On the topological invariance of the algebraic multiplicity of holomorphic foliations

TL;DR

This paper proves that the algebraic multiplicity of holomorphic foliations is a topological invariant in various cases, including in higher dimensions and for specific classes of foliations, extending previous results and addressing key conjectures.

Contribution

It generalizes the topological invariance of algebraic multiplicity to higher dimensions and broader classes of foliations, including saddle-nodes and those with certain local homeomorphisms.

Findings

Algebraic multiplicity equals one is topologically invariant in dimension two.

Saddle-nodes are topologically invariant under certain conditions.

The invariance holds for several classes like generalized curves and second type foliations.

Abstract

In this paper, we address one of the most basic and fundamental problems in the theory of foliations and ODEs, the topological invariance of the algebraic multiplicity of a holomorphic foliation. For instance, we prove an adapted version of A'Campo-L\^e's Theorem for foliations, i.e., the algebraic multiplicity equal to one is a topological invariant in dimension two. This result is further generalized to higher dimensions under mild conditions; as a consequence, we prove that saddle-nodes are topologically invariant. We prove that the algebraic multiplicity is a topological invariant in several classes of foliations that contain, for instance, the generalized curves and the foliations of second type. Additionally, we address a fundamental result by Rosas-Bazan, which states that the existence of a homeomorphism extending through a neighborhood of the exceptional divisor of the first blow-up implies the topological invariance of the algebraic multiplicity. We show that the result holds if the homeomorphism extends locally near a singularity, even if it does not extend over the entire divisor.