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Not an answer to your question, but just to point out that there must be a great to deal to add at complex analytic ∞-groupoid and elsewhere relating to general structures worked out for complex analytic cohesion. The discussion for the page is here.
There’s also a fragment of discussion here, and ’Deligne-Beilinson cohomology (in the holomorphic case)’ gets a mention is a discussion starting here.
I suppose what you are after is the statement that is alternatively Prop. 3.11 here at Deligne cohomology, which states how the Deligne complex is a homotopy fiber product of the coefficients for integral cohomology with those for closed differential forms.
To prove this (as proven at Deligne cohomology) one just needs a Poincaré lemma, stating that the sheaf of complexes of differential forms in question is locally exact. This goes through in the complex case.
Thanks David and Urs. Would we simply define $\widehat{(\flat \mathbf{B}^{n+1}\mathbb{C})}_\bullet$ analogously to Definition 2.11 in Deligne cohomology? Any comments on what happens with this problem of pseudo-connections? (I’m new to this subject, and still searching for references, so I have quite a few elementary questions I’m afraid.)
Would we simply define $\widehat{(\flat \mathbf{B}^{n+1}\mathbb{C})}_\bullet$ analogously to Definition 2.11 in Deligne cohomology?
Yes, with the Dolbeault complex of holomorphic forms instead.
Any comments on what happens with this problem of pseudo-connections?
Higher pseudo-connections is the name for cocycles in the chain complex $\widehat{\mathbf{B}^{n+1} \mathbb{Z}}$ from this def..
This may be used to model the homotopy fiber product for the Deligne complex by an ordinary fiber product.
You see that $\widehat{\mathbf{B}^{n+1} \mathbb{Z}}$ from this def. already looks like the Deligne complex (top row) but then in addition it also contains a kind of shifted copy of the Deligne complex (bottom row). This is such that a Cech cocycle with values in this sheaf of complexes starts out looking like a Cech-Deligne cocycle, only that at each stage the cocyle condition on the connection data is allowed to “fail”, with the failure being picked up by the data from the bottom row.
For low $n$ one sees that this yields the structure that in some part of the literature is known as “pseudo-connections”. But here we see (the following lemma) that this really serves as a fibration resolution of the “Chern character map” on ordinary cohomology, namely the “real-ification” map along which the Deligne complex is going to be a homotopy pullback.
OK, I think I’m starting to make sense of this now, thank you for all the explanations. So this $\widehat{(\flat \mathbf{B}^{n+1}\mathbb{C})}_\bullet$ is really an explicit fibrant replacement that we can use to calculate the homotopy pullback as a ’true’ pullback, if I’m summarising correctly.
As a naive question, we know that the real Deligne complex arises from looking at $U(1)$-principal bundles in some sense (made explicit in infinity-Chern-Weil theory introduction) in that the Deligne complex is given by $\mathbb{B}^n U(1)_\mathrm{conn}$. What happens in the complex case? Is $U(1)$ replaced by $\mathbb{C}^\times$ or some such complex analogue?
Is $U(1)$ replaced by $\mathbb{C}^\times$ or some such complex analogue?
Yes, one takes the holomorphic form complex with the inclusion of some constant sheaf of groups $A \hookrightarrow \mathcal{O}$ at the far left.
This is, as you seemed to have noted right at the beginning, what Deligne had done almost half a century back in the first place. I thought your question was whether these holomorphic Deligne complexes still may be obtained as homotopy pullbacks in complexes of sheaves over a site of complex manifolds.
Sorry, my original question was slightly confused, but it was indeed about this homotopy pullback construction. I suppose a final question is if there are any recommended references for this (i.e. ’differential’ cohomology in a cohesive topos, but in the complex-analytic setting), or if not, is it simply because everything works pretty much the same as in the smooth case (despite this partition of unity argument not working)?
is it simply because everything works pretty much the same as in the smooth case (despite this partition of unity argument not working)?
Yes, so what I meant to indicate is that instead of appealing to partition of unity, one may just appeal to the Poincaré lemma, hence the local exactness of the complex of differential forms being used, and that does work in the holomorphic case, doesn’t it.
Myself, I didn’t get around to writing much about the complex-analytic case of cohesive differential cohomology. It would be great if somebody did!
I’ve been thinking about this a lot recently, and I am really struggling to get to grips with how this all works in the complex-analytic case. Primarily, global connections on bundles don’t always exist, and secondly, the Chern classes should live in Deligne cohomology of degree (2k,k), and not (2k,2k). I’ve had a look at Cech Cocycles for Characteristic Classes by Brylinksi and McLaughlin, which (as I have read, but don’t fully understand) agrees with the abstract construction in Differential Cohomology in a Cohesive Topos, but they don’t really discuss the complex-analytic case much. Any further pointers?
It may not be relevant, but there is some trouble with setting up cohesion in the setting of algebraic geometry. In view of GAGA, would it perhaps be a bit surprising if one could do it in complex analytic geometry?
Edit: I see that in fact cohesion in the complex analytic case is discussed at complex analytic ∞-groupoid. Interesting. I see that it uses some things that would not translate in an obvious way through GAGA, such as polydiscs. I should return again to the question of cohesion in (some modification of) the étale topos.
There’s also the page higher complex analytic geometry, announced in this discussion, which is a stub but points to a talk by Urs, Differential cohomology is Cohesive homotopy theory. There’s a little bit there on the complex analytic case.
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