Manifolds with an Su(2)-action on the Tangent Bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over CP 1 with the normal bundle of each section isomorphic to O(k)⊗ Cn. Any hypercomplex manifold can be constructed as a space of sections of a complex manifold Z fibering over CP . The normal bundle of each section must be the sum of O(1)’s, and this suggests that interesting geometric structures can be obtained if we replace O(1) with other line bundles. Such structures have been introduced by many authors [1, 2, 3, 7, 8, 11, 13] and are variously known as conic, Grassman, paraconformal or P-structures. Spaces of sections with normal bundle O(n)⊕O(n) have been studied recently, in detail, by Maciej Dunajski and Lionel Mason [7, 8], and much of the present paper can be viewed as a translation of their work from spinor language. We adopt the point of view that spaces of sections of complex manifolds fibering over CP 1 are manifolds with a fibrewise linear action of SU(2) on the tangent bundle. The integrability condition says that the ideal in (Ω∗M)C generated by the highest weight 1-forms is closed, for any Borel subgroup. We call such manifolds generalised hypercomplex manifolds (or k-hypercomplex manifolds if the representation of SU(2) splits into copies of the k-th symmetric power of the standard representation). These manifolds seem to be interesting from several points of view, quite apart from “intrinsic worth”. On the one hand, they provide a natural setting for certain integrable systems, “monopoles”, as we discuss in sections 4 and 5. These comprise the Bogomolny hierarchy of Mason and Sparling [14] and the self-dual hierarchy. On the other hand, even if one is interested primarily in hypercomplex or hyperkähler manifolds, one may wish to consider these generalised hypercomplex manifolds, since (for odd k) they are foliated by hypercomplex submanifolds (see section 7; this observation goes back to Gindikin [10]; see also [7, 8]). The paper is organised as follows: in the next section we recall some basic facts about foliations and distributions. In section 2 we define the generalised hypercomplex manifolds and their twistor spaces. Section 3 is devoted to an interpretation of a construction of D. Quillen [16], which we need later on. In sections 4 and 5 we discuss the Ward correspondence and monopoles on k-hypercomplex manifolds. In the following section we show that a k-hypercomplex manifold M has, analogously to hypercomplex manifolds, an S-worth of certain integrable structures. For odd k, M is always a complex manifold, but the S-worth of complex structures are Received by the editors September 29, 2003 and, in revised form, June 16, 2004. 2000 Mathematics Subject Classification. Primary 53C26, 53C28. c ©2005 American Mathematical Society Reverts to public domain 28 years from publication