Abstract

Let \(\Omega\) be a domain in \({\mathbb C^n}\). The cone of functions (respectively negative functions) plurisubharmonic in \(\Omega\) is denoted by \(\text{PSH}(\Omega) (\text{PSH}^{-}(\Omega)).\) A set \(E\subset {\mathbb C^n}\) is called pluripolar if for every \(a\in E\) we can find neighborhood \(U_a\) of \(a\) and \(u\in \text{PSH}(U_a)\) such that \(u(z)= -\infty\) if \(z\in E\cap U_a\) and \( u(z)\not{\equiv} -\infty .\) A set \(E\subset \Omega\) is called complete pluripolar in \(\Omega\) if \(E=u^{-1}(\{-\infty\}), u\in \text{PSH}(\Omega)\) and \(u(z)\not{\equiv} -\infty.\) It is known (theorem of Josefson) that every pluripolar set \(E\) is a subset of some complete pluripolar set in \({\mathbb C^n}.\) The following pluripolar hulls of \(E\) are considered: \[ E_\Omega ^\ast = \cap \{ z\in \Omega: u(z)=-\infty, u\in \text{PSH}(\Omega), u\left| \right. _E =-\infty, u(z)\not{\equiv} -\infty\} , \] \[ E_\Omega ^{-} = \cap \{ z\in \Omega: u(z)=-\infty, u\in \text{PSH}^{-}(\Omega), u\left| \right. _E =-\infty, u(z)\not{\equiv} -\infty\} , \] \[ \text{ Ẽ} = \cap \{ z\in {\mathbb C^n}: u(z)=-\infty, u\in L({\mathbb C^n}), u\left| \right. _E =-\infty, u(z)\not{\equiv} -\infty\} , \] where \(L({\mathbb C^n})\) is the space of the functions plurisubharmonic in \({\mathbb C^n}\) and of logarithmic growth. Among the results of the article we see the following. Let \(D\) be a pseudo-convex domain in \({\mathbb C^n}\) and \(f\) be a holomorphic function in \(D\backslash A, A=\{z: g(z)=0\},\) where \(g\) is holomorphic on \(D\). Denote by \(E\) the graph of \(f\) in \((D\backslash A)\times{\mathbb C}.\) Then \(Z=E\cup (A\times {\mathbb C})\) is complete pluripolar in \(D\times {\mathbb C}.\) The authors prove numerous results concerning pluripolar hulls. The base for proofs is previous results of Bedford and Taylor (1988), Edigarian and Wiegerinck (2003), Levenberg and Poletsky (1999), Wiegerinck (2000), (2002), Zeriahi (1989).