Pluripolar hulls and complete pluripolar sets
For a domain \(D\subset\mathbb C^n\) and a pluripolar set \(E\subset D\), let \(E_D^\ast:=\{z\in D: \forall_{u\in\mathcal{PSH}(D)}:\; u|_E\equiv-\infty \Longrightarrow u(z)=-\infty\}\). The main results of the paper are the following theorems: {\parindent4mm \begin{itemize}\item[{\(\bullet\)}] Let \(D\subset\mathbb C^n\) be a bounded domain, let \(E\subset D\) be a closed pluripolar set, and let \(T\) be a positive current of bidimension \((n-1,n-1)\) on \(D\setminus E\) such that \(T=0\) in \(D'\setminus E\), where \(D'\) is a neighborhood of \(\partial D\), \(dd^cT\) is of order zero, and \(\int_{D\setminus E}\|dd^cT\|<+\infty\). Then, for every complete pluripolar set \(F\subset D\) with \(E\subset F\), we have \(\int_{D\setminus F}dd^cT\geq0\). \item[{\(\bullet\)}] Let \(D\subset\mathbb C^n\) be a domain, let \(E\subset D\) be a closed complete pluripolar set, and let \(F\subset D\setminus E\) be complete pluripolar. Then: (a) \((E\cup F)\cap D'\) is complete pluripolar in \(D'\) for every subdomain \(D'\Subset D\), (b) if \(D\) is pseudoconvex and \(F\) is of type \(\mathcal F_\sigma\) and \(\mathcal G_\delta\), then \(E\cup F\) is complete pluripolar; see also [\textit{A.~Edigarian} and \textit{J.~Wiegerinck}, Indiana Univ. Math. J. 52, 1663--1680 (2003; Zbl 1047.32021)]. \item[{\(\bullet\)}] Let \(D\subset\mathbb C^n\) be a pseudoconvex domain, let \(E\subset D\) be a closed complete pluripolar set, let \(F\subset D\setminus E\) be a complex subvariety, and let \(h\in\mathcal O(D)\). Then \(F\) is complete pluripolar in \(D\) if the following conditions are satisfied: {\parindent10mm \begin{itemize}\item[(a)] \(D\setminus E\) is pseudoconvex and \(F\) has a finite number of irreducible components meeting a neighborhood of \(E\) in \(D\), \item[(b)] \(h\) is non-constant on each irreducible component of \(F\), \item[(c)] \(h(E)\) is a closed polar subset of \(h(D)\), \(h(E)\cap h(F)=\emptyset\), \item[(d)] there exists an exhaustion \(D_j\nearrow D\) by relatively compact hyperconvex open sets such that for any \(z\in E\), \(j\in\mathbb N\), either \(h(z)\notin\partial(h(F\cap D_j))\) or \(h(z)\) is a regular boundary point of \(h(F\cap D_j)\); see also [\textit{A.~Edigarian} and \textit{J.~Wiegerinck}, Ann. Inst. Fourier 54, No.~6, 2085--2104 (2004; Zbl 1083.32009)]. \end{itemize}} \item[{\(\bullet\)}] Let \(D\subset\mathbb C^n\) be a pseudoconvex domain, \(n\geq2\), and let \(f\in\mathcal O(D)\), \(f\not\equiv0\), be such that \(S:=f^{-1}(0)\) is a complex manifold. Then for every pluripolar set \(E\subset D\setminus S\), the set \(E_D^\ast\cap S\) is pluripolar in \(S\). \end{itemize}}
scientific article; zbMATH DE number 5373451