On functoriality of Baum-Bott residues

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 $