On defining functions and cores for unbounded domains I

Pluricomplex Green Functions on Manifolds

“…The set c(M ) of all points w ∈ M , where every function of P SH cb (M ) fails to be smooth and strictly plurisubharmonic near w, is called the core of M . This notion was introduced by Harz-Shcherbina-Tomassini in [18].…”

A Kobayashi and Bergman complete domain without bounded representations

“…Then n converges uniformly on compact subsets of ℂ 2 ⧵E to a pluriharmonic function ∶ ℂ 2 ⧵E → ℝ , and lim (z,w)→(z 0 ,w 0 ) (z, w) = −∞ for every (z 0 , w 0 ) ∈ E (see [19,Lemma 5.1]). In particular, applying [11,Chapter I,4.15] to the decreasing to sequence of plurisubharmonic functions ̃k ∶= max{ , −k}, k = 1, 2, … , we see that has a unique extension to a plurisubharmonic function on the whole of ℂ 2 and the set E = { = −∞} is complete pluripolar.…”

Plurisubharmonically separable complex manifolds

“…The main obstruction to separation is the set c(M ) of all points w ∈ M , where every function of P SH cb (M ) fails to be smooth and strictly plurisubharmonic near w. It was first introduced and systematically studied by Harz-Shcherbina-Tomassini in [2]- [4] and was called the core of M . Observe that directly from the definition one concludes that c(M ) is a closed subset of M .…”

“…Note that in this paper we are dealing mainly with the core defined using continuous plurisubharmonic on M functions (the core c 0 (M ) in terminology of [4]), while the main object of the study in [2]- [4] was the core c(M ) defined using smooth plurisubharmonic on M functions. Note also, that another proof of Theorem II for cores defined by smooth plurisubharmonic functions was obtained by Slodkowski [11] using essentially different methods.…”