Evgeny A. Poletsky, Nikolay Shcherbina
Abstract: Let $M$ be a complex manifold and $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when 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 1-pseudoconcave in the sense of Rothstein sets $E_j$ with the following Liouville property: every function from $PSH^{cb}(M)$ is constant on each of $E_j$.
Journal: Proc. Amer. Math. Soc.
DOI: 10.1090/proc/14222