On branched coverings of quadrics (Q1924950)

Using the ``\(k\)-ampleness'' (in the sense of Sommese) of the vector bundle associated with a branched covering, \(f : X \to {\mathcal Q}_n\), of the smooth quadric \({\mathcal Q}_n\), we prove that \(f\) induces \(\mathbb{C}\)-cohomology isomorphisms between \(X\) and \({\mathcal Q}_n\) in a certain range of cohomology and homotopy dimensions, provided the degree of the covering is small enough.