From: lrudolph@panix.com (Lee Rudolph)
Newsgroups: sci.math
Subject: Re: Stein curves
Date: 11 Nov 1995 16:28:19 -0500

rojas@math.mit.edu (J. Maurice Rojas) writes:

>	Is there a simple characterization for a complex 
>algebraic curve to be a Stein manifold?  

No compact components.  (I.e., it's a complex *affine*
algebraic curve, not a complex *projective* algebraic 
curve.)

>[Mod. note:  I don't really know the topic, but I was under
>the impression that for starters you have to have a non-compact
>complex manifold. -Greg]

Yup.  Not by definition, but the Embedding Theorem for Stein
manifolds (cf. Gunning & Rossi, VII.C.13) says that if X is
a Stein manifold of complex dimension n, then almost every
(2n+2)-tuple of complex-analytic functions on X gives an
embedding of X as a closed complex submanifold of C^{2n+2}; 
by exercising care, one can embed X in C^{2n+1}.  Conversely,
any closed complex submanifold of C^N is a Stein manifold.

Of course, you can have a Stein manifold which, as it's given
to you, isn't a closed submanifold of affine space, for 
instance, any open subset of C, or (my current favorite
example) the interior of the strictly pseudoconvex domain
{(z,w)\in C^2 : (|z+w^g|-2)^2+(|z-w^g|-2)^2 \le 1},
which is diffeomorphic to S^1 times a solid handlebody
of genus g.

Lee Rudolph