Baum-Bott residue currents
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910359553572864 |
|---|---|
| author | Kaufmann, Lucas Lärkäng, Richard Wulcan, Elizabeth |
| author_facet | Kaufmann, Lucas Lärkäng, Richard Wulcan, Elizabeth |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08887 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Baum-Bott residue currents Kaufmann, Lucas Lärkäng, Richard Wulcan, Elizabeth Complex Variables Algebraic Geometry Differential Geometry 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. |
| title | Baum-Bott residue currents |
| topic | Complex Variables Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2302.08887 |