Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910596400676864 |
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| author | Gehlawat, Sahil Nguyên, Viêt-Anh |
| author_facet | Gehlawat, Sahil Nguyên, Viêt-Anh |
| contents | Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of $\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the following properties:
1. for every $\mathcal{F} \in \mathcal{S}_d(\mathbb{P}^n),$ all singular points of $\mathcal{F}$ are linearizable hyperbolic.
2. If, moreover, $d \geq 2,$ then every $\mathcal{F}$ does not possess any invariant algebraic curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$ Gehlawat, Sahil Nguyên, Viêt-Anh Complex Variables Differential Geometry 37F75, 37A30 (Primary) 57R30 (Secondary) Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of $\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the following properties: 1. for every $\mathcal{F} \in \mathcal{S}_d(\mathbb{P}^n),$ all singular points of $\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d \geq 2,$ then every $\mathcal{F}$ does not possess any invariant algebraic curve. |
| title | Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$ |
| topic | Complex Variables Differential Geometry 37F75, 37A30 (Primary) 57R30 (Secondary) |
| url | https://arxiv.org/abs/2409.06052 |