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 $V\longrightarrow Z$ 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.