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For any constructible sheaf (or D-module) $\mathcal{F}$ over a smooth variety $X$ over $\mathbb{C}$, there is a notion of singular support $SS(\mathcal{F})$ that lives in the cotangent bundle $T^{*}X$ of $X$.

Now, suppose that $X$ is singular (quasi-projective for simplicity), can we define in the same way some singular support that will have the same functoriality property as in the smooth case. Of course, we can embed $X$ into a smooth $X'$, but can we make it independent of the choice and would $SS(\mathcal{F})$ will still live in $T^{*}X$?

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  • $\begingroup$ Won't any two embeddings of the same dimension be locally equivalent in the analytic or etale topology? Independence should follow from that. $\endgroup$
    – Will Sawin
    Commented Mar 13, 2020 at 16:05
  • $\begingroup$ the drawback of such a definition is for instance if one would have functoriality w.r.t to proper morphisms, because there is no reason that it extends to the embedding. $\endgroup$
    – prochet
    Commented Mar 13, 2020 at 23:57
  • $\begingroup$ For $f: X \to Y$ proper, embed $X$ into $A$, embed $Y$ into $B$, and then embed $X$ into $A \times B$ diagonally via $f$. This works fine if $X$ and $Y$ are quasiprojective, I think. I agree it's beneficial to have a general theory though. $\endgroup$
    – Will Sawin
    Commented Mar 14, 2020 at 0:06
  • $\begingroup$ @WillSawin, do you know if the containment $SS(F) \subset T^*X$ holds in the singular case? $\endgroup$
    – user125639
    Commented Apr 16, 2021 at 20:12
  • $\begingroup$ @user125639 If we embed $X \subset A$ then $T^* X$ is not a subspace of $T^* A $, but rather a quotient, so the inclusion $SS(F) \subset T^* X$ would not be well-defined. $\endgroup$
    – Will Sawin
    Commented Apr 16, 2021 at 20:49

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