A Dolbeault-type Double Complex on Quaternionic Manifolds

Abstract

It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. This defines an action of the group Sp(1) of unit quaternions on the cotangent space, which induces an action of Sp(1) on the space of k-forms. A double complex is obtained by decomposing the k-forms into irreducible representations of Sp(1), resulting in new 'quaternionic Dolbeault' operators and cohomology groups. Links with previous work in quaternionic geometry, particularly the differential complex of Salamon and the q-holomorphic functions of Joyce, are demonstrated.