A characterization of 1-convex spaces
This paper in several complex variables deals with the Andreotti-Grauert theory of \(1\)-complex spaces, and with Stein spaces. The following theorems are given. Theorem~1. If \(X\) is a finite dimensional reduced complex space satisfying the following three conditions, then \(X\) is \(1\)-convex. (1) \(X\) is holomorphically spreadable at infinity (i.e., the set of points of \(X\) at which \(X\) is not spreadable is compact in \(X\).) (2) The cohomology group \(H^1(X,{\mathcal O}_X)\) is a finite dimensional complex vector space. (3) There is a holomorphic function \(f\in{\mathcal O}(X)\) with \(f^{-1}(t)\) \(1\)-convex for all \(t\in{\mathbb C}\). Via some lemmas Theorem~1 is deduced from Theorem~2. Theorem 2. A finite-dimensional reduced complex space \(X\) is Stein if \(\dim_{\mathbb C}H^1(X,{\mathcal O}_X)<\infty\), and there is a holomorphic function \(f\in{\mathcal O}(X)\) with \(f^{-1}(t)\) Stein for all \(t\in{\mathbb C}\). A corollary of Theorem 2 is Theorem 3. Theorem~3. Let \(\pi:E\to X\) be an affine space over a complex space \(X\). If \(\dim_{\mathbb C}H^1(E,{\mathcal O}_E)<\infty\), and there is an \(f\in{\mathcal O}(X)\) with all fibers Stein, then \(E\) is Stein. In \S2 the author reduces Theorem~1 to Theorem~2. Here the main tools are the Remmert reduction and Grauert exceptional compact sets. Also it is shown here that the total space of an affine space over a Stein space is itself Stein, the proof of which fact is similar to the usual proof of the same for vector bundles. \S3 proves Theorem~2, which proof depends on an extension result with \(\sup\)-norm bounds for holomorphic functions, which in turn is deduced from a bounded lifting lemma of \textit{J.-E. Fornaess} and \textit{R. Narasimhan} [Math. Ann. 248, 47--72 (1980; Zbl 0411.32011)]. Theorem~1 claims also to fill a gap in [\textit{Nguyen Van Khue} and \textit{Nguyen Dinh Sing}, Rev. Roum. Math. Pures Appl. 31, 237--245 (1986; Zbl 0602.32004)], and Theorem~2 in [\textit{R. Jennane}, Math. Ann. 268, 305--316 (1984; Zbl 0586.32017)]. The paper is carefully written and pleasant to read.
scientific article; zbMATH DE number 2164962
scientific article