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arXiv · 1601.07767

Stable singularities of holomorphic vector fields

Victor Leon·Bruno Scardua
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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.

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Victor Leon, Bruno Scardua. 2016-01-28. Stable singularities of holomorphic vector fields. https://arxiv.org/abs/1601.07767

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