Abstract
In this work, we study analytic deformations of a pencil [Formula: see text] where [Formula: see text] and [Formula: see text] are holomorphic functions defined in [Formula: see text], and [Formula: see text] are relatively prime integers. We consider analytic deformations given by integrable [Formula: see text]-forms [Formula: see text] of the pencil form [Formula: see text] We prove that, under mild hypotheses, [Formula: see text] admits a meromorphic first integral if and only if it is dicritical.
In the local two-dimensional case the fixed axes need not remain invariant: dicriticality produces analytic moving axes and a pencil-type meromorphic first integral. Our results apply, in particular, to deformations of the local pencil [Formula: see text], with [Formula: see text], and to foliations defined by homogeneous [Formula: see text]-forms of the same degree as [Formula: see text], where [Formula: see text] and [Formula: see text] are homogeneous and satisfy [Formula: see text].
We show that, in the projective homogeneous case, when [Formula: see text] is a branched Lefschetz pencil, [Formula: see text], [Formula: see text], and the projective dimension is at least three, projective deformations remain homogeneous branched pencils whose generators may move with the parameter. Under the fixed-level homological condition, namely the vanishing of the integrals of [Formula: see text] along all admissible cycles of the levels [Formula: see text], every coefficient of the deformation is an infinitesimal moving-generator deformation.