Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/10640
Title: On Grothendieck's counterexample to the Generalized Hodge Conjecture
Authors: Portelli, Dario
Keywords: Cohomology classessupportsgeneralized Hodge conjecture
Issue Date: 23-Dec-2014
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
46 (2014)
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 $N^{1} H^{3}(X,Q )\subsetneq F^{1} H^{3}(X,C )\cap H^{3}(X,Q ).$
Description: 
Dario Portelli, "On Grothendieck's counterexample to the Generalized Hodge Conjecture", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 46 (2014), pp.237-254
Type: Article
URI: http://hdl.handle.net/10077/10640
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.46 (2014)

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