A generalization of the \(q\)-convex case of a theorem of Fornæss and Narasimhan
An upper semicontinuous function \(\varphi: D\rightarrow[-\infty,+\infty)\), where \(D\) is an open subset of \(\mathbb C^n\), is called subpluriharmonic if for every relatively compact subset \(G\subset\subset D\) and for every function \(u\) pluriharmonic in a neighborhood of \(\overline G\), if \(\varphi\leq u\) on \(\partial G\), then \(\varphi\leq u\) on \(\overline G\). An upper semicontinuous function \(\varphi: D\rightarrow[-\infty,+\infty)\) is called \(q\)-plurisubharmonic if for every \(q\)-dimensional affine subspace \(V\subset\mathbb C^n\) the function \(\varphi|_{D\cap V}\) is subpluriharmonic. An upper semicontinuous function \(\varphi: X\rightarrow[-\infty,+\infty)\), where \(X\) is a complex space, is called \(q\)-plurisubharmonic if for every point \(x\in X\) there exist a local embedding \(i:U\rightarrow\widetilde U\subset\mathbb C^n\), \(x\in U\), and a \(q\)-plurisubharmonic function \(\widetilde\varphi\) on \(\widetilde U\) such that \(\widetilde\varphi\circ i=\varphi\). An upper semicontinuous function \(\varphi: X\rightarrow[-\infty,+\infty)\) is called weakly \(q\)-plurisubharmonic if for every holomorphic mapping \(f: G\rightarrow X\), where \(G\) is open in \(\mathbb C^q\), the function \(\varphi\circ f\) is subpluriharmonic. The author proves that every continuous weakly \(q\)-plurisubharmonic function on a reduced complex space is \(q\)-plurisubharmonic. In the case \(q=1\) the result reduces to the ``continuous'' case of the Fornæss-Narasimhan theorem (saying that for any complex space, every weakly plurisubharmonic function is plurisubharmonic).
scientific article; zbMATH DE number 1943232