Orthogonal bundles over curves in characteristic two

TL;DR

This paper constructs examples of stable orthogonal bundles over algebraic curves in characteristic two, providing counterexamples to Behrend's conjecture on principal G-bundles for certain orthogonal groups.

Contribution

It introduces explicit stable orthogonal bundles with unstable underlying bundles in characteristic two, challenging existing conjectures in the theory of principal G-bundles.

Findings

Counterexamples to Behrend's conjecture for SO(n), n ≥ 7

Existence of stable orthogonal bundles with unstable underlying vector bundles in characteristic two

Insights into the structure of principal G-bundles in positive characteristic

Abstract

Let X be a smooth projective curve of genus g \geq 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 \geq 7.