On principal bundles over a projective variety defined over a finite field

Abstract

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξon M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomorphic to (F^a_M)^*E_G, where F_M is the absolute Frobenius morphism of M. The principal G-bundle E_G is strongly semistable, degree(c_2(ad(E_G))c_1(ξ)^{d-2}) = 0, where d = \dim M, and degree(c_1(E_G(χ))c_1(ξ)^{d-1}) = 0 for every character χof G, where E_G(χ) is the line bundle over $M$ associated to $E_G$ for χ. The equivalence between the first statement and the third statement was proved by S. Subramanian under the extra assumption that dim(M) = 1 and $G$ is semisimple.