Unbranched Riemann Domains over Q-Complete Spaces
It is proved that if Π : X → Ω is an unbranched Riemann domain and locally r-complete morphism over a q-complete space Ω, then X is cohomo- logically (q +r −1)-complete, if q ≥ 2. We have shown in [1] that if Π : X → Ω is an unbranched Riemann domain and locally q-complete morphism over a Stein space Ω, then X is cohomologically q-complete with respect to the struc- ture sheaf. In section 4 of this article, we prove by means of a counterexample that that there exists for each integer n ≥ 3 an open subset Ω ⊂ Cn which is locally (n − 1)-complete but Ω is not (n − 1)-complete. The counterexample we give is obtained by making a slight modification of a recent example given by the author [2].
