Topological rigidity for holomorphic foliations

TL;DR

This paper investigates the rigidity of holomorphic foliations in complex projective planes, establishing conditions under which they are either Darboux or unfoldings, and explores the algebraic structure of certain subgroups of diffeomorphisms.

Contribution

It provides a topological rigidity result for holomorphic foliations, linking their global classification to local singularity behavior and introduces new insights into subgroup solvability.

Findings

Holomorphic foliations are either Darboux or unfoldings under certain conditions.

A connection between global foliation classification and local singularities is established.

Subgroups of polynomial growth diffeomorphisms are shown to be solvable.

Abstract

We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{\mathcal{F}_t\}_{t \in \mathbb{D}_{\epsilon}}$ be topological trivial (in $\mathbb{C}^2$) analytic deformation of a foliation $\mathcal{F}_0$ on $\mathbb{C}^2$. We show that under some dynamical restriction on $\mathcal{F}_0$, we have two possibilities: $\mathcal{F}_0$ is a Darboux (logarithmic) foliation, or $\{\mathcal{F}_t\}_{t \in \mathbb{D}_{\epsilon}}$ is an unfolding. We obtain in this way a link between the analytical classification of the unfolding and the one of its germs at the singularities on the infinity line. Also we prove that a finitely generated subgroup of $\mathrm{Diff}(\mathbb{C}^n,0)$ with polynomial growth is solvable.