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On functoriality of Baum-Bott residues

Maurício Corrêa, Tatsuo Suwa

TL;DR

The paper establishes a functorial framework for Baum–Bott residues beyond compact settings by deriving a localized pullback formula under transversality and degeneracy hypotheses. It then applies this to obtain a sharp lower bound ${\rm dim}\,{\rm Sing}(\mathscr F)\ge k-1$ for foliations with $k\le n/2$, solving long-standing questions of Baum–Bott, Cerveau–Lins Neto, and Druel in non-compact contexts. The results also confirm the Beauville–Bondal conjecture for Poisson degeneracy loci when the generic rank $r\le n/2$, unifying topological residue techniques with relative Čech–de Rham theory and BM-homology. Overall, the work provides new tools to compute Baum–Bott residues in general, non-compact settings and demonstrates significant geometric consequences for foliations and Poisson structures.

Abstract

We establish the functoriality of Baum--Bott residues under certain conditions. As an application, we show that if $\mathcal{F}$ is a holomorphic foliation, of dimension $k\leq n/2$, on a (possibly non-compact) complex manifold $X$ of dimension \(n\), then its singular set $Sing(\mathcal{F})$ has dimension $\dim(Sing(\mathcal{F}))\geq k-1$. This result addresses a longstanding question by Baum and Bott regarding the functoriality of residues. Also, This provides answers to questions posed by Cerveau and Lins Neto concerning foliations of dimension 2 in $\mathbb{C}^4$ and Druel regarding holomorphic foliations on projective manifolds. Furthermore, it confirms the Beauville-Bondal conjecture for the maximal degeneracy locus of Poisson structures. Specifically, if $X$ is a (possibly non-compact) complex Poisson manifold with generic rank $ r \leq n/2$, and the degeneracy locus $X \setminus X_r$ is non-empty, then it contains a component of dimension $ > r - 2 $

On functoriality of Baum-Bott residues

TL;DR

The paper establishes a functorial framework for Baum–Bott residues beyond compact settings by deriving a localized pullback formula under transversality and degeneracy hypotheses. It then applies this to obtain a sharp lower bound for foliations with , solving long-standing questions of Baum–Bott, Cerveau–Lins Neto, and Druel in non-compact contexts. The results also confirm the Beauville–Bondal conjecture for Poisson degeneracy loci when the generic rank , unifying topological residue techniques with relative Čech–de Rham theory and BM-homology. Overall, the work provides new tools to compute Baum–Bott residues in general, non-compact settings and demonstrates significant geometric consequences for foliations and Poisson structures.

Abstract

We establish the functoriality of Baum--Bott residues under certain conditions. As an application, we show that if is a holomorphic foliation, of dimension , on a (possibly non-compact) complex manifold of dimension , then its singular set has dimension . This result addresses a longstanding question by Baum and Bott regarding the functoriality of residues. Also, This provides answers to questions posed by Cerveau and Lins Neto concerning foliations of dimension 2 in and Druel regarding holomorphic foliations on projective manifolds. Furthermore, it confirms the Beauville-Bondal conjecture for the maximal degeneracy locus of Poisson structures. Specifically, if is a (possibly non-compact) complex Poisson manifold with generic rank , and the degeneracy locus is non-empty, then it contains a component of dimension
Paper Structure (11 sections, 8 theorems, 120 equations, 1 figure)

This paper contains 11 sections, 8 theorems, 120 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Singular holomorphic foliations
  3. Singular holomorphic foliations
  4. Baum-Bott residues for non-compact components
  5. Dualities
  6. Čech-de Rham theory
  7. Relative Čech-de Rham cohomology
  8. Localized intersection product with a map
  9. Baum-Bott residues
  10. Proof of Theorem \ref{['Main-Theorem']}
  11. Proof of Theorem \ref{['teo2']}

Key Result

Theorem 1.1

Let $X$ be a complex manifold of dimension $n$ and $\mathscr F$ a singular holomorphic foliation of dimension $k\le n-2$ on $X$. Let $V$ be a complex manifold of dimension $n'\ge n-k+1$ and $f:V\to X$ a holomorphic map generically transverse to $\mathscr F$ whose degeneracy locus has codimension $\g For any homogeneous symmetric polynomial $\phi$ of degree $d\ge n-k+1$, the Baum--Bott residues ar

Figures (1)

  • Figure 1:

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Conjecture : Druel
  • Conjecture
  • Theorem 1.3
  • Corollary 1.4
  • Conjecture : Beauville-Bondal
  • Theorem 1.5
  • Corollary 1.6
  • Example 2.1
  • ...and 10 more