Baum-Bott residue currents

TL;DR

This paper constructs explicit residue currents for holomorphic foliations' Baum-Bott residues, providing a new analytical approach that generalizes classical residue formulas, especially for singularities of higher codimension.

Contribution

It introduces a method to explicitly represent Baum-Bott residues as currents using resolutions and connections, extending classical residue formulas to more complex singularities.

Findings

Explicit construction of residue currents supported on singular sets.

Independence of residue currents from metrics under certain conditions.

Recovery of classical Baum-Bott residues in the case of isolated singularities.

Abstract

Let $\mathscr{F}$ be a holomorphic foliation of rank $\kappa$ on a complex manifold $M$ of dimension $n$, let $Z$ be a compact connected component of the singular set of $\mathscr{F}$, and let $\Phi \in \mathbb C[z_1,\ldots,z_n]$ be a homogeneous symmetric polynomial of degree $\ell$ with $n-\kappa < \ell \leq n$. Given a locally free resolution of the normal sheaf of $\mathscr{F}$, equipped with Hermitian metrics and certain smooth connections, we construct an explicit current $R^\Phi_Z$ with support on $Z$ that represents the Baum-Bott residue $\text{res}^\Phi(\mathscr{F}; Z)\in H_{2n-2\ell}(Z, \mathbb C)$ and is obtained as the limit of certain smooth representatives of $\text{res}^\Phi(\mathscr{F}; Z)$. If the connections are $(1,0)$-connections and $\text{codim} Z\geq \ell$, then $R^\Phi_Z$ is independent of the choice of metrics and connections. When $\mathscr{F}$ has rank one we give a more precise description of $R^\Phi_Z$ in terms of so-called residue currents of Bochner-Martinelli type. In particular, when the singularities are isolated, we recover the classical expression of Baum-Bott residues in terms of Grothendieck residues.