This is an amplification of the previous post to the case where $C$ is a curve in an arbitrary linear system $\Delta$ on an arbitrary toric surface $X_\Sigma$. In other words, its purpose is to explain why the result of that computation was “obvious.” Write $\Lambda_1$ for the subset of $\Lambda_\Sigma$ corresponding to the 1-dimensional cones of $\Sigma$, so $\Lambda_1$ is a Legendrian link and $\Lambda_\Sigma$ is the union of $\Lambda_1$ and the cocircle at 0. The convex lattice polygon (= linear system) $\Delta$ is determined by the “lengths” of its edges, where we say the length of an edge is the number of primitive vectors of which it is composed (i.e. one more than the number of interior lattice points on that edge). The lengths assign a nonnegative integer to each component of $\Lambda_1$, and we say an object of $\mathrm{Sh}(\Lambda_1)$ has microlocal rank $\Delta$ if it has the corresponding microlocal rank on each component.
Let’s pin down exactly how $F = CCC(\mathcal{O}(\Delta))$ fails to live in $\mathrm{Sh}(\Lambda_1)$. Say we want to compute the microlocal stalk at some point $p$ in the codirection $\theta$ (let’s choose an identification $M_\mathbb{R} \cong \mathbb{C}$ for convenience). Let $U_{-\epsilon} \subset U_{+\epsilon}$ be small open sets such that the microlocal stalk is $\mathrm{Cone}\,(F(U_{+\epsilon}) \to F(U_{-\epsilon}))$. (Say, $U$ is a neighborhood of $p$, and $U_{+\epsilon} = U \cap f^{-1}(-\infty,\epsilon)$, $U_{-\epsilon} = U \cap f^{-1}(-\infty,-\epsilon)$, where $f$ is a linear function on $U$ with $f(p) = 0$ and $\mathrm{arg}\,(df) = \theta$.) Since $F$ is the proper pushforward along $\pi: M_\mathbb{R} \to M_\mathbb{R}/M$ of the standard sheaf $\mathbb{C}_\Delta$ on $\Delta$, we will have $$\mathrm{Cone}\,(F(U_{+\epsilon}) \to F(U_{-\epsilon})) = \bigoplus_{p’ \in \pi^{-1}(p)}\mathrm{Cone}\,(\mathbb{C}_\Delta(U’_{+\epsilon}) \to \mathbb{C}_\Delta(U’_{-\epsilon})),$$
where $U’_{-\epsilon}\subset U’_{+\epsilon}$ are the preimages of $U_{-\epsilon} \subset U_{+\epsilon}$ containing $p’$.
The key point is that the only way for one of the righthand summands to be nonzero is if $\langle q-p’, e^{i \theta} \rangle \geq 0$ for all $q \in \Delta$, in which case that summand is rank 1 (i.e. you hold a straight edge at angle $\theta$ in the plane, slide it up until it touches $\Delta$, then look at all the lattice points it’s touching). Hence the rank of the microlocal stalk of $F$ at $(p,\theta)$ is the number of $p’ \in \pi^{-1}(p)$ satisfying this property. If $p \neq 0$ but $p$ lives on the projection of some component of $\Lambda_1$, this is 0 unless $\theta$ is the inward-pointing conormal to the corresponding edge of $\Delta$, in which case it is the length of that edge. Thus where $p \neq 0$ $F$ itself is already microlocal rank $\Delta$. On the other hand, at $p=0$ the microlocal stalk in any codirection is 1 more than it should be to have microlocal rank $\Delta$; in particular the SS of $F$ includes the entire cocircle at 0.
Now $CCC(\mathcal{O}_C)$ is going to be the cone over a map from $F$ to the skyscraper $\mathbb{C}_0$. So up to a shift $CCC(\mathcal{O}_C)$ is the subsheaf of $F$ whose stalk at 0 is the kernel of some linear form on $F_0 = \bigoplus_{p’ \in \pi^{-1}(0)} (\mathbb{C}_\Delta)_{p’}$. That subsheaf lives in $\mathrm{Sh}(\Lambda_1)$ (and automatically has microlocal rank $\Delta$) when it no longer has SS at 0 in any codirections that are not inward-pointing conormals to edges of $\Delta$. This is the same as saying that $(\mathbb{C}_\Delta)_{p’} \subset F_0$ is not in the kernel of the relevant linear form when $p’$ is a corner of $\Delta$. But this is the same as saying that the divisor $C$ does not contain the corresponding $T$-fixed point.
Nice work. For a general toric surface $X$, a generic covector at the identity of $M_{\mathbf{R}}/M$ determines a maximal cone in the fan of $X$, which in turn determines a $T$-fixed point of $X$. I think the microlocal stalk matches either the fiber or the cofiber at the fixed point under the CCC. (“Cofiber” means, Ext from the skyscraper sheaf, instead of Tor.) So you should be able to conclude the same for other coherent sheaves supported on $C$ besides $\mathcal{O}_C$.
I’m interested to see what the composition from Pic(C) to coherent sheaves on X to constructible sheaves on M_R is like. If $L$ is a line bundle on $C$, is there a simple resolution of $i_* L$, not too much worse than O(-Delta) \to O?
