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wrote out the definition In complex geometry
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.
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
but staring at this with a more higher topos theoretic point of view seems to induce some resonances. Not sure yet, though.
That second map in the short exact sequence (7.9)
$0 \to J^{k+1}(\Sigma) \to H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_D) \to Hg^{k+1}(\Sigma) \to 0$of the above article is curious:
The Deligne cohomology $H^{2k+2}(\Sigma, \mathbb{Z}(k+1)_D)$ itself is the 0-truncation of the homotopy pullback of the diagram
$\array{ && Maps(\Sigma, (\Omega^{k+1} \stackrel{\partial}{\to}\cdots)[-2k-2]) \\ && \downarrow \\ Maps(\Sigma,\mathbb{Z}(k+1)[-2k-2]) &\longrightarrow& Maps(\Sigma, \mathbb{C}[-2k-2]) }$On the other hand, the ordinary pullback of the 0-truncation of this diagram is the Hodge cohomology classes $Hg^{k+1}(\Sigma)$.
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….
to state this more pronouncedly:
let $\hat E$ be any differential cohomology theory with its canoncial decomposition as a homotopy fiber product:
$\hat E \simeq \Pi \hat E \underset{\Pi \flat_{dR} \hat E}{\times} \flat_{dR} \hat E \,.$Then set
$J \coloneqq ker\left( \tau_0\left(\Pi \hat E \underset{\Pi \flat_{dR} \hat E}{\times} \flat_{dR} \hat E\right) \longrightarrow \tau_0(\Pi \hat E) \underset{\tau_0(\Pi \flat_{dR} \hat E)}{\times} \tau_0(\flat_{dR} \hat E) \right)$where evaluation on some $X$ is left implicit (this is a measure for the failure of the 0-truncation $\tau_0$ to preserve the homotopy fiber product).
Then for $\hat E$ the complex analytic $\mathbb{Z}(p)$ Deligne-cohomology in degree 2p, this reduces to Deligne’s characterization of the intermediate Jacobian.
It seems.
gave the entry some actual Idea section, inspired in parts by the above observation.
(Have to call it quits for today.)
Have people come across this Jacobian construction for other $\hat E$?
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 $\hat {KU}$ into the above. But that would be something to think about.
@Urs Your Witten 96 link points back to this page.
Thanks, fixed.
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.
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)
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):
Do these $\hat{E}$-Hodge filtrations crop up frequently?
Yes. In all the examples where $\flat_{dR}\hat E$ 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.
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.
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.
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.
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
After
This modulates G-principal connections
The quotient $\sim$ should outside the braces { }
After
transgressed to the circle it yields the WZW gerbe
$[S^1,L_{WZW}]$ should be $[S^1,L_{CS}]$
section 5
generak
and
Deligen
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.)
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 :
The last section is the one that briefly states the axiomatic characterization of higher (i.e. intermediate) Jacobians.
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 $J^1(X)$ (from their page 154 (11 of 76) on).
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
Just made some small additions to the page Abel-Jacobi map.
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