On functoriality of Baum-Bott residues
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915631487516672 |
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| author | Corrêa, Maurício Suwa, Tatsuo |
| author_facet | Corrêa, Maurício Suwa, Tatsuo |
| contents | We establish the functoriality of Baum--Bott residues under certain conditions. As an application, we show that if $\mathcal{F}$ is a holomorphic foliation, of dimension $k\leq n/2$, on a (possibly non-compact) complex manifold $X$ of dimension \(n\), then its singular set $Sing(\mathcal{F})$ has dimension $\dim(Sing(\mathcal{F}))\geq k-1$. This result addresses a longstanding question by Baum and Bott regarding the functoriality of residues. Also, This provides answers to questions posed by Cerveau and Lins Neto concerning foliations of dimension 2 in $\mathbb{C}^4$ and Druel regarding holomorphic foliations on projective manifolds. Furthermore, it confirms the Beauville-Bondal conjecture for the maximal degeneracy locus of Poisson structures. Specifically, if $X$ is a (possibly non-compact) complex Poisson manifold with generic rank $ r \leq n/2$, and the degeneracy locus $X \setminus X_r$ is non-empty, then it contains a component of dimension $ > r - 2 $ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15133 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On functoriality of Baum-Bott residues Corrêa, Maurício Suwa, Tatsuo Complex Variables Algebraic Geometry Differential Geometry Dynamical Systems Symplectic Geometry We establish the functoriality of Baum--Bott residues under certain conditions. As an application, we show that if $\mathcal{F}$ is a holomorphic foliation, of dimension $k\leq n/2$, on a (possibly non-compact) complex manifold $X$ of dimension \(n\), then its singular set $Sing(\mathcal{F})$ has dimension $\dim(Sing(\mathcal{F}))\geq k-1$. This result addresses a longstanding question by Baum and Bott regarding the functoriality of residues. Also, This provides answers to questions posed by Cerveau and Lins Neto concerning foliations of dimension 2 in $\mathbb{C}^4$ and Druel regarding holomorphic foliations on projective manifolds. Furthermore, it confirms the Beauville-Bondal conjecture for the maximal degeneracy locus of Poisson structures. Specifically, if $X$ is a (possibly non-compact) complex Poisson manifold with generic rank $ r \leq n/2$, and the degeneracy locus $X \setminus X_r$ is non-empty, then it contains a component of dimension $ > r - 2 $ |
| title | On functoriality of Baum-Bott residues |
| topic | Complex Variables Algebraic Geometry Differential Geometry Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2501.15133 |