Deformations of Hypercomplex Structures

Twistor theory shows that deformations of a hypercomplex manifold correspond to deformations of a canonically constructed holomorphic map from the twistor space of the hypercomplex manifold onto the complex projective line. Applying Horikawa's theory of deformations of maps, we calculate deformations of compact quotients and twists of the associated bundles of quaternionic manifolds. In particular , deformations of hypercomplex structures on Stiefel manifolds are discussed. x1 Introduction Three integrable complex structures on a manifold form a hypercomplex structure if they satisfy the relations of the quaternions (2.1). Recent interest in hyper-complex structures is inspired by S. Salamon's work 31]. The subject is studied along with hyperKK ahler structures 10], quaternionic KK ahler structures 30], quater-nionic structures 31], self-dual conformal structures 1] and most recently together with 3-Sasakian structures 4]. All these subjects are deeply related to twistor theory. It is known that a particular type of Hopf surfaces and their high dimensional analogue are hypercomplex 3] 15]. The collection of examples of hypercomplex manifolds is greatly enriched by Joyce's work, following ideas from physics 32], on the construction and classiication of homogeneous hypercomplex manifolds 12]. Joyce gives various natural geometric constructions of hypercomplex structures such as hypercomplex quotients, twists of associated bundles and he produces more non-trivial examples 12] 26]. From a diierent, yet closely related, perspective Boyer,