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 $V$ by means 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$can be locally parameterized (in that sense), around anyone of its points, by a an onto holomorphic mapping from an open set of some$\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$lies in an affine space. I thank the community in advance for any help/comment/rebuke ! 3 deleted 101 characters in body Let$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$by means of a proper, onto map$M\rightarrow V$, where$M$is a complex manifold. Therefore$V$can be locally parameterized, around anyone of its points, by a holomorphic mapping from an open set of some$\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$lies in an affine space. The furthest I can get to is saying that$M$can be embedded in some$\mathbb{C}^\ell$, but I'm not sure that it helps. I thank the community in advance for any help/comment/rebuke ! 2 added 6 characters in body Let$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$by means of a proper, onto map$M\rightarrow V$, where$M$is a complex manifold. Therefore$V$can be locally parameterized, around anyone of its points, by a holomorphic mapping from an open set of some$\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$? The answer to this question boils down to knowing whether any complex manifold can be the range of a holomorphic map from some open set of$\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$lies in an affine space. As far as The furthest I can get to is saying that$M$can be embedded in some$\mathbb{C}^\ell\$, but I'm not sure that it helps.