Topology of singular holomorphic foliations along a compact divisor

TL;DR

This paper studies the topology of singular holomorphic foliations near a compact divisor, establishing conditions for incompressibility of leaves and classifying foliations via fundamental group representations and topological conjugation deformations.

Contribution

It introduces a topological classification framework for singular holomorphic foliations near a divisor, including incompressibility results and deformation of conjugations extending to singularity reductions.

Findings

Existence of tubular neighborhoods with incompressible leaves.

Representation of fundamental groups into automorphism groups.

Deformation of topological conjugations to extend over singularities.

Abstract

We consider a singular holomorphic foliation $\uF$ defined near a compact curve $\uC$ of a complex surface. Under some hypothesis on $(\uF,\uC)$ we prove that there exists a system of tubular neighborhoods $U$ of a curve $\underline{\mc D}$ containing $\uC$ such that every leaf $L$ of $\uF_{|(U\setminus \underline{\mc D})}$ is incompressible in $U\setminus \underline{\mc D}$. We also construct a representation of the fundamental group of the complementary of $\underline{\mc D}$ into a suitable automorphism group, which allows to state the topological classification of the germ of $(\uF,\uD)$, under the additional but generic dynamical hypothesis of transverse rigidity. In particular, we show that every topological conjugation between such germs of holomorphic foliations can be deformed to extend to the exceptional divisor of their reductions of singularities.