# Singular support of an irreducible perverse sheaf

I was studying Sheaves on Manifolds by Kashiwara and Schapira, and while the singular support seems like a complicated invariant I cannot seem to find a counterexample to the following:

Let $$X$$ be a smooth complex variety and $$\mathcal{F}=IC(U,\mathcal{L})$$ be an irreducible perverse sheaf, where $$\mathcal{L}$$ is a local system on $$U\subset X$$. Then $$SS(\mathcal{F})=T_{\overline{U}}^*X$$, where the latter means the conormal bundle at $$\overline{U}$$.

This seems too easy of an answer to be true, but I still cannot find either a counterexample or a proof, and I cannot think of how to get an explicit answer using the Riemann-Hilbert correspondance either. Any help?

PS. I have asked the question already in Stack Exchange but it wasn't answered and I thought it may be more appropriate here after all?

Take $$X=\mathbb C$$, $$U=\mathbb C^\times$$, and $$\mathcal L$$ a nontrivial rank 1 local system (with monodromy $$\mu \neq 1$$, say).

Then the singular support of $$IC(U,\mathcal L)$$ is the union $$T^\ast _X X \cup T^\ast_0 X$$ of the zero section (which is what your conjecture would give in this case) with the cotangent fiber.

Note that here we have $$j_\ast \mathcal L[1] \cong IC(U,\mathcal L) \cong j_! \mathcal L[1]$$. One can compute the singular support either using the sheaf definition from Kashiwwara-Schapira, or by considering the associated $$D$$-module $$D_{\mathbb C}/D_{\mathbb C}(x\partial_x - \log(\mu))$$ under the Riemann-Hilbert correspondence.

• I didn't see your answer when I wrote mine, but its seems to be the same. May 16 '20 at 8:45
• Yes, except your answer has the proper cohomological shifts! I will add those for clarity... May 16 '20 at 8:52
• Thank you both very much! I am sorry I could only accept one solution. Another small question, it seems though as if every irreducible component is of that form. Maybe this is true in more generality? May 16 '20 at 10:04
• @IoannisZolas It is known that the singular support of a constructible complex is contained in a union of conormal bundles (to the strata). I believe this containment is not necessarily an equality in general, though I don't have a good example off the top of my head. May 16 '20 at 11:08

My impression is that the singular support of a perverse sheaf is the characteristic variety of the regular holonomic $$D$$-module corresponding to it under Riemann-Hilbert. Assuming that's the case, it is possible to answer this in the negative. Let $$X$$ be the disk or the affine line if you prefer, and $$U=X-\{0\}$$. Choose a nontrivial rank one local system $$\mathcal{L}$$ on $$U$$. By Riemann-Hilbert, there is regular connection $$\nabla$$ on $$\mathcal{O}_X$$ such that $$\mathcal{L}=\ker\nabla$$ over $$U$$. Then $$IC(U,L)= j_*L[1]$$ (or a translate depending on your convention), and the corresponding $$D$$-module $$M$$ is the minimal extension of $$\nabla$$. The characteristic variety of $$M|_U$$ is the zero section of $$T_U^*$$. Therefore the characteristc variety of $$M$$ is either the zero section of $$T^*_X$$ or the zero section union $$T_0^*$$. If it's the former, then $$M$$ would have to be a connection which contradicts our choice, so it must be the latter.