Splitting aspects of holomorphic distributions with locally free tangent sheaf
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914826783031296 |
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| author | da Costa, Raphael Constant |
| author_facet | da Costa, Raphael Constant |
| contents | In this work, we mainly deal with a two-dimensional singular holomorphic distribution $\mathcal{D}$ defined on $M$, in the two situations $M=\mathbb{P}^n$ or $M=(\mathbb{C}^n,0)$, tangent to a one-dimensional foliation $\mathcal{G}$ on $M$, and whose tangent sheaf $T_{\mathcal{D}}$ is locally free. We provide sufficient conditions on $\mathcal{G}$ so that there is another one-dimensional foliation $\mathcal{H}$ on $M$ tangent to $\mathcal{D}$, such that their respective tangent sheaves satisfy the splitting relation $T_{\mathcal{D}}=T_{\mathcal{G}} \oplus T_{\mathcal{H}}$. As an application, we show that if $\mathcal{F}$ is a codimension one holomorphic foliation on $\mathbb{P}^3$ with locally free tangent sheaf and tangent to a nontrivial holomorphic vector field on $\mathbb{P}^3$, then $T_{\mathcal{F}}$ splits. Some division results for vector fields and differential forms are also obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Splitting aspects of holomorphic distributions with locally free tangent sheaf da Costa, Raphael Constant Complex Variables Commutative Algebra Algebraic Geometry 32M25, 32S65 In this work, we mainly deal with a two-dimensional singular holomorphic distribution $\mathcal{D}$ defined on $M$, in the two situations $M=\mathbb{P}^n$ or $M=(\mathbb{C}^n,0)$, tangent to a one-dimensional foliation $\mathcal{G}$ on $M$, and whose tangent sheaf $T_{\mathcal{D}}$ is locally free. We provide sufficient conditions on $\mathcal{G}$ so that there is another one-dimensional foliation $\mathcal{H}$ on $M$ tangent to $\mathcal{D}$, such that their respective tangent sheaves satisfy the splitting relation $T_{\mathcal{D}}=T_{\mathcal{G}} \oplus T_{\mathcal{H}}$. As an application, we show that if $\mathcal{F}$ is a codimension one holomorphic foliation on $\mathbb{P}^3$ with locally free tangent sheaf and tangent to a nontrivial holomorphic vector field on $\mathbb{P}^3$, then $T_{\mathcal{F}}$ splits. Some division results for vector fields and differential forms are also obtained. |
| title | Splitting aspects of holomorphic distributions with locally free tangent sheaf |
| topic | Complex Variables Commutative Algebra Algebraic Geometry 32M25, 32S65 |
| url | https://arxiv.org/abs/2405.17415 |