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An extension of a theorem by Ohsawa and applications
 

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 1-complete 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 1-convex 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

  

Source: Abbondandolo, Alberto - Scuola Normale Superiore of Pisa

 

Collections: Mathematics