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May 2022 Complete complex hypersurfaces in the ball come in foliations
Antonio Alarcón
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J. Differential Geom. 121(1): 1-29 (May 2022). DOI: 10.4310/jdg/1656005494

Abstract

In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n (n \geq 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required.

Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q \in \lbrace 1, \dotsc , n-1 \rbrace$.

Citation

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Antonio Alarcón. "Complete complex hypersurfaces in the ball come in foliations." J. Differential Geom. 121 (1) 1 - 29, May 2022. https://doi.org/10.4310/jdg/1656005494

Information

Received: 21 February 2020; Accepted: 2 September 2020; Published: May 2022
First available in Project Euclid: 24 June 2022

Digital Object Identifier: 10.4310/jdg/1656005494

Subjects:
Primary: 32E10 , 32E30 , 32H02 , 53C12

Rights: Copyright © 2022 Lehigh University

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Vol.121 • No. 1 • May 2022
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