Timeline for Weight filtration over the integers
Current License: CC BY-SA 2.5
20 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 22, 2010 at 6:04 | history | edited | algori | CC BY-SA 2.5 |
typos
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| Jul 9, 2010 at 6:25 | history | edited | algori | CC BY-SA 2.5 |
grammar
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| Jun 25, 2010 at 18:12 | comment | added | algori | YBL -- that's exactly what is a bit confusing: in that paragraph Voisin considers the integral cohomology proper (i.e. not modulo torsion) of the complement of a normal crossing divisor, but it is not stated what the ambient variety is. As to the counterexamples to the Hodge conjecture (for K\"ahler manifolds or over the integers): they may be useful, but I don't see how: they all involve compact manifolds, for which the weight filtration is trivial. | |
| Jun 25, 2010 at 17:19 | comment | added | AFK | Voisin only considers integral cohomology modulo torsion. In her definition a mixed Hodge structure the underlying abelian group is supposed to be free (at least in the French version). I recall Atiyah and Hirzebruch gave a counter example to the Hodge conjecture with integral coefficients. Can't the same reasoning apply to this problem? | |
| Jun 25, 2010 at 1:56 | comment | added | JS Milne | algori. Thanks! you have helped clarify things for me. | |
| Jun 25, 2010 at 1:00 | history | edited | algori | CC BY-SA 2.5 |
typos!!
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| Jun 25, 2010 at 0:54 | comment | added | algori | JS -- so am I. The way I understand it, in that paragraph both $U$ and its compactification are assumed algebraic. There may be another preferred equivalence class of compactifications (other than algebraic) which Voisin may be referring to, but I have no idea what this could be (K\"ahler is excluded by the same example). | |
| Jun 25, 2010 at 0:35 | history | edited | algori | CC BY-SA 2.5 |
typos!
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| Jun 25, 2010 at 0:33 | comment | added | JS Milne | Actually, I'm still confused by Voisin, since in that section she seems to be talking about complex manifolds, not algebraic varieties. Perhaps the assumption is hidden somewhere. | |
| Jun 25, 2010 at 0:32 | comment | added | algori | Donu -- will do. But I'm still hoping that someone comes and posts it here. | |
| Jun 25, 2010 at 0:06 | history | edited | algori | CC BY-SA 2.5 |
specified where the weights come from in the question
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| Jun 24, 2010 at 23:59 | comment | added | algori | ... and for any two algebraic compactifications there is a third one that dominates them both. In the analytic category this is not true, as the example in the posting shows. | |
| Jun 24, 2010 at 23:53 | history | edited | algori | CC BY-SA 2.5 |
removed a confusing question
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| Jun 24, 2010 at 23:52 | comment | added | algori | JS -- thanks! Yes, the last question is confusing and I'll remove it. For algebraic varieties the weight filtration is defined on the integral cohomology and it is an invariant (in the algebraic but not in the analytic category) since there is a preferred class of compactifications: the algebraic ones. | |
| Jun 24, 2010 at 23:35 | comment | added | JS Milne | In 8.36 (p.214 of the English version) of her book, Voisin defines a mixed Hodge structure to have a weight filtration on the integral cohomology. She then says that Deligne's theorem shows the existence a mixed Hodge structure on the integral cohomology of the complement $U$ of a normal divisor crossing, and adds that "One can show that this mixed Hodge structure depends only on $U$ and not on its compactification." This seems to contradict what you are saying. Did Voisin mean to define mixed Hodge structure as Deligne does, with a weight filtration defined on the rational cohomology? | |
| Jun 24, 2010 at 22:12 | comment | added | Donu Arapura | I suspect you may be right that the integral weight filtration is not strict, but I don't have a feeling for what a counterexample would look like, beyond the fact that $x$ would have to be torsion. If you do find an example, let us know. | |
| Jun 24, 2010 at 20:50 | history | edited | algori | CC BY-SA 2.5 |
typo in title
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| Jun 24, 2010 at 20:44 | history | edited | algori | CC BY-SA 2.5 |
typo
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| Jun 24, 2010 at 20:33 | history | asked | algori | CC BY-SA 2.5 |