# Is the characteristic cycle map for perverse sheaves injective?

Let $$X$$ be a smooth irreducible complex variety. Is the characteristic cycle map from the Grothendieck group of perverse sheaves (with complex coefficients) on $$X$$ to the free abelian group generated by Lagrangian conic subvarieties of $$T^*X$$ injective ? If not, is there some explicit counterexample ?

• Clearly not, as (shifted) local systems of rank $r$ are perverse sheaves whose characteristic cycle is $r$ times the zero section. Nov 5, 2019 at 10:00
• what Piotr is saying is that if you consider perverse sheaves on e.g. C^*, then any two local systems have the same characteristic cycle, but different classes in the Grothendieck group. This is the case in the constructible derived category, so Piotr is correct there. However if you take sheaves with stratification in some triangulation of your manifold, then the CC map is injective. This is somewhere in Kashiwara-Schapira if I recall correctly. Nov 5, 2019 at 10:08
• I'm sorry to quote my own paper, but in 3.5 of "characteristic cycles and decomposition numbers" there is the statement that CC provides an isomorphism between subanalytically constructible sheaves and $\mathbb{R}_+$-constructible subanalytic Lagrangian cycles with integral coefficients. So CC is an isomorphism in this case. Nov 5, 2019 at 10:11
• Thanks a lot for your answer. I deleted my previous comment since I misunderstood something really obvious. Nov 5, 2019 at 10:12