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brief entry holomorphic line 2-bundle, just to have the link and to record the reference there
Do you know of any examples, excepting Brylinski’s as defined in http://arxiv.org/abs/math/0002158?
Thanks for the pointer, have added that to the entry, too.
And in reply to your question, I have added the following to the entry:
This means that the moduli stack of holomorphic line 2-bundles on a complex analytic space or more generally on a complex analytic ∞-groupoid $X$ is the Brauer stack $\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]$ (the line 2-bundle itself is the associated ∞-bundle to the $\mathbf{B}\mathbb{G}_m$-principal ∞-bundle which is the homotopy fiber of a given map $X \to \mathbf{B}^2 \mathbb{G}_m$). In particular equivalence classes of holomorphic line 2-bundles form the elements of the bigger Brauer group of $X$ (the Brauer group proper if they are torsion).
And of course there is the old
and the applications to higher twistor transforms.
More recent discussion of examples connecting explicitly to the Brauer groups includes this here:
Discussion connecting explicitly to the holomorphic Brauer group includes
Oren Ben-Bassat, Gerbes and the Holomorphic Brauer Group of Complex Tori, Journal of Noncommutative Geometry, Volume 6, Issue 3 (2012) 407-455 (arXiv:0811.2746)
Edoardo Ballico, Oren Ben-Bassat, Meromorphic Line Bundles and Holomorphic Gerbes, Math. Res. Lett. 18 (2011), 6, 1-14 (arXiv:1101.2216)
There is also the equivariant hermitian holomorphic gerbe constructed in http://arxiv.org/abs/math/0601337. I will add it to the page.
David, coming back to this issue here, here is an ignorant question: is it clear that the torsion of $H^n(-,\mathbb{G}_m)$ is actually a problem? (Probably it is, but I am just trying to check if we really thought it through to the end.) Without an exponential sequence in sight, does it necessarily imply that no element in there may be a suitable algebraic WZW 2-bundle?
And even if torsion: might we have an idea of which elements in $H^n(G,\mathbb{G}_m)$ might be multiplicative? (For reductive $G$.) A torsion CS-bundle on $\mathbf{B}G$ would still be interesting…
By the way, in here
is a theorem 4.11 of the form we’d be after as the next best thing, identifying for reductive algebraic $G$
$H^4(B G_{\mathbb{C}}, \mathbb{Z}) \simeq H^2(\mathbf{B}G, \mathcal{K}_2)$for $\mathcal{K}_2$ some K-theory sheaf.
Ah, here is the MO discussion of this fact.
I should move discussion of this to another thread. Notes now at Chern-Simons 3-bundle – For reductive algebraic group
One ’problem’ is that in the case that $H^2(-,\mathbb{G}_m)=\mathbb{Z}$, we more or less have a canonical generator, but in the torsion case I have no idea what could possibly happen. If we are happy to say ’take a generator of $H^2$…’, then that issue is non-existent, but one would hope there are no (or essentially no) choices.
Regarding the $\mathcal{K}_2$ extension, this is what Andre Henriques mentioned at my question on the central extension of the algebraic loop group. Note that there still is a group in ind-schemes that is the central extension of $\Omega G$ and this corresponds to the usual central extension. This is ideally the transgression of the things living over $G$ or $\mathbf{B}G$ that we are after. But if we need to go to ind-algebraic groups (associated as: (ind-algebraic) groups) to get a (presumably non-torsion) class in $H^1(\Omega G,\mathbb{G}_m)$, then do we need something like an ind-algebraic gerbe? I mention in the comments at my question the idea that log geometry may be what we need, but this is still pure speculation. Are ind-algebraic (2-)gerbes classified by a slightly different cohomology? On a different site? I can imagine the gerbe being given by a groupoid in ind-schemes, which surely don’t behave quite the same as schemes…
I had earlier missed the fact that the idea of $K_2$-extension was not just an idea but is already well-documented in the literature (as noted now here). I’d think one should just run from there.
The big question next seems to be: given the $K_2$-extension $\mathbf{B}G \to \mathbf{B}^2 K_2$ over the algebraic site, how, after base change, does it map to the actual string extension $\mathbf{B}G \to \mathbf{B}^3 \mathbb{G}_m$ over the complex-analytic site.
