Graphs of holomorphic functions with isolated singularities are complete pluripolar
For a domain \(D\subset\mathbb{C}^{n}\), let \(\text{PSH}(D)\) denote the set of plurisubharmonic functions on \(D\). A subset \(E\) of \(D\) is said to be pluripolar if there exists \(u\in \text{PSH}(D)\), \(u\neq -\infty\), such that \(E\subset\{z\in D: u(z)=-\infty\}\), whereas \(E\) is complete pluripolar if there exists \(u\in\text{ PSH}(D)\) such that \(E=\{z\in D: u(z)=-\infty\}\). In the paper the author proves the following Theorem: Suppose that \(D\) is a domain in \(\mathbb{C}\) and that \(A\) is a sequence of points in \(D\) without density point in \(D\). Let \(f\) be holomorphic on \(D\setminus A\), and let \(E\) denote the graph of \(f\) in \((D\setminus A)\times\mathbb{C}\). Then \(E\) is complete pluripolar in \(D\times\mathbb{C}\). The proof of the theorem uses the following result of \textit{A. Zeriahi} [Ann. Pol. Math. 50, No. 1, 81-91 (1989; Zbl 0688.32004)]: Let \(E\) be a pluripolar subset of a pseudoconvex domain \(D\) in \(\mathbb{C}^n\). If the pluripolar hull of \(E\) is equal to \(E\), and \(E\) is both a \(G_{\delta}\) and \(F_{\sigma}\) set, then \(E\) is complete pluripolar in \(D\).
scientific article; zbMATH DE number 1548341