Topological equivalence of holomorphic foliation germs of rank $1$ with isolated singularity in the Poincar\'e domain

TL;DR

This paper demonstrates that the topological classification of certain holomorphic foliation germs with isolated singularities can be reduced to the classification of their intersection with a sphere, leading to a complete classification in the plane.

Contribution

It introduces a Reconstruction Theorem that links the topological class of foliation germs to their intersection foliation, enabling a full classification in two dimensions.

Findings

Topological class determined by intersection foliation

Complete classification of plane holomorphic foliation germs of Poincaré type

Discussion of conjecture for higher dimensions

Abstract

We show that the topological equivalence class of holomorphic foliation germs with an isolated singularity of Poincar\'e type is determined by the topological equivalence class of the real intersection foliation of the (suitably normalized) foliation germ with a sphere centered in the singularity. We use this Reconstruction Theorem to completely classify topological equivalence classes of plane holomorphic foliation germs of Poincar\'e type and discuss a conjecture on the classification in dimension $\geq 3$.