Smoothly parameterized Cech cohomology of complex manifolds

A Stein covering of a complex manifold may be used to realize its analytic cohomology in accordance with the Cech theory. If, however, the Stein covering is parameterized by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.