On holomorphic flows on Stein surfaces: transversality, dicriticalness and stability

TL;DR

This paper classifies holomorphic flows on Stein surfaces, especially those biholomorphic to ^2, and analyzes their singularities, periodic orbits, and integrability, with applications to algebraic surfaces.

Contribution

It provides necessary and sufficient conditions for Stein surfaces to be biholomorphic to ^2 based on vector field properties and classifies flows with many periodic orbits.

Findings

Stein surface N is biholomorphic to ^2 if certain cohomological and singularity conditions are met.

Flows with many periodic orbits either have a meromorphic first integral or a special periodic orbit with non-resonant holonomy.

Algebraic flows on affine surfaces are given by rational linear forms or admit rational first integrals.

Abstract

We study the classification of the pairs $(N, \,X)$ where $N$ is a Stein surface and $X$ is a complete holomorphic vector field with isolated singularities on $N$. We describe the role of transverse sections in the classification of $X$ and give necessary and sufficient conditions on $X$ in order to have $N$ biholomorphic to $\mathbb C^2$. As a sample of our results, we prove that $N$ is biholomorphic to $\mathbb C^2$ if $H^2(N,\mathbb Z)=0$, $X$ has a finite number of singularities and exhibits a non-nilpotent singularity with three separatrices or, equivalently, a singularity with first jet of the form $\lambda_1 \, x\frac{\partial }{\partial x} + \lambda_2 \, y\frac{\partial}{\partial y}$ where $\lambda_1 / \lambda_2 \in \mathbb Q_+$. We also study flows with many periodic orbits, in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant non-linearizable diffeomorphism map. Finally, we apply our results together with more classical techniques of holomorphic foliations on algebraic surfaces to study the case of flows on affine algebraic surfaces. We suppose the flow is generated by an algebraic vector field. Such flows are then proved, under some undemanding conditions on the singularities, to be given by closed rational linear one-forms or admit rational first integrals.