Resumen de Dynamics and Integrability for Germs of Complex Vector Fields with Singularity in Dimension Two

Bruno Scárdua

• In this paper we prove classification results for singularities of holomorphic vector fields under some dynamical hypotheses on their orbits regarding their separatrices.

  We consider germs for which the set of separatrices is finite. We prove that if there is no orbit accumulating properly at the set of separatrices then the corresponding holomorphic foliation admits a holomorphic first integral or it is a linear logarithmic foliation of real type (cf. Theorem 1.1). Then we conclude (cf. Theorem 1.1 and Corollary 1.2) that the germ of a holomorphic foliation, with a finite number of separatrices, admits a holomorphic first integral iff it admits some closed leaf and no other leaf accumulates at the singularity. More generally, we prove (cf. Theorem 1.5) that if the germ admits some closed leaf then either it admits a holomorphic first integral or it is associated to a Pérez-Marco singularity (see Example 1.4). 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.