CONICET | Buscador de Institutos y Recursos Humanos

We study codimension one foliations in projective space $mathbb{P}^n$ over $mathbb{C}$ 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 $mathbb{U}(omega )$, 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 $mathbb{U}(omega )$, 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 $mathbb{U}(omega )$.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.