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

Published online by Cambridge University Press:  26 October 2009

Indranil Biswas
Indranil Biswas
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India, indranil@math.tifr.res.in
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Abstract

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 homomorphism (M,x0)→G.

2. There are integers b > a ≥ 1, such that the principal G-bundle (FbM)* EG is isomorphic to (FaM) * EG where FM is the absolute Frobenius morphism of M.

3. The principal G-bundle EG is strongly semistable, the degree(c2(ad(EG))c1 (ξ)d−2 = 0, where d = dimM, and the degree(c1(EG(χ))c1(ξ)d−1) = 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.

Keywords

Fundamental group-schemeprincipal bundlefinite field
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 2 , October 2009 , pp. 209 - 221
Copyright
Copyright © ISOPP 2009

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