Timeline for Nearby cycles without a function
Current License: CC BY-SA 4.0
14 events
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| May 8, 2023 at 7:49 | history | edited | Geordie Williamson | CC BY-SA 4.0 |
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| May 8, 2023 at 7:40 | history | edited | Geordie Williamson | CC BY-SA 4.0 |
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| Apr 13, 2021 at 21:47 | comment | added | Geordie Williamson | @naf: fixed thanks! | |
| Apr 13, 2021 at 21:47 | history | edited | Geordie Williamson | CC BY-SA 4.0 |
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| Apr 13, 2021 at 8:18 | comment | added | naf | A minor quibble: the map $f$ has to be proper for the cohomology of the nearby cycles to always agree with that of a nearby fibre. | |
| Apr 13, 2021 at 0:50 | history | edited | Geordie Williamson | CC BY-SA 4.0 |
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| Apr 9, 2021 at 16:11 | history | edited | Sam Hopkins |
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| Apr 9, 2021 at 0:31 | history | edited | Geordie Williamson | CC BY-SA 4.0 |
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| Apr 9, 2021 at 0:28 | comment | added | Geordie Williamson | @DonuArapura: I don't find this so troubling as the nearby cycles only sees the limit mixed Hodge structure on the nearby fibre. (A nice simple example is a generic cubic in $\mathbb{P}^2$ degenerating to the coordinate axes $xyz = 0$.) | |
| Apr 8, 2021 at 23:50 | comment | added | Donu Arapura | If there was truly natural procedure for getting $Gr_W\psi_f(\mathbb{Q})$ from the central fibre, then it would presumably work in the category of mixed Hodge modules. But then it seems almost too much to expect, especially if $Z$ is Hodge-Tate type, and the nearby fibre isn't. I realize, I may be reading more into the question than what you asked, but I thought I'd point this out anyway. | |
| Apr 8, 2021 at 22:47 | comment | added | Geordie Williamson | @WillSawin: good point, in 2) I should have added "for any stratification for which the associated graded of $i^*\mathbb{Q}_X$ is constructible". However such a stratification is certainly "known" if I'm able to compute the weight filtration on $i^*\mathbb{Q}_X$. | |
| Apr 8, 2021 at 22:18 | comment | added | sdr | A trivial comment that addresses the stated question but does not address its emphasis: if you know Z but not X, you can Verdier specialize $\mathbb{Q}_X$ to the normal bundle of $Z$. This sheaf carries all the information of the nearby and vanishing cycles in the presence of the function $f$ (using it to trivialize the normal bundle). | |
| Apr 8, 2021 at 22:06 | comment | added | Will Sawin | For 2 one has to be a bit careful: depending on how you define stratification, a perverse sheaf can be constructible for a stratification without its associated graded being constructible for the stratification. But of course even being sufficiently careful this is remarkable (if true). | |
| Apr 8, 2021 at 21:31 | history | asked | Geordie Williamson | CC BY-SA 4.0 |