Let be an arithmetic variety?, that is: a quasi-projective flat? regular scheme over an arithmetic ring?. In (Gillet-Soule) the arithmetic Chow groups of , denoted , are defined as groups whose elements are equivalence classes of pairs consisting of a codimension subvariety of together with a Green current? for it. Later, in (Burgos Gil 97), an alternative definition was given in terms of a Deligne complex of differential forms with logarithmic singularities along infinity, that computes a version of ordinary differential cohomology groups.
Arithmetic intersection theory was introduced in
Generalization are discussed in
Articles that discuss the relation of arithmetic Chow groups to ordinary differential cohomology include
Henri Gillet, Christoph Soulé, Arithmetic Chow groups and differential characters in Rick Jardine (ed.) Algebraic K-theory: Connections with Geometry and Topology, Springer (1989)
J. I. Burgos Gil, Arithmetic Chow rings, Ph.D. thesis, University of Barcelona, (1994).
J. I. Burgos Gil, Arithmetic Chow rings and Deligne-Beilinson cohomology, J. Alg. Geom. 6 (1997), 335–377.