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| Chow group | |
|---|---|
| Name | Chow group |
| Field | Algebraic geometry |
| Introduced by | William Fulton |
| First appeared | 1950s |
Chow group The Chow group is an algebraic invariant attached to an algebraic variety that encodes algebraic cycles modulo rational equivalence. It appears in the work of Jean-Pierre Serre, André Weil, Alexander Grothendieck, and William Fulton and plays a central role in intersection theory, motives, and arithmetic geometry. The theory connects to classical results of Emmy Noether, modern developments by Pierre Deligne and Vladimir Drinfeld, and computational approaches influenced by David Mumford and János Kollár.
For a separated scheme of finite type over a field, one defines groups of algebraic cycles of given codimension and then quotients by rational equivalence to obtain the Chow groups; this construction generalizes ideas from Bernard Riemann's work, the Riemann–Roch theorem and the program of Grothendieck in the SGA seminars. The basic properties include functoriality for proper pushforward and flat pullback, localization sequences modeled on the excision techniques of Jean Leray and Alexander Grothendieck's cohomological formalism, and compatibility with the cycle class map to Betti cohomology and ℓ-adic cohomology developed by Pierre Deligne and Alexander Beilinson. For smooth projective varieties over a field such as those studied by André Weil and Serre, Chow groups are finitely generated in many classical cases, and they satisfy relations predicted by the Hodge conjecture and the Bloch–Kato conjecture formulated by Spencer Bloch and Kazuya Kato.
Computations of Chow groups for curves recover classical results about divisors on curves treated by Riemann and Roch; for a smooth projective curve C, the Chow group of codimension one is the Picard group studied by Niels Abel and Jacques Hadamard. For projective spaces examined by Élie Cartan and Hermann Weyl, Chow groups are generated by linear subspaces and are isomorphic to Z in each relevant degree, mirroring computations in Schubert calculus developed by Hermann Schubert and later formalized by Bertram Kostant. Grassmannians and flag varieties arising in the work of Élie Cartan and Claude Chevalley have Chow rings described by Schubert classes; these calculations parallel results in the theory of Schubert polynomials initiated by Anders Buch and Richard Stanley. For singular varieties studied by Oscar Zariski and John Milnor, one often passes to resolutions of singularities using techniques of Heisuke Hironaka to compute Chow groups via pullback and pushforward, while for arithmetic schemes like models of elliptic curves in the work of Andrew Wiles and Gerd Faltings, Chow groups of zero-cycles connect to the Birch and Swinnerton-Dyer conjecture.
The cycle class map relates Chow groups to Betti cohomology for varieties over C and to étale cohomology and ℓ-adic cohomology developed by Jean-Pierre Serre and Pierre Deligne. Motivic cohomology introduced by Vladimir Voevodsky and Maxim Kontsevich provides a framework in which Chow groups appear as motivic homology groups, linking to the conjectures of Beilinson and Bloch. Comparisons with K-theory studied by Daniel Quillen and Friedhelm Waldhausen yield cycle class maps from higher Chow groups introduced by Spencer Bloch to algebraic K-groups, and these maps are central in the Bloch–Quillen identification and the study of the Quillen–Lichtenbaum conjecture pursued by Voevodsky and Suslin.
Chow groups admit pushforward for proper morphisms and pullback for flat morphisms, echoing the functorial features in the frameworks developed in SGA and in Grothendieck's general formalism. For smooth morphisms one gets Gysin maps related to the excess intersection formula used by William Fulton and Robert MacPherson, and localized Chern class operations connect to characteristic classes studied by Chern, Hirzebruch, and Atiyah–Singer index theory. External products make Chow groups into bi-graded theories compatible with products of schemes studied by Alexander Grothendieck and Jean-Louis Verdier, and specialization maps compare fibers in families as in deformation theory by Michael Artin and David Mumford.
When the scheme is smooth and projective, the sum over codimensions of Chow groups acquires a graded ring structure via the intersection product; this intersection theory was axiomatized by William Fulton and draws on classical intersection calculations by André Weil and Oscar Zariski. The intersection product satisfies projection formulae used in enumerative geometry problems popularized by Schubert and modernized by Kontsevich in Gromov–Witten theory. The structure of the Chow ring for homogeneous spaces such as flag varieties reflects representation-theoretic data studied by Pierre Deligne and Robert Langlands, while the study of algebraic cycles modulo numerical equivalence touches on results and conjectures of Grothendieck and Jannsen.
Chow groups are applied in enumerative geometry problems pursued by David Hilbert and Hermann Schubert and in the formulation of conjectures about algebraic cycles such as those of Hodge, Tate, and Bloch–Beilinson. They appear in the study of motives initiated by Grothendieck and advanced by Yves André and Alexander Beilinson and play a role in Arakelov theory as developed by Henri Gillet and Christophe Soulé for Diophantine applications related to the Mordell conjecture proved by Gerd Faltings. In moduli problems for varieties and sheaves investigated by David Mumford, Simon Donaldson, and Maxim Kontsevich, Chow groups control cycle classes of families and contribute to enumerative invariants used in modern mirror symmetry studied by Shing-Tung Yau and Kontsevich.