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Have added more of the original (“historical”) References with brief comments and further pointers.
added a lightning quick paragraph at Properties – GAGA on a sufficient condition for a complex-analytic Deligne cocycle to lift to a complex algebraic one. (Thanks to David Roberts for the pointer!)
Added the reference
which introduces an interesting generalization of Deligne cohomology, by reformulating it via simplicial presheaves.
I also started some notes at Higher regulators and values of L-functions but didn’t get very far yet.
I have expanded a good bit the Idea section.
I am spelling out details at Deligne cohomology in the expository style (or at least that’s the intention) of geometry of physics. So far there is now a section Definition – In smooth differential geometry.
Wrote a section Characteristic maps out of and into Deligne cohomology spelling out the construction of all the relevant maps (curvatue, underlying class etc.) as zig-zags of chain maps. From this will nicely follow all the exactness yoga, which I am now writing out in the next section.
Wrote a section Deligne cohomology – Porperties – Chern character which spells out how the Deligne complex is the homotopy pullback of closed differential forms along the “higher abelian Chern character” map.
(All this has been at circle n-bundle with connection for a long time, but, looking back now, this looks quite messy, whereas the new version now should be much more readable. I hope.)
Wrote a section Deligne cohomology – Properties – The exact sequences for curvature and characteristic classes which spells out the proof of the curvature and characteristic class short exact sequence as it follows naturally from the above homotopy pullback characterization.
Wrote a section Deligne cohomology – Properties – The exact differential cohomology hexagon concluding the above discussions.
Did a little more polishing at Deligne cohomology, rearranged the sub-sections slightly to flow more naturally. Copied the definition of Cech hypercohomology over to the subsection on Preliminaries on sheaf cohomology in order to be a tad more self-contained.
added a more illustrative picture of the Cech-Deligne double complex here
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