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nLab > Latest Changes: holomorphic line 2-bundle

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014

    brief entry holomorphic line 2-bundle, just to have the link and to record the reference there

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 30th 2014

    Do you know of any examples, excepting Brylinski’s as defined in http://arxiv.org/abs/math/0002158?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014
    • (edited May 30th 2014)

    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 XX is the Brauer stack Br(X)≔[X,B 2𝔾 m]\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m] (the line 2-bundle itself is the associated ∞-bundle to the B𝔾 m\mathbf{B}\mathbb{G}_m-principal ∞-bundle which is the homotopy fiber of a given map X→B 2𝔾 mX \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 XX (the Brauer group proper if they are torsion).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014

    And of course there is the old

    and the applications to higher twistor transforms.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2014

    More recent discussion of examples connecting explicitly to the Brauer groups includes this here:

    Discussion connecting explicitly to the holomorphic Brauer group includes

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 5th 2014

    There is also the equivariant hermitian holomorphic gerbe constructed in http://arxiv.org/abs/math/0601337. I will add it to the page.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2014
    • (edited Sep 3rd 2014)

    David, coming back to this issue here, here is an ignorant question: is it clear that the torsion of H n(−,𝔾 m)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,𝔾 m)H^n(G,\mathbb{G}_m) might be multiplicative? (For reductive GG.) A torsion CS-bundle on BG\mathbf{B}G would still be interesting…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2014
    • (edited Sep 3rd 2014)

    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 GG

    H 4(BG ℂ,ℤ)≃H 2(BG,𝒦 2) H^4(B G_{\mathbb{C}}, \mathbb{Z}) \simeq H^2(\mathbf{B}G, \mathcal{K}_2)

    for 𝒦 2\mathcal{K}_2 some K-theory sheaf.

    Ah, here is the MO discussion of this fact.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014

    I should move discussion of this to another thread. Notes now at Chern-Simons 3-bundle – For reductive algebraic group

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2014

    One ’problem’ is that in the case that H 2(−,𝔾 m)=ℤH^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 2H^2…’, then that issue is non-existent, but one would hope there are no (or essentially no) choices.

    Regarding the 𝒦 2\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 ΩG\Omega G and this corresponds to the usual central extension. This is ideally the transgression of the things living over GG or BG\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(ΩG,𝔾 m)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…

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014
    • (edited Sep 4th 2014)

    I had earlier missed the fact that the idea of K 2K_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 2K_2-extension BG→B 2K 2\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 BG→B 3𝔾 m\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)→H^ 2(X,ℤ(1))K_2(X) \to \hat H^2(X,\mathbb{Z}(1)).

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2014

    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 BK 2→B 2𝔾 m\mathbf{B}K_2 \to \mathbf{B}^2\mathbb{G}_m over the complex analytic site (or at worst, the base change of BK 2\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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014

    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.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014
    • (edited Sep 4th 2014)

    Of course all we really need to check is that Brylinski’s (or others’s) map on cohomology groups

    H 1(−,K 2)⟶H 2(−,𝔾 m) 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.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2014

    What site are you thinking of in #14?

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014

    Right, the “U(1)U(1)” was a typo, have changed it to 𝔾 m\mathbb{G}_m now.

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2014

    I was serious in #15. Do you mean over the site of polydiscs?

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014

    Oh, I see. Yes, if on that site we could show that c 1,2c_{1,2} is presented by a morphism of presheaves of 2-groupoids, then we’d be done.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014
    • (edited Sep 4th 2014)

    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=XU = X throughout. Then item (2) on p. 6 says, I’d believe, that the map c 1,2c_{1,2} of presheaves of groupoids that we are after works as follows:

    it sends for each XX an element of the group H 1(X,K 2)H^1(X,\mathbf{K}_2) which is represented by pairs consisting of

    1) divisors {D i↪X}\{D_i \hookrightarrow X\} given as zeros of functions g ig_i and 2) meromorphic functions f if_i on XX

    to the groupoid which is the full sub-groupoid of holomorphic line bundles over XX on those which are BD-cup products (f i,g i)(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 1,2c_{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)(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)(D_i, f_i) are in the kernel of the differential δ 0\delta_0 of the Gersten complex…

    (Have to interrupt, my battery dies any second now…)

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2014

    I’m a bit suspicious of the notation in the Gersten complex, namely in the middle term, one has sums of ℂ(x)\mathbb{C}(x) for xx denoting points in codimension one subvarieties (I guess that’s what codimension 1 points are?), but in the left-hand term one has simply ℂ(x)\mathbb{C}(x), which is guess is some function field, but with xx merely an unknown?

