On Principal Bundles over a Projective Variety Defined over a Finite Field

Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k–rational point x0. Let ̟(M,x0) denote the corresponding fundamental group–scheme introduced by Nori. Let EG be a principal G–bundle over M , where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξ on M . We prove that the following three statements are equivalent: (1) The principal G–bundle EG over M is given by a homomorphism̟(M,x0) −→ G. (2) There are integers b > a ≥ 1, such that the principal G–bundle (F b M )∗EG is isomorphic to (F a M )∗EG, where FM is the absolute Frobenius morphism of M . (3) The principal G–bundle EG is strongly semistable, degree(c2(ad(EG))c1(ξ) ) = 0, where d = dimM , and degree(c1(EG(χ))c1(ξ) ) = 0 for every character χ of G, where EG(χ) is the line bundle over M associated to EG for χ. In [16], the equivalence between the first statement and the third statement was proved under the extra assumption that dimM = 1 and G is semisimple.