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Let $M$ be a real analytic manifold. Let $F$ be an object of the bounded derived category of sheaves on $M$ with real constructible cohomology sheaves. Let $CC(F)$ denote the characteristic cycle of $F$ and $SS(F)$ be its singular support (I follow the terminology and notation of the book “Sheaves on manifolds” by Kashiwara and Schapira). If I understand correctly one has the inclusion: $$supp(CC(F))\subset SS(F)\,\,\,\,\, (1)$$ by formula (9.4.10) in the above book by Kashiwara-Schapira.

Does one have equality in the inclusion (1)?

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No.

Consider $M = \mathbb R$, $F$ the direct sums of the constant sheaves on the positive real numbers, negative real numbers, and $0$, extended by zero to the whole space.

Then $F$ is the associated graded of a filtration on the constant sheaf, hence has the same characteristic cycle as the constant sheaf, which doesn't contain the cotangent space of $0$.

But the singular support of $F$ clearly contains the singular support of the skyscraper sheaf at $0$, which is the cotangent space of $0$.

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  • $\begingroup$ I apologize for asking this question here. We know that the inclusion is an equality for l-adic perverse sheaves as has been proved by Saito. Is there a reference for a similar result for perverse sheaves over complex analytic manifolds? $\endgroup$
    – random123
    Commented Nov 11, 2023 at 7:21
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    $\begingroup$ @random123 I think this follows from basic results on the D-module side, i.e. that the support of the associated graded of a good filtration on a holonomic $D$-module is pure of dimension the dimension of the variety, that the characteristic cycle is the cycle class of this module, and that the singular support is the support of this module. I didn't find this statement in Kashiwara-Schapira, I'm not sure if it follows quickly from results there. $\endgroup$
    – Will Sawin
    Commented Nov 11, 2023 at 10:57
  • $\begingroup$ Thank you very much. I too had a similar thought in mind. I suppose one will need that Riemann-Hilbert correspondence commutes with characteristic cycle and singular support. That it does as described in an article of Schmid-Vilonen (Ann. Math. (2000)) on page 1115-1116, and [Theorem 11.3.3, Kashiwara-Schapira] respectively. $\endgroup$
    – random123
    Commented Nov 12, 2023 at 6:59

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