nLab
arithmetic Chow group

Contents

Idea

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

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

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, 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.

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