Singular support of an irreducible perverse sheaf

Question

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?

Answer 1

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.

Answer 2

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.