# Godbillon-Vey sequence and Francoise algorithm

## TL;DR

This paper explores the relationship between Godbillon-Vey sequences and Francoise's algorithm in analyzing deformations of foliations in complex two-dimensional space, connecting integrability conditions with computational methods.

## Contribution

It establishes a correspondence between Godbillon-Vey sequences and Francoise's algorithm, translating results on integrability from one framework to the other.

## Key findings

- Established the link between Godbillon-Vey sequences and Francoise's algorithm.
- Translated Casale's results on integrability types into the Francoise algorithm context.
- Provided insights into the structure of deformations of foliations in complex geometry.

## Abstract

We consider foliations given by deformations $dF+\epsilon\omega$ of exact forms $dF$ in $\mathbb{C}^2$ in a neighborhood of a family of cycles $\gamma(t)\subset F^{-1}(t)$. In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function $\Delta$ along $\gamma$ of such deformations. This algorithm recalls the well-known Godbillon-Vey sequences discovered in 1971 for investigation integrability of a form $\omega$. In this paper, we establish the correspondence between the two approaches and translate some results by Casale relating types of integrability for finite Godbillon-Vey sequences to the Francoise algorithm settings.