Orthogonal Bundles over Curves in Characteristic Two

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. In [Be] (Conjecture 7.6) K. Behrend conjectured that for any principal G-bundle EG over X the canonical reduction EP has no infinitesimal deformations, or equivalently, that the vector space H(X,EP × P g/p) is zero. Here p and g are the Lie algebras of P and G respectively. Behrend’s conjecture implies that the canonical reduction EP is defined over the same base field as EG. We note that this conjecture holds for the structure groups GL(n) and Sp(2n) in any characteristic, and also for SO(n) in any characteristic different from two — see [H] section 2. On the other hand, a counterexample to Behrend’s conjecture for the exceptional group G2 in characteristic two has been constructed recently by J. Heinloth in [H] section 5. In this note we focus on SO(n)-bundles in characteristic two. As a starting point we consider the rank-2 vector bundle F∗L given by the direct image under the Frobenius map F of a line bundle L over the curve X and observe (Proposition 4.4) that the SO(3)-bundle A := End0(F∗L) is stable, but that its underlying vector bundle is unstable and, in particular, destabilized by the rank-2 vector bundle F∗OX . We use this observation to show that the SO(7)-bundle F∗OX ⊕ (F∗OX) ∗ ⊕A equipped with the natural quadratic form gives a counterexample to Behrend’s conjecture. Replacing A by  = End(F∗L) we obtain in the same way a counterexample for SO(8), and more generally for SO(n) with n ≥ 7 after adding direct summands of hyperbolic planes. Note that Behrend’s conjecture holds for SO(n) with n ≤ 6 because of the exceptional isomorphisms with other classical groups. The first three sections are quite elementary and recall well-known facts on quadratic forms, orthogonal groups and their Lie algebras, as well as orthogonal bundles in characteristic two. In the last section we give an example (Proposition 6.1) of an unstable SO(7)-bundle having its canonical reduction only defined after an inseparable extension of the base field. This note can be considered as an appendix to the recent paper [H] by J. Heinloth, whom I would like to thank for useful discussions. 2000 Mathematics Subject Classification. Primary 14H60, 14H25.