No abstract

We develop a study on local polar invariants of planar complex analytic foliations at (C 2 , 0), which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the GSV -index. We apply it to the Poincaré problem for foliations on the complex projective plane P 2 C , establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve S in terms of the degree of the foliation F . We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of F over the curve S. Our method, in particular, recovers the known solution for the non-dicritical case, deg(S) ≤ deg(F ) + 2.

An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many saddle-node singularities of the complex plane that are not algebraizable.Comment: 11 page

We build a germ of singular foliation in C 2 with analytical class of separatrix and holonomy representations prescribed. Thanks to this construction, we study the link between moduli of a foliation and moduli of its separatrix.

We introduce the concept of good Saito basis for a plane curve and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we give a lower bound for the Tjurina number in terms of the Milnor number that allow us to present a positive answer for a question of Dimca and Greuel.