A Dolbeault-type Double Complex on Quaternionic Manifolds

It has long been known that differential forms on a complex manifold M2n can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for a quaternionic manifold M4n using the fact that its cotangent space T ∗ mM is isomorphic to the quaternionic vector space H. This defines an action of the group Sp(1) of unit quaternions on T M , which induces an action of Sp(1) on the space of k-forms ΛkT M . A double complex is obtained by decomposing ΛkT M 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.