 
Summary: Complex surfaces with vanishing cohomology
and projective closures
Giuseppe Tomassini and Viorel V^aj^aitu
1 Introduction
A classical result (see [2], [4]) asserts that an open subset D of C2
is Stein (or a
domain of holomorphy) if, and only if, the additive Cousin problem is always
solvable on D. Furthermore, this condition is equivalent to the vanishing of
H1
(D, O). Motivated by this kind of result in a series of papers ([18], [19],
[20], [21]) the following problem is considered. "Let M be a nonsingular
complex surface and D M an open set such that H1
(D, O
) vanishes.
Under what reasonable conditions on M it follows that D is Stein?"
A positive answer is stated for the following cases, namely M is projective,
or D is relatively compact and either M has positive holomorphic bisectional
curvature or that M is weakly 1complete.
Subsequently we shall (re)prove this result in a more general setting (we
allow also singularities), see Theorem 1 from below. Before stating it, let
