Cohomological completeness of unramified covering spaces with parameters (Q2719785)

The author proves the following result, which is a relative version for the vanishing theorems in top degrees of complex manifolds: Let \( \pi : X \rightarrow Y \) be a smooth proper surjective holomorphic map of complex manifolds and \( \sigma : \widetilde{X} \rightarrow X \) an unramified cover such that every fiber of \(\pi \circ \sigma\) has no compact component. Then the composition \(\pi \circ \sigma\) is cohomologically \(n\)-complete, where \(n =\dim X -\dim T\). This is done by proving a local statement asserting that for each point \(z\) in \(T\) there exists a neighborhood \(U\) such that \(\widetilde{X}_{U}\) is a cohomologically \(n\)-complete space.