A metric on a bundle E over X is an extension of E to some model of X over R and the arithmetic Chow group of X is the inductive limit of Chow groups of such models. Assuming resolution of singularities, we get results similar to those in the usual Arakelov theory, e.g., a Riemann-Roch theorem.

Furthermore, the analogy between special fibers and Kähler varieties leads to interesting questions related to the standard conjectures of Grothendieck. It was used by Künnemann to define a local height pairing for cycles on abelian varieties.