Complements of graphs of meromorphic functions and complete vector fields

Abstract

Given a meromorphic function $s: {\mathbb {C}}\rightarrow \mathbb {P}^{1}$, we obtain a family of fiber-preserving dominating holomorphic maps from $\mathbb {C}^{2}$ onto $\mathbb {C}^{2}\setminus \text{ graph }(s)$ defined in terms of the flows of complete vector fields of type $\mathbb {C}^{*}$ and of an entire function $h:\mathbb {C}\rightarrow \mathbb {C}$ whose graph does not meet $\text{ graph }(s)$, which was determined by Buzzard and Lu. In particular, we prove that the dominating map constructed by these authors to prove the dominability of $\mathbb {C}^{2}\setminus \text{ graph }(s)$ is in the above family. We also study the complement of a double section in $\mathbb {C}\times \mathbb {P}^{1}$ in terms of a complex flow. Moreover, when $s$ has at most one pole, we prove that there are infinitely many complete vector fields tangent to $\text{ graph }(s)$, describing explicit families of them with all their trajectories proper and of the same type ($\mathbb {C}$ or $\mathbb {C}^{*}$), if $\text{ graph }(s)$ does not contain zeros; and families with almost all trajectories non-proper and of type $\mathbb {C}$, or of type $\mathbb {C}^{*}$, if $\text{ graph }(s)$ contains zeros. We also study the dominability of $\mathbb {C}^{2}\setminus A$ when $A\subset \mathbb {C}^{2}$ is invariant by the flow of a complete holomorphic vector field.