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

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.