The Levi problem in the blow-up
The Levi problem asks whether a domain of \({\mathbb C}^n\) which is locally Stein needs to be Stein. It was solved in the affirmative by \textit{K. Oka} [Jap. J. Math. 23, 97--155 (1953; Zbl 0053.24302)] in the 1950's and generalisations of it have been actively investigated ever since. For instance, the answer is also known to be positive in the case of Stein manifolds [\textit{F. Docquier} and \textit{H. Grauert}, Math. Ann. 140, 94--123 (1960; Zbl 0095.28004)]. In another direction, \textit{R. Fujita} proved that every locally Stein subset of \({\mathbb{P}}^n\) is Stein, provided it is not \({\mathbb{P}}^n\) itself [J. Math. Soc. Japan 15, 443--473 (1963; Zbl 0138.06404)]. If \({\mathbb{P}}^n\) is replaced by the line bundle \({\mathbb{P}}^n \times {\mathbb C} = {\mathcal O}(0)\), \textit{J. Brun} showed that a locally Stein subset is Stein if and only if it contains no compact fiber \({\mathbb{P}}^n \times\{x\}\) [Manuscr. Math. 14, 217--222 (1974; Zbl 0299.32024)]. In the paper under review, the same problem is addressed for line bundles over \({\mathbb{P}}^n\). Let \(X= {\mathcal O}(r)\) and \(\Omega\) be a locally Stein subset of it. The authors give necessary and sufficient conditions for \(\Omega\) to be Stein. {\parindent4mm \begin{itemize}\item[--] If \(r<0\), the condition is that \(\Omega\) contains no open subset of the form \(U\setminus A\), where \(A\) is the zero section and \(U\) an open neighborhood of it. (In particular, when \(r=-1\), \(O(-1)\) is the blow-up of \({\mathbb C}^{n+1}\) at a point and \(A\) is the exceptional divisor.) \item[--] If \(r>0\), the condition is that \(\Omega\) contains no neighborhood of the section at infinity. \end{itemize}}
scientific article; zbMATH DE number 5848635
scientific article