Timeline for Intersection homology of orbifolds?
Current License: CC BY-SA 4.0
8 events
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| Dec 8, 2021 at 0:46 | comment | added | D.-C. Cisinski | @JohnPardon Moreover, even in the smooth case, intersection homology of the naive quotient $V/G$ will be interesting a priori, since this quotient is possibly singular. Even with coefficient in a field of characteristic zero, that gives a cute perspective on the cohomology of orbifolds vs intersection homology of their coarse moduli space. | |
| Dec 8, 2021 at 0:42 | comment | added | D.-C. Cisinski | @JohnPardon Well, the principle still holds. We can still work with a possibly singular version of orbifolds (objects which are locally quotients by the action of a finite group of stratified spaces of some sort, whether it would be pseudo-manifolds à la Banagl or conically stratified spaces or what not) and we will still get a Chen-Ruan version of intersection homology. | |
| Dec 7, 2021 at 23:56 | comment | added | John Pardon | @D.-C. Cisinski: The inertia stack of an orbifold is another orbifold, so if intersection cohomology is ordinary cohomology for orbifolds, the same holds for inertia stacks of orbifolds. | |
| Dec 7, 2021 at 21:34 | comment | added | Geordie Williamson | @Cisinski: Ahh, I had never thought of this! It is very possible that this works. | |
| Dec 7, 2021 at 20:58 | comment | added | D.-C. Cisinski | @GeordieWilliamson I do not see why this is an obstruction: Chen-Ruan cohomology is ordinary cohomology of the inertia stack (see Emily Clader's notes here www-personal.umich.edu/~eclader/OctLect1.pdf). Therefore, why not trying to consider ordinary intersection homology of the inertia stack? | |
| Dec 7, 2021 at 20:46 | comment | added | Geordie Williamson | Consider a chart for an orbifold, of the form $V/G$ for some finite group $G$. The applying the Decomposition Theorem (achtung overkill!) for $V \to V/G$ we conclude that the constant sheaf appears as a direct summand, and hence must be the IC sheaf. | |
| Dec 7, 2021 at 20:44 | comment | added | Geordie Williamson | Good question, I also thought about this when I first saw Chen-Ruan cohomology. When coefficients are characteristic 0, IH of orbifolds agrees with ordinary cohomology, so there is (sadly) nothing to see here. | |
| Dec 7, 2021 at 20:01 | history | asked | Shaoyun Bai | CC BY-SA 4.0 |