SU Mathematics Research: Plurisubharmonically separable complex manifolds

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