On Framed Instanton Bundles and Their Deformations

Abstract

We consider a compact twistor space P and assume that there is a surface S in P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space. Similar to Donaldson and Buchdahl we examine the restriction of an instanton bundle V equipped with a fixed trivialisation along F to a framed vector bundle over (S,F). First we develope inspired by Huybrechts and Lehn a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)-instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)-instanton bundles and isomorphism classes of framed vector bundles over (S,F) due to Buchdahl is actually an isomorphism of moduli spaces.