Reduction of structure group of principal bundles over aprojective manifold with Picard number one

Abstract

Let X be a complex projective manifold with Pic(X) ≅ Z. Let G be a connected reductive algebraic group. A principal G-bundle will becalled split if it admits a reductionof structure group to a maximal torus of G. This corresponds tothe usual definition of a split vector bundle as a direct sum ofline bundles. Let E be a principal G-bundle on X, and let ρ G→ G* be an injective homomorphism.We prove that if the associated principal G*-bundle E(G*) is split, then E admits a reduction to a Borel subgroupof G. Furthermore, if we assume that X is Fano or has trivialcanonical bundle, then E is also split. We apply this to obtain generalizations for principal bundles of two results about vector bundles on projective spaces. Namely, a principal G-bundle on CP is split if its restriction to a plane CP CPis split. Also, a principal G-bundle E on CP is trivial if there is a point p such that the restriction of E to every line through p is trivial.