On functoriality of Baum-Bott residues

TL;DR

This paper proves the functoriality of Baum-Bott residues, leading to new insights into the structure of singular sets of holomorphic foliations and confirming conjectures related to Poisson structures on complex manifolds.

Contribution

It establishes the functoriality of Baum-Bott residues and applies this to analyze singular sets of foliations and degeneracy loci of Poisson structures, answering longstanding open questions.

Findings

Singular set of certain holomorphic foliations has dimension at least k-1.

Confirmed the Beauville-Bondal conjecture for Poisson degeneracy loci.

Provided answers to questions by Cerveau, Lins Neto, and Druel.

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 $