    BTW, your f if_is should be meromorphic functions on the D iD_is.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014

    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.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014

    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.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2014
    • (edited Sep 4th 2014)

    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.

    • CommentRowNumber24.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 5th 2014
    • (edited Sep 5th 2014)

    Definition 4.1 seems to define Γ(ℂ(X),n)\Gamma(\mathbb{C}(X),n) but for ℂ(n)\mathbb{C}(n) replaced by an arbitrary field FF. I guess we are taking n=2n=2 in (62), so Γ(ℂ(X),n)\Gamma(\mathbb{C}(X),n) should be ℬ 2→Λ 2ℂ(X) *\mathcal{B}_2\to \Lambda^2\mathbb{C}(X)^\ast. The trick is then to see what ℬ 2\mathcal{B}_2 is.

    Going back a step, we have ℬ 2(ℂ(X))=ℤ[ℙ ℂ(X) 1]/ℛ 1(ℂ(X))\mathcal{B}_2(\mathbb{C}(X)) = \mathbb{Z}[\mathbb{P}^1_{\mathbb{C}(X)}]/\mathcal{R}_1(\mathbb{C}(X)). So what is ℛ 1(ℂ(X))\mathcal{R}_1(\mathbb{C}(X))? If we let δ 2:ℤ[ℙ ℂ(X) 1]→Λ 2ℂ(X) *\delta_2\colon \mathbb{Z}[\mathbb{P}^1_{\mathbb{C}(X)}]\to \Lambda^2\mathbb{C}(X)^\ast be given by {x}↦(1−x)∧x\{x\} \mapsto (1-x)\wedge x for x≠0,1,∞x\neq 0,1,\infty and 0 otherwise, and then 𝒜 1(ℂ(X)(t)):=kerδ 2\mathcal{A}_1(\mathbb{C}(X)(t)) := ker \delta_2. Then ℛ 1(ℂ(X))\mathcal{R}_1(\mathbb{C}(X)) is generated by {0},{∞}\{0\},\{\infty\} and α(0)−α(1)\alpha(0)-\alpha(1) for all α\alpha in 𝒜 1(ℂ(X)(t))\mathcal{A}_1(\mathbb{C}(X)(t)).

    Perhaps one can unwind this to see it is the Gersten complex…

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2014
    • (edited Sep 5th 2014)

    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

    • J. I. Burgos Gil, E. Feliu, Higher arithmetic Chow groups (arXiv:0907.5169)

    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 ℤ,ℤ)≃H 2(BG,K 2)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.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2014
    • (edited Sep 6th 2014)

    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=GL nG = \mathrm{GL}_n and take the “Chern-Simons infinity-bundle” simply to be that modulated by the map

    BGL n⟶Vect⟶K \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 K\mathbf{K}.

    To the extent that there is a second Chern class c 2c_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 Σ\Sigma a surface we still get the transgresssion to something like a “theta-bundle”, now of the form

    [Π(Σ),BGL n]⟶Ω 2K [\Pi(\Sigma), \mathbf{B}\mathrm{GL}_n] \longrightarrow \Omega^2 \mathbf{K}

    and if one wishes then following this with 0-truncation yields

    [Π(Σ),BGL n]⟶Ω 2K⟶K 2 [\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 BGL n→Unknown characterUnknown charactermathbfB 2K 2\mathbf{B}\mathrm{GL}_n \to ßmathbf{B}^2 \mathbf{K}_2 that we were talking about above.

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