On the Milnor number of non-isolated singularities of holomorphic foliations and its topological invariance

TL;DR

This paper introduces a new definition of the Milnor number for holomorphic foliations' singularities and proves its topological invariance under certain conditions on three-dimensional manifolds.

Contribution

It defines the Milnor number as an intersection number for holomorphic foliations and establishes its invariance under $C^1$ topological equivalences in specific settings.

Findings

Milnor number defined via intersection of holomorphic sections

Proved topological invariance of the Milnor number under certain conditions

Applicable to three-dimensional manifolds with compact singular set components

Abstract

We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain conditions, we prove that the Milnor number of $\mathscr{F}$ on a three-dimensional manifold with respect to $C$ is invariant by $C^1$ topological equivalences.