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 !

parametrizationmeans something likesurjective holomorphic map. Regards, $\endgroup$4more comments