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 classes; supports; generalized 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) |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
RIMUT_46_Portelli.pdf | 394.23 kB | Adobe PDF | ![]() View/Open |
CORE Recommender
Page view(s) 50
837
checked on Jan 29, 2023
Download(s) 50
383
checked on Jan 29, 2023
Google ScholarTM
Check
This item is licensed under a Creative Commons License