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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 9th 2014
   • (edited Jun 9th 2014)
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   Author: DavidRoberts
   Format: MarkdownItexWith all this discussion of complex analytic things, here's a question which may clear up a number of points for me: over $\mathbb{C}^\ast\times\mathbb{C}^\ast$ there is a holomorphic line bundle with curvature $dx/x \wedge dy/y$. It is trivialised by pulling back along $exp\times id$. Clearly this does not naively go through to the algebraic world. Is there some sort of analogous canonical algebraic line bundle on $\mathbb{G}_m\times\mathbb{G}_m$? If one allows things like log schemes, which I don't understand, then I guess it might work.

   With all this discussion of complex analytic things, here’s a question which may clear up a number of points for me: over $\mathbb{C}^\ast\times\mathbb{C}^\ast$ there is a holomorphic line bundle with curvature $dx/x \wedge dy/y$. It is trivialised by pulling back along $exp\times id$. Clearly this does not naively go through to the algebraic world. Is there some sort of analogous canonical algebraic line bundle on $\mathbb{G}_m\times\mathbb{G}_m$? If one allows things like log schemes, which I don’t understand, then I guess it might work.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 12th 2014
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   Author: DavidRoberts
   Format: MarkdownItexTo record a reference that needs to go in the lab, but I don't know where, yet: Grothendieck, in Le groupe de Brauer II, proposition 1.4 says that for Noetherian schemes satisfying a condition (satisfied, for instance, by smooth varieties), all algebraic $\mathbb{G}_m$-gerbes (in the étale topology) are torsion, and indeed all such $n$-gerbes are torsion. Now I don't know how this works when we take Deligne cohomology instead of just sheaf cohomology, but it seems relevant for Urs' recent interest in the interface between the analytic and algebraic worlds. In particular this shows that the standard gerbes on a nice Lie group (e.g. simple, simply-connected and compact/reductive), hence the String 2-groups, will never lift to the algebraic world (Andre Henriques told me this last year, but I've just now finally tracked down this reference). How much we need to have in some sense a union of the analytic and algebraic world when using these 2-groups, I don't know.

   To record a reference that needs to go in the lab, but I don’t know where, yet:

   Grothendieck, in Le groupe de Brauer II, proposition 1.4 says that for Noetherian schemes satisfying a condition (satisfied, for instance, by smooth varieties), all algebraic $\mathbb{G}_m$-gerbes (in the étale topology) are torsion, and indeed all such $n$-gerbes are torsion. Now I don’t know how this works when we take Deligne cohomology instead of just sheaf cohomology, but it seems relevant for Urs’ recent interest in the interface between the analytic and algebraic worlds.

   In particular this shows that the standard gerbes on a nice Lie group (e.g. simple, simply-connected and compact/reductive), hence the String 2-groups, will never lift to the algebraic world (Andre Henriques told me this last year, but I’ve just now finally tracked down this reference).

   How much we need to have in some sense a union of the analytic and algebraic world when using these 2-groups, I don’t know.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2014
   • (edited Jun 12th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> I don’t know where, At _[[holomorphic line n-bundle]]_ or, if you wish to create it, _[[algebraic line n-bundle]]_. > this shows that the standard gerbes on a nice Lie group (e.g. simple, simply-connected and compact/reductive), hence the String 2-groups, will never lift to the algebraic world One wouldn't want to make the gerbe on the Lie group $G$ itself holomorphic/algebraic. One wants to make the transgression of the 2-gerbe on $\mathbf{B}G$ to the moduli stack of (flat) $G$-connections over a complex/algebraic surface homolomorphic/algebraic, because that is part of the program of quantizing 3d Chern-Simons theory and 2d [[self-dual higher gauge theory]]. The resulting system of holomorphic/algebraic bundles induced by the string gerbe is what Lurie calls "2-equivariant elliptic cohomology". More on this is [here](http://ncatlab.org/nlab/show/equivariant%20elliptic%20cohomology#InterpretationInQuantumFieldTheory).

   I don’t know where,

   At holomorphic line n-bundle or, if you wish to create it, algebraic line n-bundle.

   this shows that the standard gerbes on a nice Lie group (e.g. simple, simply-connected and compact/reductive), hence the String 2-groups, will never lift to the algebraic world

   One wouldn’t want to make the gerbe on the Lie group $G$ itself holomorphic/algebraic. One wants to make the transgression of the 2-gerbe on $\mathbf{B}G$ to the moduli stack of (flat) $G$-connections over a complex/algebraic surface homolomorphic/algebraic, because that is part of the program of quantizing 3d Chern-Simons theory and 2d self-dual higher gauge theory. The resulting system of holomorphic/algebraic bundles induced by the string gerbe is what Lurie calls “2-equivariant elliptic cohomology”. More on this is here.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added some form of the statetement that you are referring to (together with a pointer to the MO discussion) at [holomorphic line n-bundle -- Properties](http://ncatlab.org/nlab/show/holomorphic+line+n-bundle#Properties). But please expand, if possible. I have also cross-linked to this from _[Brauer group -- Relation to etale cohomology](http://ncatlab.org/nlab/show/Brauer+group#RelationToEtaleCohomology)_.

   I have added some form of the statetement that you are referring to (together with a pointer to the MO discussion) at holomorphic line n-bundle – Properties. But please expand, if possible. I have also cross-linked to this from Brauer group – Relation to etale cohomology.

