Generalized Hodge Conjecture

The rational generalized Hodge conjecture is a support assertion for rational Hodge substructures in the cohomology of smooth projective complex varieties. For such a variety , the geometric coniveau subspace consists of sums of classes vanishing outside closed algebraic subsets of codimension at least . The conjecture asserts that this subspace equals the largest rational sub-Hodge structure whose types satisfy . For this includes the rational Hodge conjecture.