Wild holomorphic foliations of the ball
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911134900027392 |
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| author | Alarcon, Antonio |
| author_facet | Alarcon, Antonio |
| contents | We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$. In particular, every leaf of $\mathcal F$ is both a limit of complete leaves of $\mathcal F$ and a limit of incomplete leaves of $\mathcal F$. This gives the first example of a holomorphic foliation of $\mathbb{B}_n$ by connected closed complex hypersurfaces having a complete leaf that is a limit of incomplete ones. We obtain an analogous result for foliations by complex submanifolds of arbitrary pure codimension $q$ with $1\le q<n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_04950 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Wild holomorphic foliations of the ball Alarcon, Antonio Complex Variables Differential Geometry We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$. In particular, every leaf of $\mathcal F$ is both a limit of complete leaves of $\mathcal F$ and a limit of incomplete leaves of $\mathcal F$. This gives the first example of a holomorphic foliation of $\mathbb{B}_n$ by connected closed complex hypersurfaces having a complete leaf that is a limit of incomplete ones. We obtain an analogous result for foliations by complex submanifolds of arbitrary pure codimension $q$ with $1\le q<n$. |
| title | Wild holomorphic foliations of the ball |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2106.04950 |