X and the direct sum operation. We construct a neutral Tannakian category using the vector bundles with the property that there are two distinct polynomials f and g with non-negative integer coefficients such that f^-(E) = g^-(E). We also investigate the group scheme defined by this neutral Tannakian category.