On the compactification of a Stein surface

Abstract

Let M and S be complex manifolds of the same dimension, with M open and S compact. We shall say that S is a compactification of M if S contains a closed complex analytic subvariety F such that S-F is biholomorphically equivalent to M. It seems natural to ask what manifolds can occur as compactifications if various conditions are imposed on M. In particular, if M is a Stein manifold, what conclusions may be drawn concerning S? As a small first step toward answering this question, we take up the case of dimension two, and prove the following: Theorem. Let S be a compact, complex analytic surface, and let F be a closed subvariety of S such that S - F is a Stein manifold. Then either (i) S is an algebraic surface, or (if) the first Betti number bl(S)= 1 and S admits no nonconstant merornorphic function. The proof, which is a simple application of KODAIRA'S theory of surfaces ([4], [5]), will be given in the next section. First, however, we will show by an example that alternative (if) above does indeed occur. To this end, let W = C 2 - (0, 0), and let g : W-+ W be the mapping