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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 21st 2014

    wrote out the definition In complex geometry

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2014

    have expanded at intermediate Jacobian: maded clearer how the complex structure of the the Griffith-version is induced and added discussion of the Weil-version.

    This will need polishing. But right now I am on the train with shaky connection and will leave it the way it is for the moment.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2014
    • (edited Jun 4th 2014)

    added “Deligne’s theorem”, the characterization of the intermediate Jacobian as the “Hodge-trivial” shifted/truncated Deligne cohomology group, here.

    For the time being I am really just extracting some highlights from

    • Hélène Esnault, Eckart Viehweg, section 7 of Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)

    but staring at this with a more higher topos theoretic point of view seems to induce some resonances. Not sure yet, though.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2014
    • (edited Jun 4th 2014)

    That second map in the short exact sequence (7.9)

    0J k+1(Σ)H 2k+2(Σ,(k+1) D)Hg k+1(Σ)0 0 \to J^{k+1}(\Sigma) \to H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_D) \to Hg^{k+1}(\Sigma) \to 0

    of the above article is curious:

    The Deligne cohomology H 2k+2(Σ,(k+1) D)H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_D) itself is the 0-truncation of the homotopy pullback of the diagram

    Maps(Σ,(Ω k+1)[2k2]) Maps(Σ,(k+1)[2k2]) Maps(Σ,[2k2]) \array{ && Maps(\Sigma, (\Omega^{k+1} \stackrel{\partial}{\to}\cdots)[-2k-2]) \\ && \downarrow \\ Maps(\Sigma,\mathbb{Z}(k+1)[-2k-2]) &\longrightarrow& Maps(\Sigma, \mathbb{C}[-2k-2]) }

    On the other hand, the ordinary pullback of the 0-truncation of this diagram is the Hodge cohomology classes Hg k+1(Σ)Hg^{k+1}(\Sigma).

    So that second map in the above exact sequence is just that induced on pullbacks by 0-truncation of the underlying cospan diagram of mapping stacks/mapping spectra (unless I got my degrees mixed up).

    Hm….

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2014
    • (edited Jun 4th 2014)

    to state this more pronouncedly:

    let E^\hat E be any differential cohomology theory with its canoncial decomposition as a homotopy fiber product:

    E^ΠE^×Π dRE^ dRE^. \hat E \simeq \Pi \hat E \underset{\Pi \flat_{dR} \hat E}{\times} \flat_{dR} \hat E \,.

    Then set

    Jker(τ 0(ΠE^×Π dRE^ dRE^)τ 0(ΠE^)×τ 0(Π dRE^)τ 0( dRE^)) J \coloneqq ker\left( \tau_0\left(\Pi \hat E \underset{\Pi \flat_{dR} \hat E}{\times} \flat_{dR} \hat E\right) \longrightarrow \tau_0(\Pi \hat E) \underset{\tau_0(\Pi \flat_{dR} \hat E)}{\times} \tau_0(\flat_{dR} \hat E) \right)

    where evaluation on some XX is left implicit (this is a measure for the failure of the 0-truncation τ 0\tau_0 to preserve the homotopy fiber product).

    Then for E^\hat E the complex analytic (p)\mathbb{Z}(p) Deligne-cohomology in degree 2p, this reduces to Deligne’s characterization of the intermediate Jacobian.

    It seems.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2014

    gave the entry some actual Idea section, inspired in parts by the above observation.

    (Have to call it quits for today.)

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 5th 2014

    Have people come across this Jacobian construction for other E^\hat E?

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2014
    • (edited Jun 5th 2014)

    Starting in (Witten 99) and then more prominently in 2000 Witten discussed the quantization of the self-dual “RR-fields” in 10d carried out analogously to the quantization of the self-dual “B-field” in 6d as in (Witten 96). This produces a lattice of K-theory classes equipped with a symplectic form etc. in direct analogy to the intermediate Jacobians, but with ordinary cohomology replaced by K-theory.

    I don’t know yet if that is reproduced by feeding KU^\hat {KU} into the above. But that would be something to think about.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 5th 2014

    @Urs Your Witten 96 link points back to this page.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2014

    Thanks, fixed.

