On functoriality of Baum-Bott residues

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 $k\le n\mathrm{/}2$, on a (possibly non-compact) complex manifold $X$ of dimension $n$, then its singular set $Sing(\mathcal{F})$ has dimension $\dim (Sing(\mathcal{F}))\ge k-1$. 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 $r\le n\mathrm{/}2$, and the degeneracy locus $X\setminus X_r$ is non-empty, then it contains a component of dimension $>r-2$