nLab arithmetic Chow group




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.



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.

Last revised on August 17, 2013 at 16:50:06. See the history of this page for a list of all contributions to it.