Ahh ok that’s a handy fact about the (co)fibers indeed — I assume this is an exercise I do by checking it on Theta sheaves then arguing from Cech resolutions?
I think at this point my guess for moving forward on the rest of Pic(C) is to say: well, if $L$ is high enough degree then I can present it as a cone over a map $\mathcal{O}_C^{\oplus n} \to \mathcal{O}_C^{\oplus n+1}$, and now we know what those trivial bundles look like across $CCC$. Of course, the cone over a random such map won’t be a line bundle, but maybe this at least gets us some kind of grip on $L$. Like if we can push the condition “is injective on stalks” across $CCC$, we’d be in business. But then maybe that’s not much easier than trying to directly write down $CCC$ of the condition “the stalks on $C$ are rank 1…
Another idea, which I’m not that confident about, would be to use the Abel-Jacobi map, since we know what skyscraper sheaves on $X$ look like across $CCC$. But this only gets you a few line bundles, and you have to take tensor products of these to get all of at least some component of Pic. And my impression that actually computing convolution products on the constructible side is difficult in practice. (TBH, I still don’t really understand how to compute why $\mathcal{O}(1)$ and $\mathcal{O}(-1)$ are inverses on the constructible side).
A related question: do you know offhand what twisting by $\mathcal{O}(1)$ does on the constructible side?
Algebraic geometers say there is no simple resolution of $i_*L.$
the map on moduli is weird. like, on the coherent side, moduli are algebraic completely integrable systems. on the constructible side, moduli are knotty character varieties. we’re doing mirror symmetry, but it looks like nonabelian hodge theory. what’s that about?
also: it will be really interesting to do this for the total space of a line bundle over an orbifold line. this is because the corresponding completely integrable systems are the ones which Alexei and Zhiwei were studying and out of which made the rational DAHA reps, and on the other side, the knot is going to be a torus knot on a torus. so, this might give some sneaky way of identifying those moduli spaces bypassing nonabelian hodge theory (or secretly doing it in some weird way).
Ok. I need to jump in here. This is Jason, from Postmatic.
Just wanted to say that I love that I have absolutely NO IDEA what it is
that you are all talking about over here. It cracks me up and therefore
I think you are all awesome.
Glad to see you found a pretty good solution in our plugin. Enjoy using
it and keep the geek flowing. I’ll probably butt out now. Let me know if
you need anything else.
PS – We looked into some sort of way we could help your math display
better in your email templates but came up short. Sorry 🙁
Jason
Hey Jason, Thanks for trying! Do you know whether private posts get announced correctly over email? We had some concerns that this was not happening, though we would like it to happen.
Hey Harold,
At this time posts which are marked as private in the WordPress-native
sense of the word are not sent out over email via Postmatic.
The notion of private in WordPress is kind of tricky (and often times
not that useful). What is it that you are trying to do? There is usually
a simple way to do things like lock down the full contents of a site
(while still sending posts by email and getting comments back). Is that
the sort of thing you are into?
If it is more appropriate you could reply to me via
support@gopostmatic.com.
Best, Jason
Comment 1) Our progress on the toric case suggests the following lesson: whenever you meet an algebraic integrable system, you should interrogate it by thinking of it as a space of sheaves on the ambient variety where its spectral curves live, and then asking it what its mirror description is. For the toric integrable systems, this question provides some sort of answer to an analogue of one of our starting questions, which was the precise relation between Hitchin spectral curves and microlocalization. The fact that we happened to already know about the CCC let us say in the toric/relativistic/5d case “oh, the relation is that the spectral curves and the exact Lagrangians are mirror.” Is there a sense in which this is also the relationship in the Hitchin case? To begin to ponder that I would need to know how mirror symmetry works for cotangent bundles of Riemann surfaces, but I will have to wait for you guys to educate me on that…
Comment 2) The above discussion does something a little worrying in that it neglects any discussion of the Poisson structure. Why should it be that the constructible sheaves whose associated coherent sheaves have the same support form a holomorphic Lagrangian with respect to some form related to the intersection pairing on the exact Lagrangians? It bothers me that I have no vision for an extension of the above argument past trivial line bundles that would also give insight into this; I’d guess that the “right” proof would make it obvious. It may even be presumptuous to think that the map from constructible sheaves to the support of their coherent associates is necessarily an integrable system: recall that these moduli spaces have a crazy smooth automorphism, which is an avatar of nonabelian Hodge in the world of monopoles. It at least merits consideration that the “Hamiltonians” furnished by applying CCC and taking support are off from the true Hamiltonians by this automorphism (this can probably be ruled out by CCC being suitably holomorphic, but you take my point: in the Hitchin case there’s no way around thinking about the hyperkahler structure)
Comment 3) My attempt to extract anything useful from Alexei and Zhiwei’s paper was not successful. Part of your statement (which Eric partially passed on) is that the cotangent bundle of a weighted projective line is a toric surface? Something is fishy here, as the toric systems and Hitchin systems are really disjoint classes of systems, for example the former are not what KS call “integrable systems with central charge” though the latter are. That is, the analogue of the Liouville form for the toric systems is multivalued, something like (log(x)/y)dy.
Re 3) Wait, I said that only cotangents of weighted projective lines with up to two stacky points would be toric. Any more than two and you’re out of luck.