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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?

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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.

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  • $\begingroup$ I didn't see your answer when I wrote mine, but its seems to be the same. $\endgroup$ – Donu Arapura May 16 at 8:45
  • $\begingroup$ Yes, except your answer has the proper cohomological shifts! I will add those for clarity... $\endgroup$ – Sam Gunningham May 16 at 8:52
  • $\begingroup$ 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? $\endgroup$ – Ioannis Zolas May 16 at 10:04
  • $\begingroup$ @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. $\endgroup$ – Sam Gunningham May 16 at 11:08
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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.

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