A Steinness criterion for Stein fibrations (Q2882750)

Since \textit{J.-P. Serre}'s question in 1953 [``Quelques problèmes globaux rélatifs aux variétés de Stein'', Centre Belge Rech. math., Colloque fonctions plusieurs variables, Bruxelles du 11 au 14 mars 1953, 57--68 (1953; Zbl 0053.05302)], the problem of finding sufficient conditions on a Stein fibration over a Stein space to be Stein has aroused much interest. This short article provides a criterion in a quite general setting.NEWLINENEWLINELet \(X\) be a complex space that admits a map to \(\mathbb{C}^k\), for some \(k\), with Stein fibres. If \(H^i(X,\mathcal{O})\) is finite-dimensional for \(1\leq i\leq k\), then \(X\) is Stein. As a corollary, a branched Riemann domain \(X\) over a \(k\)-dimensional Stein space whose cohomology groups satisfy the former condition is Stein.NEWLINENEWLINEThe author finally derives another result: a \(k\)-dimensional complex space satisfying the same cohomological conditions is 1-convex (i.e., a proper modification of a Stein space in a fiber number of points) if and only if it is holomorphically spreadable at infinity (for every point, there exists a map to some \(\mathbb{C}^N\) such that the given point is isolated in its fiber).