On Grothendieck counterexample to the Generalized Hodge Conjecture

For a smooth complex projective variety X, let N^p the subspace of the cohomology space H^i(X, Q) of the classes supported by an algebraic subvariety of codimension at least p. Grothendieck showed that a conjectural description of this space given by Hodge is false, by an explicit example. Recently the point of view of Hodge was somewhat refined (Portelli, 2014), and we aimed to use this refinement to revisit Grothendieck’s example. We explicitly compute the classes in this second space which are not in N^1H^3(X, Q). We also get a complete clarification that the representation of the homology customarily used for complex tori does not allow to apply the methods of (Portelli, 2014) to give a different proof that N^1H^3(X, Q) is different from the space conjectured by Hodge.