2016/01/28 by Leon, Victor, Scardua, Bruno
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1601.07767
We consider germs of holomorphic vector fields with an isolated singularity at the origin 0∈ℂ2. We introduce a notion of stability, similar to "Lyapunov stability". For such a germ, called L-stable singularity, either the corresponding foliation admits a holomorphic first integral, or it is a real logarithmic foliation singularity. A notion of L-stability is also naturally introduced for a leaf of a foliation. In the complex codimension one case, for holomorphic foliations, the holonomy groups of L-stable leaves are proved to be abelian, of a suitable type. This implies the existence of local closed meromorphic one-forms defining the foliation, in a neighborhood of L-stable leaves.