 
Summary: An extension of a theorem by Ohsawa and
applications
GT and VV
1 Introduction
Throughout this paper complex spaces are assumed to be reduced and with
countable topology. A curve, surface, etc., will be complex spaces of the
appropriate dimension.
Let X be a complex space. We say that X is weakly 1complete if there
exists a plurisubharmonic (psh) function : X  [, ), not necessarily
continuous, such that is exhaustive, i.e. for every c R the sublevel set
{x X ; (x) < c} is relatively compact in X. If we may choose strictly
plurisubharmonic (spsh) outside a compact subset of X, then X is called 1
convex. In this case it can be shown that there exists an exhaustion function
: X  [, ) which is spsh on the whole space X; see [4].
For 1convex spaces X one has several equivalent descriptions: The stan
dard one states that X is a proper modification of a Stein space at a finite
number of points. More precisely, there is a Stein space Y , a proper holo
morphic map : X  Y with (OX) OY (in particular is surjective
and has connected fibers) and a finite set B Y such that induces a bi
holomorphism between X \1
