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- In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from that of divisors. For example, every curve has a constant N such that every divisor of degree zero is linearly equivalent to a difference of two effective divisors of degree at most N. David Mumford proved that, on a smooth complete complex algebraic surface S with positive geometric genus, the analogous statement for the group of rational equivalence classes of codimension two cycles in S is false. The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group contains transcendental information, and in effect Mumford's theorem implies that, despite having a purely algebraic definition, it shares transcendental information with . Mumford's theorem has since been greatly generalized. The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The Hodge conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algebraic cycles yield enough cycles to construct his category of motives and would imply that algebraic cycles play a vital role in any cohomology theory of algebraic varieties. Conversely, Alexander Beilinson proved that the existence of a category of motives implies the standard conjectures. Additionally, cycles are connected to algebraic K-theory by Bloch's formula, which expresses groups of cycles modulo rational equivalence as the cohomology of K-theory sheaves. (en)
- En géométrie algébrique, les cycles sont des combinaisons formelles de fermés irréductibles d'un schéma donné. Le quotient du groupe des cycles par une relation d'équivalence convenable aboutit aux (en) qui sont des objets fondamentaux. Tous les schémas considérés ici seront supposés noethériens de dimension finie. (fr)
- 数学では、代数多様体 V の上の代数的サイクル(algebraic cycle)とは、大まかには、V 上のホモロジー類(homology class)であり、V の部分多様体の線型結合により表されるものを言う。従って、V 上の代数的サイクルは、代数幾何学に直接関係する V の代数トポロジーである。1950年代から1960年代にかけて、いくつかの基本的な予想が提示され、代数的サイクルの研究が、一般的な多様体の代数幾何学の主要な対象のひとつとなった。 代数的サイクルの持つ難しさは、全く簡単なことであり、代数的サイクルの存在を予想することは容易であるが、それらを構成する今日の方法が不十分である。代数的サイクルの主な予想は、ホッジ予想やテイト予想を含んでいる。ヴェイユ予想の証明の研究から、アレクサンドル・グロタンディーク(Alexander Grothendieck)やエンリコ・ボンビエリは代数的サイクルの標準予想として現在知られていることを定式化した。 代数的サイクルは、代数的K-理論に密接に関連していることが示されている。 良く使われる交叉理論のためには、様々な(equivalence relations on algebraic cycles)が使われる。特に重要なことは、いわゆる有理的同値(rational equivalence)である。有理同値を無視してのサイクルは、次数付き環、(Chow ring)を形成し、積は交叉積により与えられる。さらに基本的な関係には、代数的同値(algebraic equivalence)、数値的同値(numerical equivalence)やホモロジカル同値(homological equivalence)がある。一部は予想に過ぎないが、これらはモチーフの理論への応用を持っている。 (ja)
- 대수기하학에서 대수적 순환(代數的循環, 영어: algebraic cycle)은 어떤 대수다양체 V의 부분 다양체들의 선형결합으로 나타내어지는 호몰로지류이다. 이를 이용하여, 대수적 위상수학과 대수기하학을 연관시킬 수 있다. (ko)
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