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

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

]]>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

]]>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).

]]>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.

]]>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.)

]]>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

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

]]>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.

]]>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.

]]>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. 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.

]]>Do these $\hat{E}$-Hodge filtrations crop up frequently?

]]>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)

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)

]]>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.

]]>Thanks, fixed.

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

]]>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.

]]>Have people come across this Jacobian construction for other $\hat E$?

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

(Have to call it quits for today.)

]]>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.

]]>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….

]]>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.

]]>