On Milnor and Tjurina Numbers of Foliations

We study the relationship between the Milnor and Tjurina numbers of a singular foliation F, in the complex plane, with respect to a balanced divisor of separatrices B for F. For that, we associate with F a new number called the χ-number and we prove that it is a C1 invariant for holomorphic foliations. We compute the polar excess number of F with respect to a balanced divisor of separatrices B for F, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of B and the χ-number of F. On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of F. Finally, we state numerical relationships between some classic indices, as Baum–Bott, Camacho–Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.