Hyperholomorphic bundles

Abstract

Hyperholomorphic bundle is a bundle with connection defined over a hyperkaehler manifold such that this connection is holomorphic with respect to all complex structures induced by a hyperkaehler structure. A hyperholomorphic connection is Yang-Mills. If a stable bundle has first two Chern classes invariant with respect to ${\Bbb H}$, then it admits a hyperholomorphic connection. Deformation spaces of hyperholomorphic bundles are hyperkaehler. There are no higher obstructions to a deformation besides Ioneda product. This deformation space does not depend on a base manifold.