Chern invariants of some flat bundles in the arithmetic Deligne cohomology

In this note, we investigate the cycle class map between the rational Chow groups and the arithmetic Deligne cohomology, introduced by Green–Griffiths and Asakura–Saito. We show nontriviality of the Chern classes of flat bundles in the arithmetic Deligne Cohomology in some cases and our proofs also indicate that generic flat bundles can be expected to have nontrivial classes. This provides examples of non-zero classes in the arithmetic Deligne cohomology which become zero in the usual rational Deligne cohomology