Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?


It should be a construction by Bloch using his cycle complex, but it seems I can't find on the web any paper spelling it out.



Motivic vs Deligne cohomology
The existence of a *cycle class map* from motivic cohomology is a general fact to every cohomology satisfying certain axioms. For example, to every *mixed Weil theory* in the terminology of Cisinski-Déglise (see [here](https://www.sciencedirect.com/science/article/pii/S0001870811003823), [here](https://arxiv.org/abs/0712.3291) the arXiv preprint) or more generally to every cohomology represented by a spectrum satisfying certain axioms (cf *Theorem* in the Introduction [here](https://www.cambridge.org/core/journals/journal-of-the-institute-of-mathematics-of-jussieu/article/the-rigid-syntomic-ring-spectrum/F7CC2419296F69BF10AB77A1FB43B21C), preprint [here](https://arxiv.org/abs/1211.5065)). As you see, for these results you need your cohomology to be given by a spectrum in Voevodsky's stable homotopy category. The Deligne cohomology is known to be given by a spectrum, thanks to Holmstrom-Scholbach in [this paper](http://www.ams.org/journals/jag/2015-24-04/S1056-3911-2015-00648-8/) (preprint [here](https://arxiv.org/abs/1012.2523)). The cycle class map and the Chern character for Deligne cohomology are defined in Definition 3.7 of Holmstrom-Scholbach.


If you are interested in the computation that such maps coincide with the classic definition, you should also check [this paper](https://academic.oup.com/jtopol/article-abstract/3/2/229/1148105?redirectedFrom=fulltext) by Riou (preprint [here](https://arxiv.org/abs/0907.2710)). In Definition 6.2.3.9. and Remark 6.2.3.10. Riou shows that the canonical map in spectra from $K$-theory spectrum to Beilinson's motivic cohomology spectrum composed with the canonical isomorphism to the Eilenberg-McLane spectrum with rational coefficients coincides with the classical Chern character. 


