2
$\begingroup$

Let us consider $g\colon X\to Y$ holomorphic, where $X$ is a complex manifold and $Y$ is a Stein manifold.

I am searching for all the pairs $(X,Y)$ such that we can find some non constant $g$ with bounded image.

Clearly $X$ cannot be Euclidean, as we could apply Liouville, and any holomorphic mapping $g$ with dense image would be constant. What do I have to ask to $X$ to prevent this? I think at something like $X$ non Stein manifold, as (roughly speaking) this implies that every mapping defined on it could be holomorphically extended (even if there is no sourrounding space...but that's the point of Stein manifold: to give such an intrinsic charaterization) so the image of that space would not the image of the "biggest possible space", so we would not going for images of entire mappings, in a way.

Is my intuition meaningful? How can I get it rigorous? Thanks

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ I don't think I realy understand what kind of answer are you looking for. For example if you let $X$ be a bounded open subset of $\mathbb{C}^n$ than you should always be able to find many bounded non-constant maps from $X$ to $Y$ right? $\endgroup$
    – Luka Thaler
    Commented Oct 9, 2021 at 20:11
  • $\begingroup$ Ouch, You're right $\endgroup$
    – Joe
    Commented Oct 10, 2021 at 20:46

1 Answer 1

Reset to default
2
$\begingroup$

In what concerns covers of compact complex manifolds there are results of Lin and Lin - Zaidenberg. In particular, nilpotent (more generally, FC-hypernilpotent) covers are always Liouville, whereas there are non-Liouville polycyclic covers (examples are provided by the Inoue surfaces).

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .