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

TL;DR

This paper establishes the equivalence of three conditions for principal G-bundles over a smooth projective variety over a finite field, linking fundamental group schemes, Frobenius pullbacks, and stability conditions.

Contribution

It generalizes previous results by proving the equivalence of these conditions in higher dimensions and for reductive groups, extending the understanding of principal bundles over finite fields.

Findings

Equivalence of principal bundle conditions over finite fields.

Characterization of strongly semistable principal G-bundles.

Connection between Frobenius pullbacks and fundamental group schemes.

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 \xi on 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(\xi)^{d-2}) = 0, where d = \dim M, and degree(c_1(E_G(\chi))c_1(\xi)^{d-1}) = 0 for every character \chi of G, where E_G(\chi) is the line bundle over $M$ associated to $E_G$ for \chi. 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.