Let $V$ be an analytic subvariety of some open set of $\mathbb{C}^n$ (intersection of finitely many zero-locii of analytic functions). Hironaka's desingularization theorem provides a parameterization of $V$, that is: there exists a proper, onto map $M\rightarrow V$, where $M$ is a complex manifold, which is a biholomorphism outside the strict transform of the singular set of $V$. Therefore $V$ can be locally parameterized (in that sense), around anyone of its points, by an onto holomorphic mapping from an open set of some $\mathbb{C}^m$.


My question is the following: is it true that $V$ can be globally parameterized by an onto holomorphic mapping from an open set of $\mathbb{C}^m$?


The answer to this question boils down to knowing whether any complex manifold (obtained this way) can be the range of a holomorphic map from some open set of $\mathbb{C}^m$, which is a statement that I unfortunately don't know how to handle. Neither was I able to find in the litterature any specifics regarding the conformal nature of the manifold $M$ when $V$ lies in an affine space.


I thank the community in advance for any help/comment/rebuke !



Parameterization of complex analytic subvarieties
The answer to your question is yes, even in a much broader setting and moreover the parametrizing open set can be chosen always to be the unit polydisc!


**Theorem. (Fornaess-Stout '77).** If $X$ is an $n$-dimensional complex manifold, then there exists a locally biholomorphic map $\Phi$ from the open unit polydisc in $\mathbb C^n$ onto $X$ with the property that for each $x\in X$, the fiber $\Phi^{-1}(x)$ consists of no more than $(2n+1)4^n+2$ points.


Of course, if $X$ is just a complex space, you have to desingularize before, so that your map is no more finite, but still surjective of course.


Observe finally, that there is no assumption here on $X$, it works in the abstract case also, you don't need it to be an analytic subset of an open set of some complex vector space!


The paper where the theorem is has been indicated to me by T.-C. Dinh. 


