Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$

My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known?

Of course if $X$ is compact $\mathcal O(X)=\mathbb C$ and that dimension is $0$.

There are also quite a lot of non-compact manifolds with $\mathcal O(Z)=\mathbb C$:

For example if $X$ is connected of dimension $\geq 2$ and $Y\subset X$ is an analytic subset of codimension at least $2$ ( or a small compact ball) , you will still have $\mathcal O(X\setminus Y)=\mathbb C$ .

But apart from these trivial examples I can't compute a single Krull dimension $dim \mathcal O(X)$ for, say, Stein manifolds of positive dimension.

Just in order to ask something definite, let me pose the ridiculous-sounding question:

Does there exist a connected holomorphic manifold $X$ with $0\lt \dim \mathcal O(X)\lt \infty$ ?

seemsto be relatively easy to read. $\endgroup$ – Georges Elencwajg Apr 19 '12 at 17:20