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Proceedings of the American Mathematical Society

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Plurisubharmonically separable complex manifolds


Authors: Evgeny A. Poletsky and Nikolay Shcherbina
Journal: Proc. Amer. Math. Soc. 147 (2019), 2413-2424
MSC (2010): Primary 32U05; Secondary 32F10, 32U35
DOI: https://doi.org/10.1090/proc/14222
Published electronically: March 7, 2019
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Abstract: Let $ M$ be a complex manifold and let $ PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $ M$. In this paper we study when the functions from $ PSH^{cb}(M)$ separate points. Our main results show that this property is equivalent to each of the following properties of $ M$:

  1. the core of $ M$ is empty;
  2. for every $ w_0\in M$ there is a continuous plurisubharmonic function $ u$ with the logarithmic singularity at $ w_0$.

Moreover, the core of $ M$ is the disjoint union of the sets $ E_j$ that are 1-pseudoconcave in the sense of Rothstein and have the following Liouville property: every function from $ PSH^{cb}(M)$ is constant on each $ E_j$.


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Additional Information

Evgeny A. Poletsky
Affiliation: Department of Mathematics, Syracuse University, 215 Carnegie Hall, Syracuse, New York 13244
Email: eapolets@syr.edu

Nikolay Shcherbina
Affiliation: Department of Mathematics, University of Wuppertal, 42119 Wuppertal, Germany
Email: shcherbina@math.uni-wuppertal.de

DOI: https://doi.org/10.1090/proc/14222
Keywords: Bounded plurisubharmonic functions, cores of domains
Received by editor(s): December 6, 2017
Received by editor(s) in revised form: April 22, 2018, and May 11, 2018
Published electronically: March 7, 2019
Additional Notes: The first author was partially supported by a grant from the Simons Foundation
Communicated by: Filippo Bracci
Article copyright: © Copyright 2019 American Mathematical Society