Analytic deformations of pencils and integrable one-forms having a first integral

Abstract

We consider integrable analytic deformations of codimension one holomorphic foliations near an initially singular point. Such deformations are of two possible types. The first type is given by an analytic family [Formula: see text] of integrable one-forms [Formula: see text] defined in a neighborhood [Formula: see text] of the initial singular point, and parametrized by the disc [Formula: see text]. The initial foliation is defined by [Formula: see text]. The second type, more restrictive, is given by an integrable holomorphic one-form [Formula: see text] defined in the product [Formula: see text]. Then, the initial foliation is defined by the slice restriction [Formula: see text]. In the first part of this work, we study the case where the starting foliation has a holomorphic first integral, i.e. it is given by [Formula: see text] for some germ of holomorphic function [Formula: see text] at the origin [Formula: see text]. We assume that the germ [Formula: see text] is irreducible and that the typical fiber of [Formula: see text] is simply-connected. This is the case if outside of a dimension [Formula: see text] analytic subset [Formula: see text], the analytic hypersurface [Formula: see text] has only normal crossings singularities. We then prove that, if cod sing [Formula: see text] then the (germ of the) developing foliation given by [Formula: see text] also exhibits a holomorphic first integral. For the general case, i.e. cod sing [Formula: see text], we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations [Formula: see text], of a local pencil [Formula: see text], for [Formula: see text]. For dimension [Formula: see text] we consider [Formula: see text]. For dimension [Formula: see text] we assume some generic geometric conditions on [Formula: see text] and [Formula: see text]. In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type [Formula: see text] with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form [Formula: see text] with some additional properties, provided that for [Formula: see text] the axes remain invariant for the foliations [Formula: see text].