Diogo Da Silva Machado

arXiv2025

Formulas for Residues of Type Camacho-Sad and Applications

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.

math.AG↗

arXiv2025

The Schwartz index and the residue of logarithmic foliations along a hypersurface with isolated singularities

Given a compact complex manifold $X$, we prove a Baum-Bott type formula for one-dimensional holomorphic foliations on $X$ that are logarithmic along a hypersurface with isolated singularities. We show that the residues of these foliations can be expressed in terms of the Schwartz index of the vector fields that locally define them. Furthermore, in this context, we prove that the Schwartz index is positive when $\dim(X)$ is even and that the GSV index is positive when $\dim(X)$ is odd. As application, we show that the obstruction determined by the multiplicity of the isolated singularities of the invariant hypersurface, for the solution of Poincaré's problem in holomorphic foliations on $\mathbb{P}^2$, is a more general fact, valid for holomorphic foliations defined on projective spaces of arbitrary even dimension. Additionally, we prove that the obstruction determined by the Euler characteristic for the existence of vector fields is even more comprehensive in the case of hypersurfaces with isolated singularities.

math.AG↗