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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)

    0Jk+1(Σ)H2k+2(Σ,(k+1)D)Hgk+1(Σ)0

    of the above article is curious:

    The Deligne cohomology H2k+2(Σ,(k+1)D) itself is the 0-truncation of the homotopy pullback of the diagram

    Maps(Σ,(Ωk+1)[2k2])Maps(Σ,(k+1)[2k2])Maps(Σ,[2k2])

    On the other hand, the ordinary pullback of the 0-truncation of this diagram is the Hodge cohomology classes Hgk+1(Σ).

    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 ˆE be any differential cohomology theory with its canoncial decomposition as a homotopy fiber product:

    ˆEΠˆE×ΠdRˆEdRˆE.

    Then set

    Jker(τ0(ΠˆE×ΠdRˆEdRˆE)τ0(ΠˆE)×τ0(ΠdRˆE)τ0(dRˆE))

    where evaluation on some X is left implicit (this is a measure for the failure of the 0-truncation τ0 to preserve the homotopy fiber product).

    Then for ˆE the complex analytic (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 ˆE?

    • 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 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 ˆE-Hodge filtrations crop up frequently?

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2014

    Yes. In all the examples where dRˆE 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 should outside the braces { }

    After

    transgressed to the circle it yields the WZW gerbe

    [S1,LWZW] should be [S1,LCS]

    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 J1(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.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJun 9th 2023

    re-reading this entry, I made a bunch of little cosmetic adjustments to wording, hyperlinking and formatting

    diff, v45, current