For a complex manifold \(M\) let \(\mathcal{PSH}^{cb}(M)\) denote the space of all bounded continuous plurisubharmonic functions on \(M\). Let \(\boldsymbol{c}(M)\) be the core of \(M\), i.e., the set of all points \(w\in M\) such that every function from \(\mathcal{PSH}^{cb}(M)\) fails to be smooth and strictly plurisubharmonic near \(w\). We say that a set \(E\subset M\) is \(1\)-pseudoconcave in the sense of Rothstein if, for any \(z_0\in E\), any strictly plurisubharmonic function \(\varrho\) defined in a neighborhood of \(V\) of \(z_0\), and any neighborhood \(U\subset\subset V\) of \(z_0\), there exists a \(z\in E\cap U\) such that \(\varrho(z)>\varrho(z_0)\).
The following two theorems are the main results of the paper:

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The following statements are equivalent:

(1)

\(\mathcal{PSH}^{cb}(M)\) separates the points of \(M\);

(2)

for every \(w_0\in M\) there exists a \(u\in\mathcal{PSH}^{cb}(M)\) that is smooth and strictly plurisubharmonic near \(w_0\);

(3)

for every \(w_0\in M\) there exist a negative continuous plurisubharmonic function \(v\) on \(M\) and constants \(C_1, C_2>0\) such that \(\log|z-w_0|+C_1<v(z)<\log|z-w_0|+C_2\) near \(w_0\).

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The set \(\boldsymbol{c}(M)\) is the disjoint union of sets \(E_j\), \(j\in J\), that are \(1\)-pseudoconcave in the sense of Rothstein and every function from \(\mathcal{PSH}^{cb}(M)\) is constant on each \(E_j\).