Complete nonsingular holomorphic foliations on Stein manifolds

Alarcón, Antonio (author), Forstnerič, Franc (author)

Let $X$ be a Stein manifold of complex dimension $n \ge 1$ 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 $\Phi$ of $\mathbb{C}^n$ such that the image foliation $\Phi_*\mathcal{F}$ is ${\mathfrak g}$-complete. The analogous result is obtained on every Stein manifold with Varolin's density property.

• Language: English
• Keywords: Stein manifolds, complete holomorphic foliations, density property
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.01.2024
• Year: 2024
• Number of pages: 16 str.
• Numbering: Vol. 21, iss. 1, article no. 25
• PID: 20.500.12556/RUL-154509
• UDC: 517.5
• ISSN on article: 1660-5446
• DOI: 10.1007/s00009-023-02566-0
• COBISS.SI-ID: 183749123
• Publication date in RUL: 19.02.2024

Record is a part of a journal

• Title: Mediterranean journal of mathematics
• Shortened title: Mediterr. j. math.
• Publisher: Springer Nature
• ISSN: 1660-5446
• COBISS.SI-ID: 13561433

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: Steinove mnogoterosti, kompletne holomorfne foliacije, lastnost gostote