Baum-Bott residue currents

2023/02/17 by Kaufmann, Lucas, Lärkäng, Richard, Wulcan, Elizabeth
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.08887

Abstract

Let \mathscrF be a holomorphic foliation of rank κ on a complex manifold M of dimension n, let Z be a compact connected component of the singular set of \mathscrF, and let Φ∈ \mathbb C[z₁,…,z_n] be a homogeneous symmetric polynomial of degree ℓ with n-κ< ℓ ≤ n. Given a locally free resolution of the normal sheaf of \mathscrF, equipped with Hermitian metrics and certain smooth connections, we construct an explicit current R^Φ_Z with support on Z that represents the Baum-Bott residue res^Φ(\mathscrF; Z)∈ H_2n-2ℓ(Z, \mathbb C) and is obtained as the limit of certain smooth representatives of res^Φ(\mathscrF; Z). If the connections are (1,0)-connections and codim Z≥ ℓ, then R^Φ_Z is independent of the choice of metrics and connections. When \mathscrF has rank one we give a more precise description of R^Φ_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.

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