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Lately I have been studying these two papers (first and second) that introduce a new cohomology theory called Arakelov motivic cohomology. While most of the applications presented in the papers are from a K-theoretic and homotopy theoretic viewpoint, there is still some discussion about Deligne cohomology and few other Hodge theory-related concepts as well. Since both Deligne cohomology and motives were originally invented for studying mixed Hodge theory, I'm really curious if this could provide any connection between mixed Hodge theory and Arakelov motivic cohomology.

Question: Could Arakelov motivic cohomology have any interesting applications to mixed Hodge theory?

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