Godbillon–Vey sequence and Françoise algorithm

Abstract We consider foliations given by deformations d F + ϵ ω of exact forms dF in C 2 in a neighborhood of a family of cycles γ ( t ) ⊂ F − 1 ( t ) . In 1996 Francoise gave an algorithm for calculating the first nonzero term of the displacement function Δ along γ of such deformations. This algorithm recalls the well-known Godbillon–Vey sequences discovered in 1971 for investigation of integrability of a form ω . 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.