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

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.