On the connectedness of the singular set of holomorphic foliations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912963930095616 |
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| author | Calvo-Andrade, Omegar Corrêa, Maurício Jardim, Marcos Seade, José |
| author_facet | Calvo-Andrade, Omegar Corrêa, Maurício Jardim, Marcos Seade, José |
| contents | Let $\mathcal{F}$ be a singular holomorphic foliation of dimension $k>1$ on a projective $n$-manifold $X$. Assume that the determinant of the normal sheaf of $\mathcal{F}$ is ample (as is always the case when $X=\mathbb{P}^{n}$), and that the singular set $Sing(\mathcal{F})$ has dimension $\leq k-1$. We show that the union of those irreducible components of $Sing(\mathcal{F})$ of dimension exactly $k-1$ is necessarily connected. Consequently, we obtain a Bott-type topological obstruction to the integrability of singular holomorphic distributions, echoing Bott's vanishing theorem, and we answer a question of Cerveau for codimension-one foliations on $\mathbb{P}^{3}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the connectedness of the singular set of holomorphic foliations Calvo-Andrade, Omegar Corrêa, Maurício Jardim, Marcos Seade, José Algebraic Geometry Complex Variables Differential Geometry Dynamical Systems Let $\mathcal{F}$ be a singular holomorphic foliation of dimension $k>1$ on a projective $n$-manifold $X$. Assume that the determinant of the normal sheaf of $\mathcal{F}$ is ample (as is always the case when $X=\mathbb{P}^{n}$), and that the singular set $Sing(\mathcal{F})$ has dimension $\leq k-1$. We show that the union of those irreducible components of $Sing(\mathcal{F})$ of dimension exactly $k-1$ is necessarily connected. Consequently, we obtain a Bott-type topological obstruction to the integrability of singular holomorphic distributions, echoing Bott's vanishing theorem, and we answer a question of Cerveau for codimension-one foliations on $\mathbb{P}^{3}$. |
| title | On the connectedness of the singular set of holomorphic foliations |
| topic | Algebraic Geometry Complex Variables Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2506.08942 |