Let $X$ be an arithmetic variety?, that is: a quasi-projectiveflat?regular scheme over an arithmetic ring?. In (Gillet-Soule) the arithmetic Chow groups of $X$, denoted $\hat CH^p(X)$, are defined as groups whose elements are equivalence classes of pairs consisting of a codimension $p$ subvariety of $X$ 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.

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.

Revised on August 17, 2013 16:50:06
by David Corfield
(87.112.114.219)