Complete nonsingular holomorphic foliations on Stein manifolds

Antonio Alarcón (Author), Franc Forstnerič (Author)

Abstract

Naj bo ▫$X$▫ Steinova mnogoterost kompleksne dimenzije ▫$n>1$▫ z Riemannovo metriko ▫${\mathfrak g}$▫. V članku je dokazano, da za vsako naravno število ▫$k$▫ z ▫$\left[\frac{n}{2}\right] \le k \le n-1$▫ obstaja nesingularna holomorfna foliacija dimenzije ▫$k$▫ na ▫$X$▫, katere listi so zaprti in kompletni v metriki ▫${\mathfrak g}$▫. Isto velja v primeru ▫$1\le k<\left[\frac{n}{2}\right]$▫ pod pogojem, da obstaja kompleksni epimorfizem tangentnega svežnja mnogoterosti ▫$X$▫ na trivialen sveženj ranga ▫$n-k$▫ na ▫$X$▫. Dokazano je tudi, da za vsako pravo holomorfno foliacijo ▫${\mathcal F}$▫ na kompleksnem evklidskem prostoru ▫${\mathbb C}^n$▫ ▫$(n>1)$▫ in Riemannovo metriko ▫${\mathfrak g}$▫ na ▫${\mathbb C}$▫ obstaja holomorfen avtomorfizem ▫$\Phi$▫ prostora ▫${\mathbb C}^n$▫, tako da je preslikana foliacija ▫$\Phi_*{\mathcal F}$▫ kompletna v metriki ▫${\mathfrak g}$▫. Analogen rezultat velja na vsaki Steinovi mnogoterosti z Varolinovo lastnostjo gostote.

Keywords

Steinove mnogoterosti;kompletne holomorfne foliacije;lastnost gostote;Stein manifolds;complete holomorphic foliations;density property;

Data

Language:	English
Year of publishing:	2024
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.5
COBISS:	183749123
ISSN:	1660-5446
Views:	622
Downloads:	50
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Kompletne nesingularne holomorfne foliacije na Steinovih mnogoterostih
Secondary abstract:	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.
Secondary keywords:	Steinove mnogoterosti;kompletne holomorfne foliacije;lastnost gostote;
Type (COBISS):	Article
Pages:	16 str.
Volume:	ǂVol. ǂ21
Issue:	ǂiss. ǂ1, [article no.] 25
Chronology:	Jan. 2024
DOI:	10.1007/s00009-023-02566-0
ID:	22947233