Hyperholomorphic bundles over a hyperkähler manifold.

Hyperholomorphic bundles over a hyperkähler manifold. 0. Intruduction. The main object of this paper is the notion of a hyperholomorphic bundle (Definition 2.4) over a hyperkähler manifold M (Definition 1.1). The hyperholomorphic bundle is a direct sum of holomorphic stable holomor-phic bundles. The first Chern class of a hyperholomorphic bundle is of zero degree. Roughly speaking, the hyperholomorphic bundle is a bundle which is holomorphic with respect to all complex structures induced by the hy-perkähler structure on M. It was proven (Proposition 4.1 of [V]) that if M is a complex hyperkähler surface (K3 or abelian surface) then any stable bundle whith first Chern class of zero degree is hyperholomorphic. Interesting properties of hyperholomorphic bundles include the analogue of (p,q)-decomposition, ∂ ¯ ∂-lemma and an analogue of the strong Lefshetz theorem on the holomorphic cohomology H * (B) of a hyperholomorphic B with a parallel real structure (proven in the Section 4). For a hyperkähler manifold, one can define the action of quaternions on its cohomology groups. The characteristic classes of a hyperholomorphic bundle are invariant under this action. Conversely, if B is a stable bundle with the first two Chern classes invariant under the quaternion action, the bundle B is hyperholomorphic (Theorem 2.5). We are describing a coarse moduli space of deformations of a hyperholo-morphic bundle locally (Theorem 6.2). In particular, we show that there are no obstructions for a deformation besides Yoneda pairing (Definition 6.2). This description is used to construct a hyperkähler structure on the space of deformations of a given hyperholomorphic bundle (Theorem 6.3), thus generalizing results of [M] and [Ko]. As an application, one can prove that the stable moduli space for the stable bundles with certain Chern classes do not depend on the choice of a base manifold in its deformation class (Proposition 10.3).