    (Witten 96)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2014

    Regarding the above links: I have collected them now more visibly and with a bit more of commentary at self-dual higher gauge fields – Examples – RR-field in 10d.

    I suppose this should work out, that Witten’s “K-theoretic intermediate Jacobian” comes out as in #5, but I need to look into some details.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2014
    • (edited Jun 11th 2014)

    re #7, #8:

    I see now that there is one article by mathematicians on intermediate Jacobians for complex K-theory (Müller-Stach & Peters & Srinivas 2011)

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2014

    the discussion of the intermediate Jacobian stacks for generalized cohomology I have now started typing into a pdf instead (as in the entry it became a bit tedious):

    • Intermediate Jacobian stacks (pdf)
    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 13th 2014

    Do these E^\hat{E}-Hodge filtrations crop up frequently?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2014

    Yes. In all the examples where dRE^\flat_{dR}\hat E is given by actual differential forms (which are pretty much all the examples understood well) then it’s just induced by the degree filtration on differential forms.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 13th 2014

    So we hear about them in arithmetic geometry too

    In arithmetic geometry one uses the difference between a splitting of the Hodge filtration and of the underlying rational structure of the Betti-cohomology to define the extension of mixed Hodge structures.

    Hmm, plenty of the terms from the (b) of that Ayoub quotation I reported from his ICM talk:

    The (co)homological invariants associated to an algebraic variety fall into two classes:

    (a) the algebro-geometric invariants such as higher Chow groups (measuring the complexity of algebraic cycles inside the variety) and Quillen K-theory groups (measuring the complexity of vector bundles over the variety);

    (b) the class of transcendental invariants such as Betti cohomology (with its mixed Hodge structure) and l-adic cohomology (with its Galois representation).

    The distinction between these two classes is extreme.

    I was wondering there if this distinction lines up with the products of fracturing.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2014

    Yes, these Hodge structures are everywhere and via this translation from Kähler geometry to complex-analytic sheaf cohomology one sees why: they are very much just the reflection of the grading on (holomorphic/algebraic) differential forms.

    Regarding your wondering whether there is a fracture square here: I see you point, but, honestly, I have no idea if this distinction may be formalized in terms of a fracture. I am afraid I don’t even have a good understanding of the distinction itself yet. Maybe somebody else here might have more to say about this, I’d be interested.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2014

    addded a pointer to section 1 of Griffith 12. That has an excellent survey of the Griffiths and the Weil complex structures and their relation.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2014

    later this week I will give a talk at “Higher Structures along the Lower Rhine” which might mention some intermediate Jacobian stacks, if time permits.

    A first version of my talk script is here: pdf

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 18th 2014

    After

    This modulates G-principal connections

    The quotient \sim should outside the braces { }

    After

    transgressed to the circle it yields the WZW gerbe

    [S 1,L WZW][S^1,L_{WZW}] should be [S 1,L CS][S^1,L_{CS}]

    section 5

    generak

    and

    Deligen

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2014

    Thank you!!

    Have fixed all this now, thanks again. Today I hope to produce a second “version with details” and I’ll add an acknowledgement for typo-spotting there. (The present version has no space left for acknowledgements.)

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2014
    • (edited Jun 18th 2014)

    Okay, I have polished the note a bit more. This is my talk script for tomorrow, at Higher Geometric Structures along the Lower Rhine – June 2014 :

    • Differential cohomology is Cohesive homotopy theory (pdf, 8 pages)

    The last section is the one that briefly states the axiomatic characterization of higher (i.e. intermediate) Jacobians.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2014
    • (edited Jul 14th 2014)

    added a pointer to Scheinost-Schottenloher 96 right after the pointers to Griffiths’s articles. Because it turns out Scheinost-Schottenloher discuss what is really the non-abelian version of the Griffiths structure on J 1(X)J^1(X) (from their page 154 (11 of 76) on).

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2014

    in addition to the pointer in the references I also added comments to this effect in the main text in the section Examples – k = 0

    • CommentRowNumber25.
    • CommentAuthoradeelkh
    • CommentTimeDec 2nd 2014

    Just made some small additions to the page Abel-Jacobi map.

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