Relation between characteristic cycle and singular support of constructible sheaf

Question

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)?

Answer 1

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 $0$.