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Oka complements of countable sets and nonelliptic Oka manifolds


Author: Yuta Kusakabe
Journal: Proc. Amer. Math. Soc. 148 (2020), 1233-1238
MSC (2010): Primary 32E10, 32E30, 32H02; Secondary 32M17
DOI: https://doi.org/10.1090/proc/14832
Published electronically: November 6, 2019
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Abstract: We study the Oka properties of complements of closed countable sets in $ \mathbb{C}^{n}\ (n>1)$ which are not necessarily discrete. Our main result states that every tame closed countable set in $ \mathbb{C}^{n}\ (n>1)$ with a discrete derived set has an Oka complement. As an application, we obtain nonelliptic Oka manifolds which negatively answer a long-standing question of Gromov. Moreover, we show that these examples are not even weakly subelliptic. It is also proved that every finite set in a Hopf manifold has an Oka complement and an Oka blowup.


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Additional Information

Yuta Kusakabe
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email: y-kusakabe@cr.math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/proc/14832
Keywords: Oka manifold, ellipticity, tame set, automorphism, Hopf manifold
Received by editor(s): July 28, 2019
Published electronically: November 6, 2019
Additional Notes: This work was supported by JSPS KAKENHI Grant Number JP18J20418.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2019 American Mathematical Society