Plurisubharmonically separable complex manifolds

TL;DR

This paper characterizes complex manifolds where bounded continuous plurisubharmonic functions separate points, linking this property to the emptiness of the core and the existence of specific functions with logarithmic singularities.

Contribution

It establishes equivalences between point separation by plurisubharmonic functions and geometric properties of the manifold, including the structure of the core and pseudoconcave sets.

Findings

Point separation by PSH functions is equivalent to an empty core.

Existence of PSH functions with logarithmic singularities at any point.

Core decomposes into pseudoconcave sets where PSH functions are constant.

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 $E_j$.