Complements of domains with respect to hulls of outside compact sets
The author proves the following theorem. Let \(M\) be a Stein manifold, \(n : = \dim M \geq 2\), and let \(K\), \(X\), \(\Omega \subset M\) be such that: \(K\) is compact, \(X\) is an open Stein neighbourhood of \(K\), \(\Omega\) is an open set such that \(\Omega \cap K \neq \emptyset\). Let \({\mathcal S}\) be a coherent analytic sheaf on \(M\) such that \(H^{n-1} (\Omega, {\mathcal S}) = 0\). Then \[ H^{n-1} (\Omega \backslash \widehat K_ X) = 0,\;H^{n-1} (\Omega \backslash \widetilde K_ X) = 0,\;H^{n-1} (\Omega \backslash {\mathcal H} (K)) = 0, \] where \(\widehat K_ X : = \{x \in X : | f(x) | \leq \max_ K | f |\), \(f \in {\mathcal O} (X)\}\) is the holomorphic hull of \(K\), \(\widehat K_ X : = \{x \in X : f(x) \in f(K)\), \(f \in {\mathcal O} (X)\}\) is the meromorphic hull of \(K\), \({\mathcal H} (K) : =\) (the intersection of all open Stein neighbourhoods of \(K)\) is the holomorphically convex hull of \(K\). In particular, if \(n=2\) and \(\Omega\) is an open Stein set, then the open sets \(\Omega \backslash K_ X\), \(\Omega \backslash \widetilde K_ X\), \(\Omega \backslash {\mathcal H} (K)\) are Stein. Moreover, the author shows that if \(n \geq 3\), then the groups \(H^{n-2} (\Omega \backslash \widehat K_ X)\), \(H^{n-2} (\Omega \backslash \widehat K_ X)\). \(H^{n-2} (\Omega \backslash {\mathcal H} (K))\) need not vanish.
scientific article; zbMATH DE number 561289
scientific article