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

TL;DR

This paper establishes a Baum-Bott type formula relating residues of logarithmic foliations on compact complex manifolds to the Schwartz index, revealing positivity properties and applications to Poincaré's problem across various dimensions.

Contribution

It introduces a new residue formula for logarithmic foliations, linking residues to the Schwartz index and extending obstruction results to higher-dimensional projective spaces.

Findings

Residues expressed via Schwartz index for logarithmic foliations.

Positivity of Schwartz index in even dimensions and GSV index in odd dimensions.

Obstructions related to singularities and Euler characteristic generalized to higher dimensions.

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.