Topological Moduli Space for Germs of Holomorphic Foliations III: Complete families

TL;DR

This paper constructs complete families of germs of holomorphic foliations on complex surfaces, demonstrating that all topological classes can be contained within a minimal family, which serves as a universal parameter space for equisingular deformations.

Contribution

It introduces a minimal universal family of foliation germs that encapsulates all topological classes, extending previous topological classification results.

Findings

Existence of a minimal family containing all topological classes.

Any equisingular global family factors through the minimal family.

The minimal family serves as a universal parameter space.

Abstract

In this work we use our previous results on the topological classification of generic singular foliation germs on $(\mathbb C^{2},0)$ to construct complete families: after fixing the semi-local topological invariants we prove the existence of a minimal family of foliation germs that contain all the topological classes and such that any equisingular global family with parameter space an arbitrary complex manifold factorizes through it.