Timeline for Is the category of mixed Hodge modules bi-filtered?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 18, 2016 at 9:53 | comment | added | a_g | Anycase, the refundation of Sabbah and Schnell are very useful, so thank you so much for the reference. | |
| Apr 18, 2016 at 9:53 | comment | added | a_g | I totally agree: absolutely not since the graded pieces of $F$ are no longer $D$-modules. Indeed, if my question was true, then, using that shifting we'll get that the categories of pure Hodge modules over a fixed variety are all isomorphic, no matter the weight. But this is absolutely false as can be shown, for example, using the category of pure Hodge structures of weight 0 and 1, that have a different kind of symmetry in their decompositions. However, using Tate's twisting, we get all the pure Hodge modules with even weight are isomorphic, and odd-weighted too. Is it OK? | |
| Apr 4, 2016 at 14:38 | comment | added | Donu Arapura | Oh yea good point. So no, in either case. | |
| Apr 4, 2016 at 14:29 | comment | added | Ben Webster♦ | But the associated graded for $F$ isn't a D-module, so it's obviously not a mixed Hodge module, right? Am I going crazy here? | |
| Apr 4, 2016 at 14:07 | comment | added | Donu Arapura | Maybe yes, maybe no. No using Saito's original definition, because $F$ is almost never defined over $\mathbb{Q}$. On the other hand, Sabbah and Schnell have been reworking the foundations, so that it is no longer necessary to have a $\mathbb{Q}$ "lattice" in their version. See cmls.polytechnique.fr/perso/sabbah.claude/MHMProject/mhm.html | |
| Apr 4, 2016 at 12:47 | comment | added | Geordie Williamson | It is probably helpful to think about a variation of pure Hodge structure. The associated graded for the Hodge filtration not a pure variation of Hodge structure. (Think about a non-trivial variation of Hodge structure of type (1,1) given by the $H^1$ of a family elliptic curves. One wants to think about this object as being "simple", so it shouldn't break up any further.) | |
| Apr 4, 2016 at 10:25 | history | asked | a_g | CC BY-SA 3.0 |