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

Abstract

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 Poincar\'e'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.