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

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 $\geq 3$.