On framed instanton bundles and their deformations

We consider a compact twistor space P and assume that there is a surface S ⊂ P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space ([20],[18]). Similar to [6] and [5] 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 [13] 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 [5] is actually an isomorphism of moduli spaces. 0 Introduction We consider the twistor fibration π : P → M over a real four-dimensional compact manifold M with self-dual Riemannian metric. P is a three-dimensional complex manifold with an induced antiholomorphic fixpoint free involution τ , an antipodal map on the twistor fibers (cf. [3], [4], [7]). A line on P is a complex submanifold L ⊂ P with L ∼= IPCI and normal bundle NL|P ∼= OIP1(1) . In particular, twistor fibres are lines. We denote with μ : Z → H the universal Douady-family of lines in P. The involution τ maps lines to lines and consequently induces an antiholomorphic involution on H. Then M appears as a set of fixpoints of τ and moreover as a real-analytic submanifold of H (cf. [3], [19]): H× P ∪ P = Z×HM −→ Z ν −→ P