Formulas for Residues of Type Camacho-Sad and Applications

Abstract

In this paper, we provide formulas for the sum of residues of type Camacho-Sad of a holomorphic foliation with respect to an invariant analytic subvariety. As application, in context of projective foliations, we obtain a formula that relates the sum these residues with the degree and other characteristics of the invariant subvariety. Furthermore, we establish sufficient conditions ensuring that an irreducible curve is invariant by a projective foliation on ${\mathbb{P}}^2$. In addition, we provide an adjunction formula for hypersurfaces with non-isolated singularities and obtain explicit formulas for the Milnor number of these hypersurfaces.