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.