On the \(n\)-completeness of coverings of proper families of analytic spaces (Q1598261)

Let \(\pi: X\to T\) be a proper surjection holomorphic map of complex manifolds, \(\dim X= m+ n\), \(\dim T= n\), and \(\sigma:\widetilde X\to X\) a covering map. The reviewer and \textit{M. Coltoni} [Math. Z. 237, 815-831 (2001; Zbl 0981.32009)] proved that if \(\pi\) has maximal rank, then for any \(t_0\in\Gamma\) such that the fiber \((\pi\circ\sigma)^{-1}(t_0)\) has no compact connected component, then there exists an open neighborhood \(U\) of \(t_0\) such that \((\pi\circ\sigma)^{-1}(U)\) is \(n\)-complete. The purpose of the present note is to replace the condition that \(\pi\) has maximal rank by requiring that the fiber \(\pi^{-1}(t_0)\) is reduced.