Commuting holonomies and rigidity of holomorphic foliations

TL;DR

This paper investigates how commuting holonomies of vanishing cycles influence the rigidity of holomorphic foliations with non-rational first integrals, extending previous results using Picard-Lefschetz theory and iterated integrals.

Contribution

It demonstrates that commuting deformed holonomies imply the existence of a first integral for the deformed foliation, generalizing Ilyashenko's rigidity results.

Findings

Commuting deformed holonomies imply the foliation has a first integral.

The result extends previous rigidity theorems to more general settings.

Uses Picard-Lefschetz theory and iterated integrals in the proof.

Abstract

In this article we study deformations of a holomorphic foliation with a generic non-rational first integral in the complex plane. We consider two vanishing cycles in a regular fiber of the first integral with a non-zero self intersection and with vanishing paths which intersect each other only at their start points. It is proved that if the deformed holonomies of such vanishing cycles commute then the deformed foliation has also a first integral. Our result generalizes a similar result of Ilyashenko on the rigidity of holomorphic foliations with a persistent center singularity. The main tools of the proof are Picard-Lefschetz theory and the theory of iterated integrals for such deformations.