In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley (1958)) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space.
The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.
(or more generally, a locally Noetherian normal factorial scheme [2]), this is isomorphic to the Picard group of line bundles on
, then the vanishing loci of a generic section of both line bundles defines non-equivalent cycle classes in
More generally, in various cases, intersection theory constructs an explicit cycle that represents the product
points of intersection; this is a version of Bézout's theorem, a classic result of enumerative geometry.
fits into an exact sequence Thus the Chow group of an elliptic curve
, and some of the deepest problems in number theory are attempts to understand this group.
A key computational tool for Chow groups is the localization sequence, as follows.
Therefore, we have There are several homomorphisms (known as cycle maps) from Chow groups to more computable theories.
First, for a scheme X over the complex numbers, there is a homomorphism from Chow groups to Borel–Moore homology:[6] The factor of 2 appears because an i-dimensional subvariety of X has real dimension 2i.
[7] This incorporates the Abel–Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian.
The exponential sequence shows that CH1(X) maps isomorphically to Deligne cohomology, but that fails for CHj(X) with j > 1.
For a scheme X over an arbitrary field k, there is an analogous cycle map from Chow groups to (Borel–Moore) etale homology.
[8] An (algebraic) vector bundle E on a smooth scheme X over a field has Chern classes ci(E) in CHi(X), with the same formal properties as in topology.
[9] The Chern classes give a close connection between vector bundles and Chow groups.
As part of the Grothendieck–Riemann–Roch theorem, Grothendieck showed that the Chern character gives an isomorphism This isomorphism shows the importance of rational equivalence, compared to any other adequate equivalence relation on algebraic cycles.
Some of the deepest conjectures in algebraic geometry and number theory are attempts to understand Chow groups.
For example: Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes.
[13] A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively.
The name "bivariant" refers to the fact that the theory contains both covariant and contravariant functors.
[15] Arithmetic Chow groups are an amalgamation of Chow groups of varieties over Q together with a component encoding Arakelov-theoretical information, that is, differential forms on the associated complex manifold.
The theory of Chow groups of schemes of finite type over a field extends easily to that of algebraic spaces.
The key advantage of this extension is that it is easier to form quotients in the latter category and thus it is more natural to consider equivariant Chow groups of algebraic spaces.
A much more formidable extension is that of Chow group of a stack, which has been constructed only in some special case and which is needed in particular to make sense of a virtual fundamental class.
Rational equivalence of divisors (known as linear equivalence) was studied in various forms during the 19th century, leading to the ideal class group in number theory and the Jacobian variety in the theory of algebraic curves.
For higher-codimension cycles, rational equivalence was introduced by Francesco Severi in the 1930s.
In 1956, Wei-Liang Chow gave an influential proof that the intersection product is well-defined on cycles modulo rational equivalence for a smooth quasi-projective variety, using Chow's moving lemma.
Starting in the 1970s, Fulton and MacPherson gave the current standard foundation for Chow groups, working with singular varieties wherever possible.
In their theory, the intersection product for smooth varieties is constructed by deformation to the normal cone.