Timeline for Nearby cycles for stacks
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Mar 7 at 18:01 | history | bounty ended | user492133 | ||
| S Mar 7 at 18:01 | history | notice removed | user492133 | ||
| Mar 7 at 18:01 | vote | accept | user492133 | ||
| Mar 7 at 16:18 | answer | added | Will Sawin | timeline score: 2 | |
| Mar 7 at 14:48 | comment | added | Alexey Do | To add Will Sawin, there are several choices of a theory of nearby cycles, based on the kind of covering, at least in étale cohomology, you get the unipotent one and the tame one and they coincide if the scheme is of semi-stable form, i.e. $S[x_1,...,x_n]/(x_1...x_k - \pi) \to S$ with $\pi$ a uniformizer. In general, the tame contains the unipotent as a direct summand. | |
| Mar 7 at 14:43 | comment | added | Will Sawin | The monodromy you get depends on what coverings you use: If you use Kummer morphisms you will only see tame monodromy, which is more general than unipotent, but in the mixed or equal characteristic $p$ case is less than everything. But if we're comparing with analysis we're in characteristic zero so that's irrelevant. | |
| Mar 7 at 14:41 | comment | added | Will Sawin | @Z.M It seems that the OP asked to compare two different kinds of nearby cycles but there exist so many different kinds and each commenter is familiar with a different subset of them that we might never figure out what each other are talking about, let alone answer the question. I'm not familiar with etale motives, but the nearby cycles functor defined for $\ell$-adic etale sheaves certainly sees more than just unipotent monodromy. | |
| Mar 7 at 14:32 | comment | added | Z. M | @WillSawin I am not working within this field thus maybe I misunderstood something, but here it is: if you construct the nearby cycle functor via Kummer coverings on étale motives (maybe denoted by $\operatorname{DM}^{\operatorname{ét}}$ in the literature, and recall that $\ell$-adically it is the same as $\ell$-adic coefficients), and then you specialize to the Betti world, you only see the (pro-)unipotent part. I might confuse something, since I remember that the base in question was a DVR, so maybe this only happens in the mixed characteristic case? | |
| Mar 7 at 14:15 | comment | added | Will Sawin | @Z.M In what sense is the algebraic one looking only at the unipotent part? Pulling back along the $n$th Kummer covering will detect local monodromy with eigenvalues of order n. | |
| Mar 7 at 13:23 | answer | added | Alexey Do | timeline score: 2 | |
| Mar 6 at 20:22 | comment | added | user492133 | Yes, the analytic one I am aware of is defined in terms of the exponential covering map (please correct me if I am wrong) for $\mathbb{C}^*$. And for the algebraic one, which I know almost nothing about, it is defined in SGA by Grothendieck and there are some notes by Illusie here: imo.universite-paris-saclay.fr/~luc.illusie/vanishing1b.pdf | |
| Mar 6 at 15:45 | comment | added | Z. M | @NikolaTomić The algebraic one that I am vaguely aware of is only looking at the unipotent part. Roughly speaking, for every positive integer $n$, you pull the map $f$ back along Kummer coverings $(-)^n\colon\mathbb A^1\to\mathbb A^1$, obtaining $g_n\colon Y\to\mathbb A^1$, and you look at the $C_n$-equivariant étale sheaves on the recollement determined by the closed subscheme $g_n^{-1}(0)\subseteq Y$. This is the usual way to approximate the exponential map in the algebraic setting. | |
| Mar 6 at 12:06 | comment | added | Nikola Tomić | @Z. M The analytic definition has been defined by Beilinson Berstein and Deligne I think and you can see it here : arxiv.org/abs/1211.3259. §2.3. I don't know the algebraic version though, what is it ? | |
| Mar 6 at 11:57 | comment | added | Z. M | What is the analytic nearby cycle functor? The algebraic nearby cycle is defined on the category of (algebraic) étale sheaves, but what about the analytic one? | |
| S Mar 5 at 23:01 | history | bounty started | user492133 | ||
| S Mar 5 at 23:01 | history | notice added | user492133 | Draw attention | |
| Mar 2 at 21:45 | history | asked | user492133 | CC BY-SA 4.0 |