Stein neighborhoods of graphs of holomorphic mappings (Q2865905)

Let \(N,\,M\) be complex manifolds and \(K\subset N\) a compact set. One can define several notions of holomorphic maps from \(K\) to \(M\) as follows. Let \(A(K,M)\) be the set of maps \(f:K\to M\) which are continuous on \(K\) and holomorphic on the interior of \(K\), and \(H(K,M)\) be the subclass of those \(f\) which are uniform limits on \(K\) of maps holomorphic on neighborhoods of \(K\) into \(M\). The author introduces the intermediate class \(H_{\mathrm{loc}}(K,M)\) of continuous maps \(f:K\to M\) with the property that each point \(z\in K\) has a neighborhood \(U\) such that the restriction of \(f\) to \(K\cap\overline U\) belongs to \(H(K\cap\overline U,M)\).NEWLINENEWLINE\textit{F. Forstnerič} asked in [Asian J. Math. 11, No. 1, 113--126 (2007; Zbl 1131.58007)] the following question. Let \(K\subset N\) be a compact set which has a Stein neighborhood basis and let \(f\in H(K,M)\). Does the graph of \(f\) over \(K\) have a Stein neighborhood basis? In the paper under review it is shown (Theorem 3.1) that the answer to this question is in the affirmative, even in the more general case when \(f\in H_{\mathrm{loc}}(K,M)\). This is an interesting result and it has a number of interesting consequences (Theorems 4.1, 4.3, 4.5, 4.7). The method of proof relies on a fusion technique for plurisubharmonic functions (Theorem 2.1).