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Baum-Bott residue currents

Lucas Kaufmann, Richard Lärkäng, Elizabeth Wulcan

Abstract

Let $\mathscr{F}$ 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 $\mathscr{F}$, and let $Φ\in \mathbb C[z_1,\ldots,z_n]$ be a homogeneous symmetric polynomial of degree $\ell$ with $n-κ< \ell \leq n$. 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 $R^Φ_Z$ with support on $Z$ that represents the Baum-Bott residue $\text{res}^Φ(\mathscr{F}; Z)\in H_{2n-2\ell}(Z, \mathbb C)$ and is obtained as the limit of certain smooth representatives of $\text{res}^Φ(\mathscr{F}; Z)$. If the connections are $(1,0)$-connections and $\text{codim} Z\geq \ell$, then $R^Φ_Z$ is independent of the choice of metrics and connections. When $\mathscr{F}$ 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.

Baum-Bott residue currents

Abstract

Let be a holomorphic foliation of rank on a complex manifold of dimension , let be a compact connected component of the singular set of , and let be a homogeneous symmetric polynomial of degree with . Given a locally free resolution of the normal sheaf of , equipped with Hermitian metrics and certain smooth connections, we construct an explicit current with support on that represents the Baum-Bott residue and is obtained as the limit of certain smooth representatives of . If the connections are -connections and , then is independent of the choice of metrics and connections. When has rank one we give a more precise description of 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.
Paper Structure (18 sections, 20 theorems, 130 equations)

This paper contains 18 sections, 20 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Holomorphic foliations, vector bundle complexes and characteristic classes
  3. Holomorphic foliations
  4. Vector bundle complexes, connections, and superstructure
  5. Characteristic classes and forms
  6. Baum-Bott theory
  7. Baum-Bott residues
  8. Residue currents
  9. Pseudomeromorphic currents
  10. Almost semi-meromorphic currents
  11. Residue currents of Bochner-Martinelli type
  12. Construction of connections
  13. The connections $D_{basic}$ and $\widetilde{D}_k$ on $M\setminus \sf$
  14. The connections $\widehat{D}^\epsilon_k$ on $M$
  15. Baum-Bott currents
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $M$ be a complex manifold of dimension $n$, let ${\mathscr{F}}$ be a holomorphic foliation of rank $\kappa$ on $M$, and let $\Phi\in {\mathbb C}[z_1,\ldots, z_n]$ be a homogeneous symmetric polynomial of degree $\ell$ with $n-\kappa<\ell\leq n$. Assume that the normal sheaf $N{\mathscr{F}}$ of $ exists as a current where the sum runs over the connected components $Z$ of $\text{\normalfont si

Theorems & Definitions (43)

  • Theorem 1.1
  • Lemma 2.1: baum-bott - Lemma 4.22
  • Definition 3.1
  • Theorem 3.2: Baum-Bott's Vanishing theorem, baum-bott - Proposition 3.27
  • Remark 4.1
  • Example 4.2
  • Proposition 4.3
  • Lemma 4.4: Proposition 4.16 in andersson-wulcan:asm
  • Remark 4.5
  • Example 4.6
  • ...and 33 more