Level sets of holomorphic functions

Question

It is a somewhat well known fact that any closed set (say in the plane) can be realized as the level set of a smooth ($C^\infty$) function, so level sets of smooth functions are as general as they can possibly be.

Question (very basic and probably well known, sorry) : what about level sets of holomorphic functions ?

By the isolated zero principle, any non constant holomorphic function (in one variable) must be a submersion at all but a discrete set of points, so you get level sets which are fairly reasonnable. What is the most general scenario for such sets ?

I would like to know this in those 3 levels of generality, if possible :

1) level sets of holomorphic maps from $\mathbb{C}^n$ to $\mathbb{C}^k$

2) level sets of holomorphic maps from complex manifolds to complex manifolds (finite dimensional ; all local information should be the same as case 1). Feel free to add all relevant hypotheses I wouldn't know about (Kähler, Stein, I don't know...)

3) level sets of holomorphic maps from (possibily infinite dimensional) complex Banach manifold to complex Banach manifold.

Answer 1

The level sets of holomorphic functions $\mathbb{C}^n \to \mathbb{C}^k$ are exactly the Stein spaces of finite embedding dimension.
Indeed, any Stein space of finite embedding dimension (for example a connected Stein manifold) can be embedded into some $\mathbb C^n$.
And then any analytic subspace of $\mathbb C^n$ can be defined by $n$ global holomorphic functions, i.e. is the fiber over $0\in \mathbb C^k$ of some holomorphic map $\mathbb{C}^n \to \mathbb{C}^n$. This is due to Forster-Ramspott.
These are highly non-trivial results.

Edit: some (skippable) complements
a) It sounds like a tautology that a Stein space of finite embedding dimension can be embedded into some $\mathbb C^n$, but it is actually quite a difficult result:
The embedding dimension of a complex space $X$ is the supremum for $x\in X$ of the local embeding dimensions $\text {emb}_x(X)$, and $\text {emb}_x(X)$ is the smallest $k$ such that some neighbourhood of $x$ in $X$ can be embedded as a closed analytic subspace of some open ball in $\mathbb C^k$.
For example a Stein manifold of dimension $k$ trivially has embedding dimension $k$ but it is far from trivial that it can be embedded into some $\mathbb C^n$.
Eliashberg and Gromov have proved that you can actually take $n=[\frac{3k+1}{2}]+1$

b) The algebraic analogue of Forster-Ramspott's theorem was proved independently in 1973 by Eisenbud-Evans and Storch: every algebraic subset $Y\subset \mathbb C^n$ is the zero locus of $n$ polynomials.
In other words $Y$ is the intersection of $n$ hypersurfaces of $\mathbb C^n$.
It is of some interest to note that Kronecker had already proved in 1882 that such a subvariety $Y$ is the intersection of $n+1$ hypersurfaces, so that it took near a century to gain one hypersurface in the cutting out of $Y$ from $\mathbb C^n$ !

c) A modern reference on Stein spaces is Forsternic's book.