Stable singularities of holomorphic vector fields

TL;DR

This paper studies the stability of holomorphic vector fields with isolated singularities, classifying them into specific types and analyzing the properties of their foliations and holonomy groups.

Contribution

It introduces a new notion of stability for such vector fields, characterizes $L$-stable singularities, and describes the structure of foliations and holonomy groups associated with these singularities.

Findings

$L$-stable singularities either admit a holomorphic first integral or are real logarithmic foliation singularities.

Holonomy groups of $L$-stable leaves are abelian and of a specific type.

Existence of local closed meromorphic one-forms defining the foliation near $L$-stable leaves.

Abstract

We consider germs of holomorphic vector fields with an isolated singularity at the origin $0\in\mathbb{C}^2$. 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.