Complements of graphs of meromorphic functions and complete vector fields

TL;DR

This paper constructs and analyzes holomorphic maps and complete vector fields related to the complements of graphs of meromorphic functions in complex two-space, advancing understanding of their dominability and dynamical properties.

Contribution

It introduces a family of fiber-preserving dominating maps for complements of graph of meromorphic functions and classifies complete vector fields tangent to these graphs.

Findings

Constructed explicit dominating maps for complements of graph(s) of meromorphic functions.

Proved the existing dominating map by Buzzard and Lu is within the constructed family.

Identified conditions for the existence of infinitely many complete vector fields with proper trajectories.

Abstract

Given a meromorphic function $s: \mathbb{C}\to \mathbb{P}^{1}$, we obtain a family of fiber-preserving dominating holomorphic maps from $\mathbb{C}^{2}$ onto $\mathbb{C}^{2}\setminus graph(s)$ defined in terms of the flows of complete vector fields of type $\mathbb{C}^{\ast}$ and of an entire function $h:\mathbb{C}\to\mathbb{C}$ whose graph does not meet $graph(s)$, which was determined by Buzzard and Lu. In particular, we prove that the dominating map constructed by these authors to prove the dominability of $\mathbb{C}^{2}\setminus graph(s)$ 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 $graph(s)$, describing explicit families of them with all their trajectories proper and of the same type ($\mathbb{C}$ or $\mathbb{C}^{\ast}$), if $graph(s)$ does not contain zeros; and families with almost all trajectories non-proper and of type $\mathbb{C}$, or of type $\mathbb{C}^{\ast}$, if $graph(s)$ 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.