### 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} ∈ 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}.

Original language | English (US) |
---|---|

Pages (from-to) | 2413-2424 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 147 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2019 |

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### Keywords

- Bounded plurisubharmonic functions
- Cores of domains

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*147*(6), 2413-2424. https://doi.org/10.1090/proc/14222

**Plurisubharmonically separable complex manifolds.** / Poletsky, Evgeny Alexander; Shcherbina, Nikolay.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 147, no. 6, pp. 2413-2424. https://doi.org/10.1090/proc/14222

}

TY - JOUR

T1 - Plurisubharmonically separable complex manifolds

AU - Poletsky, Evgeny Alexander

AU - Shcherbina, Nikolay

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let M be a complex manifold and let PSHcb (M) be the space of bounded continuous plurisubharmonic functions on M. In this paper we study when the functions from PSHcb (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 w0 ∈ M there is a continuous plurisubharmonic function u with the logarithmic singularity at w0. Moreover, the core of M is the disjoint union of the sets Ej that are 1-pseudoconcave in the sense of Rothstein and have the following Liouville property: every function from PSHcb (M) is constant on each Ej.

AB - Let M be a complex manifold and let PSHcb (M) be the space of bounded continuous plurisubharmonic functions on M. In this paper we study when the functions from PSHcb (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 w0 ∈ M there is a continuous plurisubharmonic function u with the logarithmic singularity at w0. Moreover, the core of M is the disjoint union of the sets Ej that are 1-pseudoconcave in the sense of Rothstein and have the following Liouville property: every function from PSHcb (M) is constant on each Ej.

KW - Bounded plurisubharmonic functions

KW - Cores of domains

UR - http://www.scopus.com/inward/record.url?scp=85071955052&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071955052&partnerID=8YFLogxK

U2 - 10.1090/proc/14222

DO - 10.1090/proc/14222

M3 - Article

AN - SCOPUS:85071955052

VL - 147

SP - 2413

EP - 2424

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 6

ER -