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 $0\lt \dim \mathcal O(X)\lt \infty$ ?
> 
> 
> 



What is the Krull dimension of the ring of holomorphic functions on a complex manifold?
It follows from the proof in Sasane's paper that Krull dimension of a (connected) complex manifold $M$ is infinite iff $M$ admits a nonconstant holomorphic function $F: M\to {\mathbb C}$. Namely, using Sard's theorem find a sequence of points $a\_k \in F(M)$ which are regular values of $F$ and so that $(a\_k)$ converges to a point in $({\mathbb C}\cup \infty) \setminus F(M)$. Then, pick regular points $b\_k\in V\_k:=F^{-1}(a\_k)$ of $F$ and define multiplicity of zero for a holomorphic function $h: M\to {\mathbb C}$ with respect to the germ of $V\_k$ at $b\_k$. (I.e., multiplicity of $h$ is determined by the largest $m$ so that $h=(F-a\_k)^m g$ on the level of germs at $b\_k$.) Now, the same proof as in Sasane's paper goes through, where you will be using functions $f\_n\circ F$ instead of Sasane's functions $f\_n$. The point is that Sasane's argument is essentially local at zeroes of the functions $f\_n$. Actually, what Sasane proves is a lemma about a commutative ring $R$ with a sequence of valuations $m\_k$ for which there exists a sequence of elements $f\_i\in R$ so that $m\_k(f\_i)$ grows slower than $m\_k(f\_{i+1})$ for every $i$ as $k\to \infty$ (more precisely, in his case, the growth rate of $m\_k(f\_i)$ is $k^{i+1}$). Under this assumption, Krull dimension of $R$ is infinite. 


**Edit**: I finally wrote a detailed proof [here](https://www.math.ucdavis.edu/~kapovich/EPR/kd.pdf). 


**Edit.** I wrote a [proof](https://www.math.ucdavis.edu/~kapovich/EPR/valuations7.pdf) that the Krull dimension of $H(M)$ (when it is positive) has cardinality at least continuum. The new proof uses surreal numbers instead of ultralimits. For the sake of completeness I am keeping the older proof as well. 


