Nondegenerate germs of holomorphic foliations with prescribed holonomy

TL;DR

This paper characterizes holonomy maps of saddle singularities in holomorphic foliations across dimensions, showing any germ can be realized as such, but with limitations on prescribing the linear part in higher dimensions.

Contribution

It extends the understanding of holonomy maps from 2D to higher dimensions, demonstrating realizability of germs and conditions for prescribing linear parts.

Findings

Any germ of holomorphic biholomorphism can be realized as a holonomy map in any dimension.

Prescribing the linear part of the saddle vector field is not always possible in higher dimensions.

A positive result is obtained under a natural condition for the holonomy map.

Abstract

We are interested in characterizing the holonomy maps associated to integral curves of non-degenerate singularities of holomorphic vector fields. Such a description is well-known in dimension 2 where is a key ingredient in the study of reduced singularities. The most intricate case in the 2 dimensional setting corresponds to (Siegel) saddle singularities. This work treats the analogous problem for saddles in higher dimension. We show that any germ of holomorphic biholomorphism, in any dimension, can be obtained as the holonomy map associated to an integral curve of a saddle singularity. A natural question is whether we can prescribe the linear part of the saddle germ of vector field provided the holonomy map. The answer to this question is known to be positive in dimension 2. We see that this is not the case in higher dimension. In spite of this, we provide a positive result under a natural condition for the holonomy map.