Timeline for Is there an approach to Gabber's theorem from the singular support of coherent sheaves?
Current License: CC BY-SA 3.0
8 events
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| Jan 31, 2016 at 0:38 | history | edited | David Treumann | CC BY-SA 3.0 |
"I heard" from somebody specific
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| Jan 30, 2016 at 23:35 | comment | added | David Ben-Zvi | 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... | |
| Jan 30, 2016 at 23:26 | comment | added | David Ben-Zvi | Great question! By the way the result you quote is from a preprint of Cohn, Nadler, Preygel and myself. | |
| Jan 30, 2016 at 22:44 | history | edited | David Treumann | CC BY-SA 3.0 |
I had "LE = E x BE" which is not quite right.
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| Jan 30, 2016 at 18:50 | comment | added | David Treumann | 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? | |
| Jan 30, 2016 at 18:17 | comment | added | Pavel Safronov | ... 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. | |
| Jan 30, 2016 at 18:17 | comment | added | Pavel Safronov | 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... | |
| Jan 28, 2016 at 22:33 | history | asked | David Treumann | CC BY-SA 3.0 |