Stein曲面上的全纯流：横截性、双临界性和稳定性
On holomorphic flows on Stein surfaces: transversality, dicriticalness and stability
摘要：
研究了$(N, \,X)$对的分类，其中$N$是Stein曲面，$X$是$N$上具有孤立奇点的全纯向量场.本文描述了横截面在$X$分类中的作用，并给出了$X$使$N$双全纯为$\mathbb C^2$的充要条件.作为我们的结果的一个例子，我们证明了$N$是双全纯$\mathbb C^2$，如果$H^2(N,\mathbb Z)=0$，$X$有有限个奇点，并表现出一个非幂零奇点与三个分离线，或等价地，奇点与第一射流的形式$λ_1 \, x\frac{\partial }{\partial x} + λ_2 \, y\frac{\partial}{\partial y}$其中$λ_1 / λ_2 \in \mathbb Q_+$。我们还研究了具有多个周期轨道的流，在某种意义上我们将弄清楚，证明它们允许一个亚纯第一积分，或者它们表现出一些特殊的周期轨道，其完整映射是一个非共振非线性化的亚纯映射。最后，我们应用我们的结果与更经典的技术全纯叶理代数曲面上研究的情况下，仿射代数曲面上的流动。我们假设流是由代数向量场生成的。然后证明了在对奇点要求不高的条件下，这类流可以由闭有理线性一元形式给出，或可由有理首次积分给出。
英文摘要：
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 $λ_1 \, x\frac{\partial }{\partial x} + λ_2 \, y\frac{\partial}{\partial y}$ where $λ_1 / λ_2 \in \mathbb Q_+$. 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.
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