The Hodge conjecture for a certain class of singular varieties (Q1358192)

Let \(X\) be a reduced separated algebraic scheme over the complex numbers. One can consider a formulation of the general Hodge conjecture (GHC) for \(X\), which extends the formulation of the GHC for smooth projective varieties (Grothendieck's amendment). In this paper, we exhibit examples of \(X\), specifically a class of hypersurfaces, where the GHC holds. The main technical ingredient is the specialization of the diagonal class \(\Delta\) on the ``generic fiber'' of a family \(\overline{\mathcal X}\to T\) to a closed fiber \(\overline X\), and to apply known results about the decomposability of \(\Delta\) for our special class of varieties.