I am guessing it must be by use of an untruncated version of the algebraic Chern-character/regulator $K_2(X) \to \hat H^2(X,\mathbb{Z}(1))$.
Brylinski talks about Holomorphic gerbes and the Beilinson regulator, but whether this is the same map as induced by base change, I don’t know.
Thinking out loud…
From what I can glean from that Brylinski paper, there’s a map $\mathbf{B}K_2 \to \mathbf{B}^2\mathbb{G}_m$ over the complex analytic site (or at worst, the base change of $\mathbf{B}K_2$ from the algebraic world over the field of meromorphic functions to the complex analytic world) - the question is whether this deloops. I guess it must, the question is whether the composition with the delooping gives the string extension. Hmm…
Note there is some 2-gerbe stuff later in the paper.
Thanks for highlighting this again.
Indeed Brylinski (and most other aurthors) discuss the regulator as a map on cohomology classes, whereas I suppose we’d need it on the level of stacks, as you indicate in your notation.
Of course the construction of Bunke-Tamme is supposed to produce just that: an incarnation of the Beilinson regulator as a map on the sheaf of algebraic K-theory spectra.
But I still need to mull over this to see if we get the statement needed fro the story of the algebraic string extension.
Of course all we really need to check is that Brylinski’s (or others’s) map on cohomology groups
$H^1(-,K_2) \longrightarrow H^2(-,\mathbb{G}_m)$lifts to a map of pre-sheaves of 2-groupoids.
This is maybe easily seen from Brylinski’s explicit formulas. But I need to go and get some breakfast first.
What site are you thinking of in #14?
Right, the “$U(1)$” was a typo, have changed it to $\mathbb{G}_m$ now.
I was serious in #15. Do you mean over the site of polydiscs?
Oh, I see. Yes, if on that site we could show that $c_{1,2}$ is presented by a morphism of presheaves of 2-groupoids, then we’d be done.
Right, so let’s talk about spelling this out. I find the construction and argument in Brylinski 94 is a fair bit scattered over the article, so let’s try to extract the pieces that we actually need:
Since we will let stackification do all the tedious work for us, we take $U = X$ throughout. Then item (2) on p. 6 says, I’d believe, that the map $c_{1,2}$ of presheaves of groupoids that we are after works as follows:
it sends for each $X$ an element of the group $H^1(X,\mathbf{K}_2)$ which is represented by pairs consisting of
1) divisors $\{D_i \hookrightarrow X\}$ given as zeros of functions $g_i$ and 2) meromorphic functions $f_i$ on $X$
to the groupoid which is the full sub-groupoid of holomorphic line bundles over $X$ on those which are BD-cup products $(f_i,g_i)$.
Moreover, we will let differential abstract nonsense do all the work of differential refinement for us much later in the story, and so for the time being we ignore all connection data that Brylinski94 discusses. That seems to allow me to skip way ahead through the article to the proof of theorem 3.3. This I read as saying that the map of presheafs of groupoids above is already the one that represents $c_{1,2}$.
If true, then all that would remain would be to check that our map of presheaves of groupoids is indeed a map of presheaves of groupal groupoids, hence of 2-groups. So sums of classes represented by $(D_i, f_i)$ should go to tensor products of line bundles. Hm. Looking at p. 19 it seems for that to happen I need to use that the elements $(D_i, f_i)$ are in the kernel of the differential $\delta_0$ of the Gersten complex…
(Have to interrupt, my battery dies any second now…)
I’m a bit suspicious of the notation in the Gersten complex, namely in the middle term, one has sums of $\mathbb{C}(x)$ for $x$ denoting points in codimension one subvarieties (I guess that’s what codimension 1 points are?), but in the left-hand term one has simply $\mathbb{C}(x)$, which is guess is some function field, but with $x$ merely an unknown?
BTW, your $f_i$s should be meromorphic functions on the $D_i$s.
Yeah, also what I said about the g-s was wrong. Then I ended up being distracted for the rest of the day.
I suppose what the statement we are after really out to be is that there is a chain map from a truncated version of the Gersten complex to the Deligne complex, or at least to a resolution of Gm[n]. Such that applying Dold-Kan and then oo-stackification would give the refined regulator.
