Complete nonsingular holomorphic foliations on Stein manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916225360068608 |
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| author | Alarcon, Antonio Forstneric, Franc |
| author_facet | Alarcon, Antonio Forstneric, Franc |
| contents | Let $X$ be a Stein manifold of complex dimension $n>1$ endowed with a Riemannian metric $\mathfrak{g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and $\mathfrak{g}$-complete. The same is true if $1\le k<\left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism $TX\to X\times\mathbb{C}^{n-k}$. We also show that if $\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^n$ $(n>1)$ then for any Riemannian metric $\mathfrak{g}$ on $\mathbb{C}^n$ there is a holomorphic automorphism $Φ$ of $\mathbb{C}^n$ such that the image foliation $Φ_*\mathcal{F}$ is $\mathfrak{g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_06030 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Complete nonsingular holomorphic foliations on Stein manifolds Alarcon, Antonio Forstneric, Franc Complex Variables Primary 32M17, 32M25, secondary 32H02, 37F75 Let $X$ be a Stein manifold of complex dimension $n>1$ endowed with a Riemannian metric $\mathfrak{g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and $\mathfrak{g}$-complete. The same is true if $1\le k<\left[\frac{n}{2}\right]$ provided that there is a complex vector bundle epimorphism $TX\to X\times\mathbb{C}^{n-k}$. We also show that if $\mathcal{F}$ is a proper holomorphic foliation on $\mathbb{C}^n$ $(n>1)$ then for any Riemannian metric $\mathfrak{g}$ on $\mathbb{C}^n$ there is a holomorphic automorphism $Φ$ of $\mathbb{C}^n$ such that the image foliation $Φ_*\mathcal{F}$ is $\mathfrak{g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property. |
| title | Complete nonsingular holomorphic foliations on Stein manifolds |
| topic | Complex Variables Primary 32M17, 32M25, secondary 32H02, 37F75 |
| url | https://arxiv.org/abs/2305.06030 |