We establish the functoriality of Baum–Bott residues under certain conditions. As an application, we demonstrate that if  \mathcal{F}  is a holomorphic foliation of dimension  k  on a (possibly non-compact) complex manifold  X  of dimension  n , then its singular set has dimension at least  k - 1 , provided that  k \leq n/2 . This result addresses questions posed by Cerveau, Lins Neto, and Druel concerning holomorphic foliations with canonical numerically trivial divisors on projective manifolds. Additionally, it confirms the Beauville–Bondal conjecture for the maximal degeneracy locus of Poisson structures. 

This is joint work with Tatsuo Suwa.