An Integral Operator into Dolbeault Cohomology

A construction is given for an integral transform from sections of a vector bundle over one manifold into Dolbeault cohomology of a (related) holomorphic vector bundle over a second manifold. It is demonstrated that the transform arises naturally in appropriate homogeneous situations where it is shown to agree with a certain representation theoretic intertwining operator due to Barchini, Knapp and Zierau. Our construction is geometric with the underlying correspondence being a double fibration of the type arising in connection withX-ray transforms. From this point of view many properties of the transform are immediate.