Cohomological q-convexity versus local q-completeness with corners

We show that if (D, π) is an unramified Riemann domain over a distinguished complex manifold X such that D is cohomologically q-convex, then π is locally q-complete with corners. We call X distinguished if for every point x of X there is a holomorphic line bundle L on X (which may depend on x) so that the global sections Γ(X,L) of L generate its 1-jets at x. Examples of distinguished complex manifolds include all complex submanifolds of Cm × Pn; in particular all Stein or projectively algebraic manifolds. © 1998, Birkhäuser Verlag, Basel.