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.