Orthogonal bundles over curves in characteristic two

Abstract

Let X be a smooth projective curve of genus g ≥ 2 defined over a field of characteristic two. We give examples of stable orthogonal bundles with unstable underlying vector bundles and use them to give counterexamples to Behrend's conjecture on the canonical reduction of principal G-bundles for G = SO(n) with n ≥ 7. Let X be a smooth projective curve of genus g ≥ 2 and let G be a connected reductive linear algebraic group defined over a field k of arbitrary characteristic. One associates to any principal G-bundle EG over X a reduction EP of EG to a parabolic subgroup P ⊂ G, the so-called canonical reduction — see e.g. (Ra), (Be), (BH) or (H) for its definition. We only mention here that in the case G = GL(n) the canonical reduction coincides with the Harder-Narasimhan filtration of the rank-n vector bundle associated to EG.