I can see how it almost works, but I am not sure yet if it really works. But this sounds like somebody after Brylinski must have already thought about.
Hopefully more tomorrow when I am more online again.
For instance glancing at
with all its chain maps labeled “regulator” might just be it. But I haven’t read it yet, need to run now.
So equation (62) in Goncharov 04 gives a regulator chain map to the Deligne complex. And the domain seems to be meant to be the Gersten complex, judging from the line over equation (25). But this article seems to forget to introduce much of its notation.
Definition 4.1 seems to define $\Gamma(\mathbb{C}(X),n)$ but for $\mathbb{C}(n)$ replaced by an arbitrary field $F$. I guess we are taking $n=2$ in (62), so $\Gamma(\mathbb{C}(X),n)$ should be $\mathcal{B}_2\to \Lambda^2\mathbb{C}(X)^\ast$. The trick is then to see what $\mathcal{B}_2$ is.
Going back a step, we have $\mathcal{B}_2(\mathbb{C}(X)) = \mathbb{Z}[\mathbb{P}^1_{\mathbb{C}(X)}]/\mathcal{R}_1(\mathbb{C}(X))$. So what is $\mathcal{R}_1(\mathbb{C}(X))$? If we let $\delta_2\colon \mathbb{Z}[\mathbb{P}^1_{\mathbb{C}(X)}]\to \Lambda^2\mathbb{C}(X)^\ast$ be given by $\{x\} \mapsto (1-x)\wedge x$ for $x\neq 0,1,\infty$ and 0 otherwise, and then $\mathcal{A}_1(\mathbb{C}(X)(t)) := ker \delta_2$. Then $\mathcal{R}_1(\mathbb{C}(X))$ is generated by $\{0\},\{\infty\}$ and $\alpha(0)-\alpha(1)$ for all $\alpha$ in $\mathcal{A}_1(\mathbb{C}(X)(t))$.
Perhaps one can unwind this to see it is the Gersten complex…
Thanks, David. I’ll think about it more now. Since the article says it is part of the “Handbook of algebraic K-theory”, maybe its notation is meant to be defined elsewhere in that book. I am however currently not at the institute nor on a good wifi connection, so I haven’t checked.
Another place where I see chain-level Beilinson regulators discussed, and maybe with more details, is
Equation (3.2) there is the sort of chain map that we are after and theorem 3.5 says that on chain cohomology and after rationalization it gives the Beilinson regulator.
Now the chain domain in (3.2) is a complex of Chow groups. But that should be all right for us. If I remember well then the proof that $H^4(BG_{\mathbb{Z}}, \mathbb{Z}) \simeq H^2(\mathbf{B}G, \mathbf{K}_2)$ goes via first showing that both sides are given by degree-2 chow classes. But I need to go back and check.
I am getting a bit, say, impatient with the literature here. Am wondering if the story should not be much more natural, like this
consider $G = \mathrm{GL}_n$ and take the “Chern-Simons infinity-bundle” simply to be that modulated by the map
$\mathbf{B} GL_n \longrightarrow \mathbf{Vect} \longrightarrow \mathbf{K}$injecting the moduli stack of rank-n-bundles into that of all vector bundles and sending that to its infinity-group completion, the sheaf of algebraic K-theory spectra $\mathbf{K}$.
To the extent that there is a second Chern class $c_2$ on algebraic K, then postcomposition would be the expected string 3-bundle. But let’s maybe not actually postcompose with any Chern class
Then for $\Sigma$ a surface we still get the transgresssion to something like a “theta-bundle”, now of the form
$[\Pi(\Sigma), \mathbf{B}\mathrm{GL}_n] \longrightarrow \Omega^2 \mathbf{K}$and if one wishes then following this with 0-truncation yields
$[\Pi(\Sigma), \mathbf{B}\mathrm{GL}_n] \longrightarrow \Omega^2 \mathbf{K} \longrightarrow \mathbf{K}_2$which ought to be just the same as if we had started with the $\mathbf{B}\mathrm{GL}_n \to ßmathbf{B}^2 \mathbf{K}_2$ that we were talking about above.
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