Timeline for intuition about perverse sheaves
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| S Sep 7, 2020 at 16:00 | history | bounty ended | CommunityBot | ||
| S Sep 7, 2020 at 16:00 | history | notice removed | CommunityBot | ||
| Aug 31, 2020 at 14:57 | comment | added | Aurelio | If you are looking for a more topological intuition, I would recommend MacPherson's lecture notes, which were mentioned in this other question. | |
| Aug 31, 2020 at 14:49 | comment | added | Donu Arapura | One difficulty with this question is that there is no way to give a short answer. So I will second Marc Besson's suggestion. Here is a link to the article arxiv.org/abs/0712.0349 | |
| Aug 31, 2020 at 3:18 | comment | added | Marc Besson | My current favorite reference is "The Decomposition Theorem and the Topology of Algebraic Maps". A very nice example if you know a little toric geometry is the section on the application of the decomposition theorem to toric varieties. In toric geometry, everything in sight consists of T-orbits and closures of T-orbits. For a very down to earth example of cases in which intersection homology works more nicely than regular homology or cohomology is that intersection is a purely combinatorial invariant of the polytope from which you construct the variety, unlike regular cohomology. | |
| Aug 30, 2020 at 20:16 | comment | added | Tim Campion | @Anonyme I've edited to make sure that you have at least one top-level tag on your question. This is considered good practice and is also a good way to maximize the number of relevant users who see your question. Since you already had 5 tags (the maximum number), I had to delete one (in this case "t-structures", which is only watched by 3 people). Feel free to re-adjust the tags as you see fit, but I do recommend having at least one top-level tag. | |
| Aug 30, 2020 at 20:13 | history | edited | Tim Campion |
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| S Aug 30, 2020 at 14:58 | history | bounty started | Amos Kaminski | ||
| S Aug 30, 2020 at 14:58 | history | notice added | Amos Kaminski | Draw attention | |
| Aug 29, 2020 at 9:10 | comment | added | Amos Kaminski | When I said in the smooth cases I am meaning smooth sheaves on essentially smooth and equidimensional. scheme ... | |
| Aug 29, 2020 at 8:37 | comment | added | Sam Gunningham | "In smooth cases perverse t-structures is approximately the same as ordinary t-structures" - This depends on what you mean by "smooth cases". For example, the perverse t-structure for (algebraically) constructible complexes on $\mathbb A^1 = \mathbb C$ is interesting - not just a shift of the usual t-structure - even though all the strata and links are themselves smooth. E.g. If $j:\mathbb C^\times \hookrightarrow \mathbb C$ is the open inclusion, $Rj_\ast \mathbb{Q}_{\mathbb{C}^\times}[1]$ is perverse, but it has ordinary cohomology in degrees -1 and 0. | |
| Aug 28, 2020 at 13:43 | comment | added | Donu Arapura | A couple of points. 1) When $X$ is smooth, a locally constant sheaf is perverse up to translation. But the converse isn't true (expect in trivial cases). 2) If you understand a bit about $D$-modules, then on the $D$-module side, under Riemann-Hilbert, perverse sheaves look very natural. | |
| Aug 28, 2020 at 13:29 | history | asked | Amos Kaminski | CC BY-SA 4.0 |