The Penrose Transform for Compactly Supported Cohomology

Let the manifold X parametrise a family of compact complex submanifolds of the complex (or CR) manifold Z . Under mild conditions the Penrose transform typically provides isomorphisms between a cohomology group of a holomorphic vector bundle V → Z and the kernel of a differential operator between sections of vector bundles over X . When the spaces in question are homogeneous for a group G the Penrose transform provides an intertwining operator between representations. The paper develops a Penrose transform for compactly supported cohomology on Z . It provides a number of examples where a compactly supported cohomology group is shown to be isomorphic to the cokernel of a differential operator between compactly supported sections of vector bundles over X . It considers also how the Serre duality pairing carries through the transform.