Dynamics and integrability for germs of complex vector fields with singularity in dimension two
Let \(X\) be a holomorphic vector field in a neighborhood \(U \subset \mathbb{C}^2\) of the origin, having an isolated singularity at this point. The associated singular holomorphic foliation is denoted by \(\mathcal{F}_X\). The author defines that an orbit of \(X\) ``accumulates properly at some separatrix'' of \(X\), if this orbit is not contained in a separatrix but contains some point of a separatrix in its closure. The main result is as follows: Let \(X\) be a holomorphic vector field as above, having a finite number of separatrices. If there is no orbit of \(X\) accumulating properly at the set of separatrices, then: (1) \(\mathcal{F}_X\) admits a holomorphic first integral, or (2) \(\mathcal{F}_X\) is a Darboux logarithmic foliation of real type (this means that the foliation is defined by a one-form of type \( \omega = \Sigma_j^r \lambda_j df_j / f_j \) where \(\lambda_j \in \mathbb{R}\) and \(f_j\) are holomorphic on \(U\)). More generally, if the germ admits some closed leaf then either it admits a holomorphic first integral or it is associated to a Pérez-Marco singularity, very roughly speaking, having holonomies as in [\textit{R. Pérez-Marco}, Ann. Sci. Éc. Norm. Supér. (4) 26, No. 5, 565--644 (1993; Zbl 0812.58051)]. \noindent In the second case, the germ has infinitely many closed leaves, the set of leaves accumulating properly at the set of separatrices (and therefore at the singularity) is not empty, and the singularity is essentially formally linearizable. The author expands the description of foliations with first integral provided in the classical work [\textit{J. F. Mattei} and \textit{R. Moussu}, Ann. Sci. Éc. Norm. Supér. (4) 13, 469--523 (1980; Zbl 0458.32005)].
scientific article; zbMATH DE number 6391429
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