Principal bundles over finite fields

Let M be an irreducible smooth projective variety defined over \bar{{\mathbb F}_p}. 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 \bar{{\mathbb F}_p}. Fix a parabolic subgroup P \subsetneq G, and also fix a strictly anti-dominant character χof P. Let E_G \to M be a principal G-bundle such that the associated line bundle E_G(χ) \to E_G/P is numerically effective. We prove that E_G is given by a homomorphism π(M, x_0)\to G. As a consequence, there is no principal G-bundle E_G \to 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\to E_G/P is a nonconstant morphism.

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