Hodge theory on some invariant threefolds of even degree (Q1364614)

The Grothendieck-Hodge conjecture for threefolds suggests that it should be possible to construct a family of curves whose Abel-Jacobi image generates the subtorus of the intermediate Jacobian spanned by the maximal rational sub-Hodge structure inside \(H^3\) which is abelian, that is, having Hodge numbers \(h^{3,0} = h^{0,3} =0\). In this paper, the author considers the universal family of the hypersurfaces of degree \(2p\) and dimension three invariant under a certain action of the group of \(p\)-th roots of unity, and proves the Grothendieck-Hodge conjecture for the general point of some special subfamilies.