On the \(n\)-concavity of covering spaces with parameters (Q1416678)

Let \(X\) and \(T\) be connected complex manifolds, \( \pi : X \rightarrow T \) be a proper and surjective holomorphic submersion and \(\sigma : \widetilde{X} \rightarrow X\) be a covering map. The author shows that if for some \(t_{0}\) the fiber \(( \pi \circ \sigma)^{-1} (t_{0})\) has at most finitely many compact components, then there exists an open neighbourhood \(U\) of \( t_{0}\) such that the restriction of \(( \pi \circ \sigma)\) is an \(n\)- concave morphism. An interesting presentation of previously obtained results on the subject accompanies the detailed proof of this result.