Plurisubharmonically separable complex manifolds

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 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 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.