Unfoldings and deformations of rational and logarithmic foliations

Abstract

We study codimension one foliations in projective space \PP^n over \CC by looking at its first order perturbations: unfoldings and deformations. We give special attention to foliations of rational and logarithmic type. For a differential form \omega defining a codimension one foliation, we present a graded module \UU(\omega), related to the first order unfoldings of \omega. If \omega is a generic form of rational or logarithmic type, as a first application of the construction of \UU(\omega), we classify the first order deformations that arise from first order unfoldings. Then, we count the number of isolated points in the singular set of \omega, in terms of a Hilbert polynomial associated to \UU(\omega). We review the notion of regularity of \omega in terms of a long complex of graded modules that we also introduce in this work. We use this complex to prove that, for generic rational and logarithmic foliations, \omega is regular if and only if every unfolding is trivial up to isomorphism.