Oka domains in Euclidean spaces

Franc Forstnerič (Author), Erlend Fornæss Wold (Author)

Abstract

V članku konstruiramo presenetljivo majhne domene Oka v kompleksnih evklidskih prostorih ▫$\mathbb C^n$▫ dimenzije ▫$n>1$▫ na meji možnega. Pokazano je, da je pod blagimi pogoji na neomejeno zaprto konveksno množico ▫$E$▫ v ▫$\mathbb C^n$▫ njen komplement ▫$\mathbb C^n\setminus E$▫ domena Oka. V posebnem obstajajo domene Oka v ▫$\mathbb C^n$▫, ki so le malo večje od polprostora. Splošneje je pokazano, da je komplement zaprte množice ▫$E$▫ v ▫$\mathbb C^n$▫ za ▫$n>1$▫, katere projektivno zaprtje ▫$\overline E$▫ ne seka kompleksne hiperravnine ▫$\Lambda\subset\mathbb{CP}^n$▫ in je polinomsko konveksno v ▫$\mathbb{CP}^n\setminus \Lambda\cong\mathbb C^n$▫, domena Oka v ▫$\mathbb C^n$▫.

Keywords

mnogoterost Oka;hiperbolične mnogoterosti;lastnost gostote;projektivno konveksna množica;Oka manifold;hyperbolic manifolds;density property;projectively convex sets;

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:	143307011
ISSN:	1687-0247
Views:	773
Downloads:	81
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Domene Oka v evklidskih prostorih
Secondary abstract:	In this paper, we find surprisingly small Oka domains in Euclidean spaces ▫$\mathbb C^n$▫ of dimension ▫$n>1$▫ at the very limit of what is possible. Under a mild geometric assumption on a closed unbounded convex set ▫$E$▫ in ▫$\mathbb C^n$▫, we show that ▫$\mathbb C^n\setminus E$▫ is an Oka domain. In particular, there are Oka domains only slightly bigger than a halfspace, the latter being neither Oka nor hyperbolic. This gives smooth families of real hypersurfaces ▫$\Sigma_t \subset \mathbb C^n$▫ for ▫$t \in \mathbb R$▫ dividing ▫$\mathbb C^n$▫ in an unbounded hyperbolic domain and an Oka domain such that at ▫$t=0$▫, ▫$\Sigma_0$▫ is a hyperplane and the character of the two sides gets reversed. More generally, we show that if ▫$E$▫ is a closed set in ▫$\mathbb C^n$▫ for ▫$n>1$▫ whose projective closure ▫$\overline E \subset \mathbb{CP}^n$▫ avoids a hyperplane ▫$\Lambda \subset \mathbb{CP}^n$▫ and is polynomially convex in ▫$\mathbb{CP}^n\setminus \Lambda\cong\mathbb C^n$▫, then ▫$\mathbb C^n\setminus E$▫ is an Oka domain.
Secondary keywords:	mnogoterost Oka;hiperbolične mnogoterosti;lastnost gostote;projektivno konveksna množica;
Type (COBISS):	Article
Pages:	str. 1801-1824
Volume:	ǂVol. ǂ2024
Issue:	ǂiss. ǂ3
Chronology:	Feb. 2024
DOI:	10.1093/imrn/rnac347
ID:	22947236