Algebraic cycles

Algebraic cycles
Name	Algebraic cycles
Field	Algebraic geometry
Introduced	19th century
Notable	Alexander Grothendieck, André Weil, Yuri Manin, John Milnor, Phillip Griffiths

Algebraic cycles are formal finite linear combinations of subvarieties on an algebraic variety, serving as fundamental objects linking geometry, arithmetic, and topology. They appear throughout modern research influenced by figures such as Alexander Grothendieck, André Weil, Alexander Mumford, Pierre Deligne, Serge Lang and are central to theories developed in contexts like the Weil conjectures, the Hodge conjecture, and the theory of motives. Algebraic cycles connect to cohomology theories studied by Jean-Pierre Serre, Grothendieck's standard conjectures, and to enumerative problems related to Hilbert scheme and Chow variety constructions.

Definition and basic examples

An algebraic cycle on a variety X over a field K is a finite formal sum sum_i n_i [V_i] with integer coefficients n_i and irreducible closed subvarieties V_i; this notion was formalized in work by Oscar Zariski and systematized by André Weil and Alexander Grothendieck. Basic examples include divisors on a curve or surface as in classical treatments by Riemann, Bernhard Riemann, and the divisor theory used by Ferdinand Georg Frobenius; zero-cycles correspond to finite subschemes and appear in the study of rational points investigated by Yuri Manin and Jean-Louis Colliot-Thélène. On projective space examples stem from linear subspaces studied in Cayley–Bacharach theorem contexts and in enumerative geometry treated by Hermann Schubert and later by David Mumford. Families of cycles are parameterized by constructions like the Chow variety and the Hilbert scheme, developed respectively by Wei-Liang Chow and Alexander Grothendieck.

Chow groups and rational equivalence

Rational equivalence, introduced in classical form by Federigo Enriques and algebraically by Oscar Zariski and André Weil, identifies cycles that differ by the divisor of a rational function on a subvariety; the resulting quotient groups are the Chow groups A_k(X) or CH^p(X), extensively used by William Fulton in his treatise on intersection theory. Chow groups are central in the work of Grothendieck on K-theory and in the formulation of the Bloch–Ogus theory and the Beilinson conjectures by Alexander Beilinson. Calculations of CH^*(X) for surfaces use classical results from Max Noether and later techniques due to Frederick C. Beilinson and Claire Voisin. The functoriality of Chow groups under proper pushforward and flat pullback is a foundational tool in the approach of Jean-Pierre Serre and J. J. Sylvester-style algebraic manipulations.

Intersection theory and cycle classes

Intersection theory assigns intersection products to cycles, formalized by William Fulton and building on ideas from Bernhard Riemann and Hermann Schubert in enumerative problems; this theory interacts with cycle class maps to cohomology theories such as singular cohomology, étale cohomology introduced by Alexander Grothendieck and Michael Artin, and de Rham cohomology studied by Jean Leray and Henri Cartan. The cycle class map sends Chow groups to cohomology rings exploited in proofs of the Weil conjectures by Pierre Deligne and in the constructions used by Grothendieck in his motive program. Intersection multiplicities and excess intersection formulae are handled via techniques related to the Grothendieck–Riemann–Roch theorem developed by Jean-Pierre Serre and L. Illusie, with operational Chow cohomology appearing in work by André Weil and Kenneth Kodaira-adjacent studies.

Homological and numerical equivalence

Equivalence relations on cycles—homological, algebraic, and numerical—were formulated in correspondence with cohomological comparisons pursued by Phillip Griffiths and Pierre Deligne; homological equivalence compares cycles by their images in singular or étale cohomology while numerical equivalence uses intersection numbers as in foundational contributions by André Weil. The comparison between these equivalences is central to Grothendieck's standard conjectures and to criteria appearing in works of Teruhisa Matsusaka and Jacob Murre. The study of Abel–Jacobi maps linking algebraic cycles to intermediate Jacobians was advanced by Rene Thom, Niels Henrik Abel-inspired classical theory, and modern treatments by Phillip Griffiths and Claire Voisin.

Motives and Hodge theory connections

The idea of motives as universal cohomology objects was proposed by Alexander Grothendieck and furthered by Yuri Manin, Pierre Deligne, and John Tate; algebraic cycles are the core correspondences defining the category of pure motives, closely tied to Hodge theory developed by Wilhelm Hodge and refined by Phillip Griffiths and Pierre Deligne. Mixed motives and regulators connect cycles to K-theory and special values of L-functions in conjectures of Alexander Beilinson and Spencer Bloch, informed by the arithmetic perspective of Gerd Faltings and Andrew Wiles. The interaction of algebraic cycles with period mappings and variations of Hodge structure is central in work by Carlos Simpson and Claire Voisin on families of varieties and in the study of normal functions originated by Henri Poincaré.

Conjectures and major open problems

Major conjectures framing contemporary research include the Hodge conjecture and the Tate conjecture, formulated by William Thomson-era successors and formalized by John Tate and Wilhelm Hodge; Grothendieck proposed the standard conjectures on algebraic cycles to underpin the Weil conjectures, and Beilinson conjectures predict deep relations between regulators, K-theory, and special values of L-function associated with motives. The Bloch–Beilinson filtration and Bloch–Kato conjectures, advanced by Spencer Bloch and Kazuya Kato, further structure expectations about CH^*(X) and are intertwined with breakthroughs by Andrew Wiles, Richard Taylor, and Peter Scholze in arithmetic geometry. Open problems include the existence of motivic t-structures conjectured by Vladimir Voevodsky and structural descriptions of algebraic cycles on higher-dimensional varieties studied by Claire Voisin and David Mumford.

Category:Algebraic geometry