On Grothendieck's counterexample to the Generalized Hodge Conjecture

Abstract

For a smooth complex projective variety X, let $N^p$ and $F^p$ denote respectively the coniveau filtration on $H^i(X,Q)$ and the Hodge filtration on $H^i(X,C).$ Hodge proved that $N^p H^i(X,Q )\subset F^p H^i(X,C )\cap H^i(X,Q ),$ and conjectured that equality holds. Grothendieck exhibited a threefold X for which the dimensions of $N^{1}H^{3}(X,Q )$ and $F^{1} H^{3}(X,C )\cap H^{3}(X,Q )$ differ by one. 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 $N^{1}H^{3}(X,Q ).$ 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 of $N^{1} H^{3}(X,Q )\subsetneq F^{1} H^{3}(X,C )\cap H^{3}(X,Q ).$