# Relation between characteristic cycle and singular support of constructible sheaf

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)?

## 1 Answer

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 $$0$$.