A STEIN CRITERION VIA DIVISORS FOR DOMAINS OVER STEIN MANIFOLDS

It is shown that a domain X over a Stein manifold is Stein if the following two conditions are fulfilled: a) the cohomology group H-i (X, O) vanishes for i >= 2 and b) every topologically trivial holomorphic line bundle over X admits a non-trivial meromorphic section. As a consequence we recover, with a different proof, a known result due to Siu stating that a domain X over a Stein manifold Y is Stein provided that H-i (X, O) = 0 for i >= 1.