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David BZ told me that the old theory of singular support of $D$-modules fits into the new theory of singular support of coherent sheaves, via the derived loop space. I wonder how to reconcile that with Gabber's theorem, which gives a strong constraint on what the singular support of a D-module can be, and with my impression that in not-so-derived situations the singular support of a coherent sheaf can be anything.

Here's an attempt to reconstruct it, probably with mistakes, and then a question:

Let $E$ be a complex vector space, and let $LE$ be its derived loop space. The set of complex points of the once-shifted cotangent bundle of $LE$ can be identified with $E \times E^*$, i.e. the usual cotangent bundle of $E$. A general construction of Benson-Iyengar-Krause produces a closed subset of $E \times E^*$ from a coherent sheaf on $LE$. Arinkin and Gaitsgory call it the singular support of the coherent sheaf.

$LE$ has an action of a circle, loop rotation. Ben-Zvi and Nadler identify the rotation-equivariant coherent sheaves on $LE$ with a category of filtered $D$-modules on $E$. The associated graded module of the filtered $D$-module is a coherent sheaf on $E \times E^*$, whose support is the singular support or characteristic variety of the $D$-module. These two notions of singular support match (I heard).

The singular support of a $D$-module is a coisotropic subset of the symplectic $E \times E^*$. Is that a consequence of any general result about singular support of coherent sheaves?

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    $\begingroup$ How are you using the circle action in the singular support of a coherent sheaf? One has $\mathrm{Coh}(\mathcal{L}E)=\mathrm{Perf}(E\times E^*[2])$ (I am abusing the notation with $E^*[2]$, hope it's clear). The singular support of a coherent sheaf is just its support in $E\times E^*[2]$ which can of course be anything. But the point is that sheaves with a non-coisotropic support do not admit $S^1$-equivariance... $\endgroup$
    – Pavel Safronov
    Commented Jan 30, 2016 at 18:17
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    $\begingroup$ ... E.g. the module $k[y]$ over $k[x,y]$ with $\mathrm{deg}(x)=0,\mathrm{deg}(y)=-1$ has zero singular support. The data of $S^1$-equivariance corresponds to an extension to a dg-module when you set $dx = uy$ ($u$ is a formal degree 2 variable) which clearly doesn't exist. This is Koszul dual to the statement that the origin in $\mathbf{A}^2$ does not quantize. $\endgroup$
    – Pavel Safronov
    Commented Jan 30, 2016 at 18:17
  • $\begingroup$ Hi Pavel. I think I am exactly asking why the equivariance implies Gabber's theorem. It doesn't seem like you get it from just any kind of circle action. Is it a special case of a theorem about coherent sheaves on a more general class of derived stacks? $\endgroup$
    – David Treumann
    Commented Jan 30, 2016 at 18:50
  • $\begingroup$ Great question! By the way the result you quote is from a preprint of Cohn, Nadler, Preygel and myself. $\endgroup$
    – David Ben-Zvi
    Commented Jan 30, 2016 at 23:26
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    $\begingroup$ It would be fascinating to get a new perspective on Gabber this way but perhaps tough. A coherent sheaf on the cotangent bundle can have any support, but only coisotropic ones can quantize. As Pavel says this is precisely Koszul dual to the statement you write, coherent sheaves on shifted tangent bundles can have any support but only coisotropics [a notion that only makes sense AFAIK in this particular situation, not for arbitrary quasismooth schemes] can admit $S^1$ equivariance. The two statements are so closely tied I'd be surprised if the dual perspective makes Gabber any easier... $\endgroup$
    – David Ben-Zvi
    Commented Jan 30, 2016 at 23:35

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