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

Diogo Da Silva Machado

TL;DR

The paper develops a Baum–Bott type residue formula for one-dimensional holomorphic foliations that are logarithmic along a hypersurface with isolated singularities, expressing residues in terms of the Schwartz index. It proves parity-dependent positivity: $Sch_p(\mathcal{F},D)>0$ when $n$ is even and $GSV_p(\mathcal{F},D)>0$ when $n$ is odd, and applies these results to Poincaré-type problems in projective spaces and to Euler-characteristic obstructions. The authors derive explicit relations among Milnor numbers, Schwartz indices, and GSV indices, leading to degree bounds for invariant hypersurfaces and generalized obstructions to the existence of global vector fields on hypersurfaces with isolated singularities. These results extend classical Baum–Bott and Poincaré–Hopf type obstructions to the setting of logarithmic foliations along singular hypersurfaces and provide tools for bounding invariant geometry in projective and compact settings.

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 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.

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

TL;DR

The paper develops a Baum–Bott type residue formula for one-dimensional holomorphic foliations that are logarithmic along a hypersurface with isolated singularities, expressing residues in terms of the Schwartz index. It proves parity-dependent positivity: when is even and when is odd, and applies these results to Poincaré-type problems in projective spaces and to Euler-characteristic obstructions. The authors derive explicit relations among Milnor numbers, Schwartz indices, and GSV indices, leading to degree bounds for invariant hypersurfaces and generalized obstructions to the existence of global vector fields on hypersurfaces with isolated singularities. These results extend classical Baum–Bott and Poincaré–Hopf type obstructions to the setting of logarithmic foliations along singular hypersurfaces and provide tools for bounding invariant geometry in projective and compact settings.

Abstract

Given a compact complex manifold , we prove a Baum-Bott type formula for one-dimensional holomorphic foliations on 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 is even and that the GSV index is positive when 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 , 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.

Paper Structure

This paper contains 10 sections, 4 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular one-dimensional holomorphic foliations
  4. The Schwartz index
  5. The GSV-index
  6. Proof of Theorem 1
  7. Proof of Theorem 2
  8. Proof of Theorem 3
  9. Proof of Theorem 4
  10. Proof of Theorem 5

Key Result

Theorem 1

Let $D \subset \mathbb{P}^2$ be an irreducible hypersurface with isolated singularities and degree $\deg(D) = k$. Let $\mathcal{F}$ be a foliation of dimension one on $\mathbb{P}^2$, with isolated singularities and degree $\deg(\mathcal{F}) = d$. If $\mathcal{F}$ leaves $D$ invariant, then

Theorems & Definitions (9)

  • Remark 1.1
  • Theorem : M. Soares, Soa
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Corollary 4.1
  • Lemma 5.1
  • Example 6.1