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It is well-known that the ordinary (singular) cohomology of orbifolds doesn't behave well for many purposes. Better approaches include the Chen-Ruan cohomology, which has an associative product structure closely tied with orbifold Gromov-Witten theory; another common approach is the Borel construction, replacing quotients by homotopy quotients.

Any orbifold has a natural stratification induced from the isomorphism type of the stabilizers. This stratification should allow one to talk about intersection homology on orbifolds. It seems that this is an area that is not fully explored. Is there any study in this direction? What kind of expectations should one have? I'm happy to restrict to algebraic orbifolds.

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    $\begingroup$ 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. $\endgroup$
    – Geordie Williamson
    Commented Dec 7, 2021 at 20:44
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    $\begingroup$ 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. $\endgroup$
    – Geordie Williamson
    Commented Dec 7, 2021 at 20:46
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    $\begingroup$ @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? $\endgroup$
    – D.-C. Cisinski
    Commented Dec 7, 2021 at 20:58
  • $\begingroup$ @Cisinski: Ahh, I had never thought of this! It is very possible that this works. $\endgroup$
    – Geordie Williamson
    Commented Dec 7, 2021 at 21:34
  • $\begingroup$ @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. $\endgroup$
    – John Pardon
    Commented Dec 7, 2021 at 23:56

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