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On Grothendieck's counterexample to the Generalized Hodge Conjecture
Portelli, Dario
2014-12-23
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 $N^{1} H^{3}(X,Q )\subsetneq F^{1} H^{3}(X,C )\cap H^{3}(X,Q ).$
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
46 (2014)
Languages
en
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