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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?

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  • $\begingroup$ What is your definition of support of a perverse sheaf? Union of support of cohomology $\mathcal{H}^*({\mathcal{F}})$? Also, are you looking for the singular support for $\mathcal{D}$-modules, which is really a subvariety of the cotangent bundle. $\endgroup$ – AG learner May 15 at 20:08
  • $\begingroup$ Not support, singular support. The singular support is defined on the level of constructible sheaves (see Kashiwara-Schapira, "Sheaves on Manifolds") and it is a subvariety of the cotangent bundle. For perverse sheaves it agrees with the singular support for D-modules under the Riemann-Hilbert corrrespondance. I guess an equivalent question is: Is the singular support for an irreducible $D$-module just the conormal bundle to its support? $\endgroup$ – Ioannis Zolas May 15 at 20:14
  • $\begingroup$ No, consider structure sheaf as a $\mathcal{D}$-module on a smooth variety $X$, it's singular support is zero section of conormal bundle. Also, Riemann-Hilbert correspondece is for those $\mathcal{D}$-modules that are holonomic and has regular singularity. Holonomic means that the dimension of the singular support is $n=\dim X$. $\endgroup$ – AG learner May 15 at 20:20
  • $\begingroup$ Well the zero section of the cotangent bundle has nothing that is conormal to it so it satisfies the pproposition right? Also yes I meant the holonomic $D$-modules with regular singularities, my bad. $\endgroup$ – Ioannis Zolas May 15 at 20:40
  • $\begingroup$ But the conormal bundle to X is indeed the zero section, isn't it? $\endgroup$ – Ioannis Zolas May 15 at 21:08
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For anybody else reading this, know that this was just answered in MathOverflow: https://mathoverflow.net/questions/360492/singular-support-of-an-irreducible-perverse-sheaf

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  • $\begingroup$ If you ask "Is the singular support for an irreducible $\mathcal{D}$-module always contained in the zero section of conormal bundle?". That will make everyone happy. $\endgroup$ – AG learner May 16 at 15:44
  • $\begingroup$ No @AGlearner that is not what I am asking. So Let $X$ be a smooth variety and $Y$ a closed subvariety. The conormal to $Y$ bundle is the subbundle of the cotangent bundle that is supported on $Y$ and is dual to $TY$. Does that make sense? For example, for the affine plane the conormal bundle to the $x$-axis is the subvariety of the cotangent bundle of covectors $(x,0,0,\lambda dy)$. $\endgroup$ – Ioannis Zolas May 16 at 17:05
  • $\begingroup$ In particular, "conormal bundle"$\neq$"cotangent bundle" $\endgroup$ – Ioannis Zolas May 16 at 17:06
  • $\begingroup$ Ah, I see. Somehow I was thinking about "cotangent bundle" when I read and write "conormal bundle". Sorry about the confusion (and I'll delete the irrelavant comments). $\endgroup$ – AG learner May 16 at 17:57

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