Smoothly Parameterised Cech Cohomology of Complex Manifolds
  A Stein covering of a complex manifold may be used to realise its analytic cohomology in accordance with the Cech theory. If, however, the Stein covering is parameterised 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. 
Toby Bailey, Michael Eastwood, and Simon Gindikin