On a class of complex spaces intermediate to Stein and compact

Abstract

In the global theory of complex spaces two extreme cases were traditionally studied: Compact and Steia A pseudoconcave space represents a variation from this traditioa However, from the point of view of global function theory, pseudoconcave spaces behave much like compact spaces. For example, on a pseudoconcave space every holomorphic function is constant and the meromorphic function field is an algebraic function field having transcendence degree over C bounded by the dimension of the space [1].The g-pseudoconvex (concave) case, which was investigated by Andreotti and Grauert [2], represents another variatioa However, the main result in the #-pseudoconvex case is that dim c if r (Z, 3F) < oo for 3F coherent and r ^ q [2]. Unless X is Stein, nothing can be said about the lower cohomology groups or, in particular, the algebra of holomorphic functions. The main result in the g-pseudoconcave case is another finiteness theorem [2], but, as we have already remarked, pseudoconcave spaces behave much like compact spaces from the function theoretic standpoint.We are therefore led to study complex spaces which at least from the global function theoretic point of view, are more clearly between the extremes of Stein and compact spaces.DEFINITION. Let X be a complex space. We say that X is k-pseudoflat if there is a relatively compact open subset Y a X such that every pedY is contained in a set S p c Y 9 where S p is an analytic subvariety of an open set in X such that codim p S p ^ fc.Of course the motivation for such a definition is a complex manifold containing a relatively compact open subset with smooth boundary which is Levi flat in a certain number of tangent directions.In [4] we prove the following theorem. THEOREM 1. Let X be a reduced, irreducible k-pseudoflat complex space. Then there are at most k analytically independent holomorphic functions onX.AM S 1969 subject classifications. Primary 3222, 3247.