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 12th 2014
   • (edited Jun 12th 2014)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI'm not sure the statement using GAGA is true. For instance, there is a nontorsion holomorphic gerbe on SL_n(C), a lift of the basic gerbe (I will discuss this in Edinburgh, among other things). And G_m is not a coherent sheaf, in any case. Edit: I should clarify: someone wrote on wikipedia something like GAGA holds for G_m, but it can't be true by the above observation, this was one thing that set me off to find the reference.

   I’m not sure the statement using GAGA is true. For instance, there is a nontorsion holomorphic gerbe on SL_n(C), a lift of the basic gerbe (I will discuss this in Edinburgh, among other things). And G_m is not a coherent sheaf, in any case.

   Edit: I should clarify: someone wrote on wikipedia something like GAGA holds for G_m, but it can’t be true by the above observation, this was one thing that set me off to find the reference.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 12th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexall right, go to _[[algebraic line n-bundle]]_ then. I have to rush off now.

   all right, go to algebraic line n-bundle then.

   I have to rush off now.

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 12th 2014
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWill do. As I said, this was just to put it in writing to get me warmed up.

   Will do. As I said, this was just to put it in writing to get me warmed up.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJun 13th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexvaguely related, I have added the following to _[[Brauer group]]_ > The observation that passing to [[derived algebraic geometry]] makes also the non-torsion elements in $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) [[Azumaya algebras]] is due to > * {#Toen10} [[Bertrand Toën]], _Derived Azumaya algebras and generators for twisted derived categories_ ([arXiv:1002.2599](http://arxiv.org/abs/1002.2599))

   vaguely related, I have added the following to Brauer group

   The observation that passing to derived algebraic geometry makes also the non-torsion elements in $H^2_{et}(-,\mathbb{G}_m)$ be represented by (derived) Azumaya algebras is due to

   • {#Toen10} Bertrand Toën, Derived Azumaya algebras and generators for twisted derived categories (arXiv:1002.2599)

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeJun 14th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexBy the way, the story of [[analytically continued Chern-Simons theory]] suggests that in the complex analytic setting one is to find an analog of the String 2-group extending $SL(n,\mathbb{C})$. Or something.

   By the way, the story of analytically continued Chern-Simons theory suggests that in the complex analytic setting one is to find an analog of the String 2-group extending $SL(n,\mathbb{C})$. Or something.

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 14th 2014
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexAs I said, wait till I see you in Scotland ;-)

    As I said, wait till I see you in Scotland ;-)

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexcoming back to #1, just for the record: the question is if the [[Deligne line bundle]] exists in [[logarithmic algebraic geometry]]

    coming back to #1, just for the record: the question is if the Deligne line bundle exists in logarithmic algebraic geometry

12. • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 30th 2014
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    Author: DavidRoberts
    Format: MarkdownItexYes, that's the better question.

    Yes, that’s the better question.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2014
    • (edited Jul 7th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThe discussion in [Saito 14](http://ncatlab.org/nlab/show/algebraic%20K-theory#Saito14), implementing a proposal stated in [Drinfeld 03](http://ncatlab.org/nlab/show/algebraic+K-theory#Drinfeld03), suggests that in the (Nisnevich) algebraic context one should replace $\mathbf{B}\mathbb{G}_m$ by the algebraic K-theory $\mathcal{K} \coloneqq \Omega^\infty K_{alg}(-)$, hence $\mathbf{B}^{n+1} \mathbb{G}_m$ by $\mathbf{B}^n \mathcal{K}$. In particular part of what Saito discusses is, it seems, a lift of the $E_\infty$ map from an algebraic loop group to $\mathbf{B}\mathbb{G}_m$ which classifies the canonical loop group extension to an $E_\infty$-map to $\mathcal{K}$. (This statement is a bit implicit in his theorem 1.13, see the lower part of the proof on p. 29 for a more explicit incarnation). I am not saying that it is clear to me that this might be the way to go about the lack of an algebraic incarnation of the Chern-Simons line 3-bundle, but maybe it's an idea to look into.

    The discussion in Saito 14, implementing a proposal stated in Drinfeld 03, suggests that in the (Nisnevich) algebraic context one should replace $\mathbf{B}\mathbb{G}_m$ by the algebraic K-theory $\mathcal{K} \coloneqq \Omega^\infty K_{alg}(-)$, hence $\mathbf{B}^{n+1} \mathbb{G}_m$ by $\mathbf{B}^n \mathcal{K}$.

    In particular part of what Saito discusses is, it seems, a lift of the $E_\infty$ map from an algebraic loop group to $\mathbf{B}\mathbb{G}_m$ which classifies the canonical loop group extension to an $E_\infty$-map to $\mathcal{K}$. (This statement is a bit implicit in his theorem 1.13, see the lower part of the proof on p. 29 for a more explicit incarnation).

    I am not saying that it is clear to me that this might be the way to go about the lack of an algebraic incarnation of the Chern-Simons line 3-bundle, but maybe it’s an idea to look into.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexIn fact, also from the physics perspective the CS line 3-bundle should be thought of as a $KU$-line 2-bundle (see slide 19 [here](https://dl.dropboxusercontent.com/u/12630719/SchreiberParis2014.pdf)), a statement that when looped twice has the form of what Saito discusses. Hm..

    In fact, also from the physics perspective the CS line 3-bundle should be thought of as a $KU$-line 2-bundle (see slide 19 here), a statement that when looped twice has the form of what Saito discusses. Hm..