Completely Integrable Foliations: Singular Locus, Invariant Curves and Topological Counterparts

TL;DR

This paper investigates the structure of completely integrable holomorphic foliations near singular points, revealing topological conditions under which these foliations admit holomorphic first integrals and characterizing certain isolated singularities.

Contribution

It introduces topological tools like the total holonomy group to characterize completely integrable foliations and provides a topological criterion for isolated singularities with separatrices.

Findings

Classification of singular points into regular, non-isolated, or with infinitely many invariant varieties.

Topological characterization of integrable singularities with isolated separatrices.

Development of tools linking topological properties to the existence of holomorphic first integrals.

Abstract

We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a regular point, a non-isolated singularity or there are infinitely many invariant analytic varieties through $P$ of the same dimension as the foliation, the so called separatrices. Moreover, we see that this phenomenon is of topological nature. Indeed, we introduce topological counterparts of completely integrable local holomorphic foliations and tools, specially the concept of total holonomy group, to build holomorphic first integrals if they have isolated separatrices. As a result, we provide a topological characterization of completely integrable non-degenerated elementary isolated singularities of vector fields with an isolated separatrix.