Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Log Baum--Bott Residues for foliations by curves

Maurício Corrêa, Fernando Lourenço, Diogo Machado

TL;DR

This work extends Baum–Bott theory to one-dimensional holomorphic foliations that are logarithmic along analytic free divisors, producing a log-residue formula that expresses global characteristic numbers of the pair $(T_X(-\log D), T_{\\mathscr F})$ as sums of BB-residues away from $D$ and log residues along $D$. It introduces log BB residues ${\\rm Res}^{\log}_{\\varphi}(\\mathscr F, D, S_{\\lambda})$ and shows, for $\varphi = \det$, that these specialize to Aleksandrov's logarithmic index (the Log residue) in the isolated-point setting, while connecting to GSV and Camacho–Sad indices on surfaces. The results extend to singular varieties via log resolutions and yield a Poincaré–Hopf–type equality with Milnor numbers and a weak global Zariski–Lipman foliated smoothness criterion, indicating geometric constraints on singular surfaces arising from foliations. Additionally, the residue theory is organized into residue-valued constructible functions, bridging the analytic residue data with a microlocal/constructible framework for further applications in singularity theory and foliation dynamics.

Abstract

We prove a Baum--Bott type residual formula for one-dimensional holomorphic foliations, and logarithmic along free divisors. More precisely, this provides a Baum--Bott theorem for a foliated triple $(X, \mathcal{F}, D)$, where $\mathcal{F}$ is a foliation by curves and $D$ is a free divisor on a complex manifold $X$. From the local point of view, we show that the log Baum--Bott residues are a generalization of the Aleksandrov logarithmic index for vector fields with isolated singularities on hypersurfaces. We also show how these new indices are related to Poincaré's Problem for foliations by curves. In the case of foliated surfaces, we show that the differences between the logarithmic residues and Baum--Bott indices along invariant curves can be expressed in terms of the GSV and Camacho--Sad indices. We also obtain a Baum--Bott type formula for singular varieties via log resolutions. Finally, we prove a weak global version of the Zariski--Lipman conjecture for compact algebraic surfaces, in the form of a foliated smoothness criterion, suggesting the appearance of saddle-nodes in the singularity reduction on singular surfaces.

Log Baum--Bott Residues for foliations by curves

TL;DR

This work extends Baum–Bott theory to one-dimensional holomorphic foliations that are logarithmic along analytic free divisors, producing a log-residue formula that expresses global characteristic numbers of the pair as sums of BB-residues away from and log residues along . It introduces log BB residues and shows, for , that these specialize to Aleksandrov's logarithmic index (the Log residue) in the isolated-point setting, while connecting to GSV and Camacho–Sad indices on surfaces. The results extend to singular varieties via log resolutions and yield a Poincaré–Hopf–type equality with Milnor numbers and a weak global Zariski–Lipman foliated smoothness criterion, indicating geometric constraints on singular surfaces arising from foliations. Additionally, the residue theory is organized into residue-valued constructible functions, bridging the analytic residue data with a microlocal/constructible framework for further applications in singularity theory and foliation dynamics.

Abstract

We prove a Baum--Bott type residual formula for one-dimensional holomorphic foliations, and logarithmic along free divisors. More precisely, this provides a Baum--Bott theorem for a foliated triple , where is a foliation by curves and is a free divisor on a complex manifold . From the local point of view, we show that the log Baum--Bott residues are a generalization of the Aleksandrov logarithmic index for vector fields with isolated singularities on hypersurfaces. We also show how these new indices are related to Poincaré's Problem for foliations by curves. In the case of foliated surfaces, we show that the differences between the logarithmic residues and Baum--Bott indices along invariant curves can be expressed in terms of the GSV and Camacho--Sad indices. We also obtain a Baum--Bott type formula for singular varieties via log resolutions. Finally, we prove a weak global version of the Zariski--Lipman conjecture for compact algebraic surfaces, in the form of a foliated smoothness criterion, suggesting the appearance of saddle-nodes in the singularity reduction on singular surfaces.

Paper Structure

This paper contains 14 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Logarithmic forms and logarithmic vector fields
  4. Singular one-dimensional holomorphic foliations
  5. Proof of Corollary \ref{['Poincare-oddn']}
  6. Logarithmic, homological and ${\rm GSV}$ indices
  7. GSV-index and Camacho--Sad indices on Surfaces
  8. The Relative $\check{\hbox{C}}$ech-de Rham cohomology
  9. Proof of Theorem \ref{['ttt1']} and Corollary \ref{['cor-princ-iso']}
  10. Proof of Corollary \ref{['dim2-poinc']}
  11. Proof of Corollary \ref{['Baum-Bott-sing']}
  12. Proof of Corollary \ref{['mu-PHGB']}
  13. Proof of Corollary \ref{['GZL']}
  14. Residue-valued constructible functions

Key Result

Theorem 1.1

Let $\mathscr F$ be a one-dimensional foliation with isolated singularities and logarithmic along the normal crossing divisor $D$ in a complex compact manifold $X$. Then where $T_{\mathscr F}$ denotes the tangent bundle of $\mathscr F$ and $\mu_x(\mathscr F)$ is the Milnor number of $\mathscr F$ at $x$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Definition 2.1
  • ...and 10 more