nLab arithmetic Chow group

Idea

Arithmetic Chow groups are refinements of ordinary Chow groups analogous to how ordinary differential cohomology refines ordinary cohomology.

Let $X$ be an arithmetic variety, that is: a quasi-projective flat? 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.

When $X$ is proper, the two definitions are naturally isomorphic.

Related concepts

• Chow group, Arakelov geometry

References

General

Arithmetic intersection theory was introduced in

• Henri Gillet, Christoph Soulé, Arithmetic intersection theory IHES Preprint (1988)

Generalization are discussed in

• J. I. Burgos Gil, J. Kramer, U. Kühn, Cohomological arithmetic Chow rings (arXiv:math/0404122v2)

• J. I. Burgos Gil, Higher arithmetic Chow groups (pdf)

Relation to differential cohomology

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.