Topological moduli space for germs of holomorphic foliations II: Universal deformations

TL;DR

This paper establishes the existence of a topological universal deformation for singular holomorphic foliations on ^2, classifying them via fixed invariants and showing how all equisingular deformations relate through this universal model.

Contribution

It constructs a topological universal deformation for germs of holomorphic foliations, providing a classification framework based on fixed invariants and functorial dependence.

Findings

Existence of a topological universal deformation for singular foliations.

Representation of the functor of topological classes of deformations.

Description of the functorial dependence of the deformation space.

Abstract

This work deals with the topological classification of singular foliation germs on $(\mathbb C^{2},0)$. Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of this representation with respect to the foliation.