The Hodge structure of the intersection of three quadrics in an odd dimensional space (Q1059119)

Let \(i: V\to {\mathbb{P}}^ n\) with n odd be the smooth base locus of a net of quadric hypersurfaces parametrized by \(\Lambda (\cong {\mathbb{P}}^ 2),\Delta \subset \Lambda\) the curve parametrizing singular quadrics, which we assume to be smooth. The degree of \(\Delta\) is \(n+1\) so we can define W to be the (irreducible) double cover of \(\Lambda\) branched over \(\Delta\) (and let \(\pi\) : \(W\to \Lambda\) be the covering map).- The purpose of the paper is to prove that there is a morphism of polarized Hodge structures \(\phi\) : \(H_{pr}^{n-3}(V,{\mathbb{C}})\to H^ 2_{pr}(W,{\mathbb{C}})\) \((H_{pr}^{n-3}\) and \(H^ 2_{pr}\) are the primitive cohomologies with respect to \(i^*H^{(n-3)/2}\) and \(\pi^*h\), where H and h are the hyperplane classes in \({\mathbb{P}}^ n\) and \(\Lambda (\cong {\mathbb{P}}^ 2)\) respectively) such that: (i) \(\phi\) is an isomorphism between \(H_{pr}^{n-3}(V,{\mathbb{Q}})\) and \(H^ 2_{pr}(W,{\mathbb{Q}})\); (ii) \(\phi\) (H\({}_{pr}^{n-3}(V,{\mathbb{Z}}))\) is a sublattice of index two in \(H^ 2_{pr}(W,{\mathbb{Z}})\).