International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-09-05 , DOI: 10.1142/s0129167x20500895 Bruno Scárdua
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 of integrable one-forms defined in a neighborhood of the initial singular point, and parametrized by the disc . The initial foliation is defined by . The second type, more restrictive, is given by an integrable holomorphic one-form defined in the product . Then, the initial foliation is defined by the slice restriction . 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 for some germ of holomorphic function at the origin . We assume that the germ is irreducible and that the typical fiber of is simply-connected. This is the case if outside of a dimension analytic subset , the analytic hypersurface has only normal crossings singularities. We then prove that, if then the (germ of the) developing foliation given by also exhibits a holomorphic first integral. For the general case, i.e. , we obtain a dimension two normal form for the developing foliation. In the second part of the paper, we consider analytic deformations , of a local pencil , for . For dimension we consider . For dimension we assume some generic geometric conditions on and . In both cases, we prove: (i) in the case of an analytic deformation there is a multiform formal first integral of type with some properties; (ii) in the case of an integrable deformation there is a meromorphic first integration of the form with some additional properties, provided that for the axes remain invariant for the foliations .