Principal Bundles Over Finite Fields

Let M be an irreducible smooth projective variety defined over . Let ϖ(M, x 0) be the fundamental group scheme of M with respect to a base point x 0. Let G be a connected semisimple linear algebraic group over . Fix a parabolic subgroup P ⊊ G, and also fix a strictly antidominant character χ of P. Let E G  → M be a principal G-bundle such that the associated line bundle E G (χ) → E G /P is numerically effective. We prove that E G is given by a homomorphism ϖ(M, x 0) → G. As a consequence, there is no principal G-bundle E G  → M such that degree(ϕ*E G (χ)) > 0 for every pair (Y, ϕ), where Y is an irreducible smooth projective curve, and ϕ: Y → E G /P is a nonconstant morphism.