Algebraic Cycle

In mathematics, an algebraic cycle on an algebraic variety V is, roughly speaking, a homology class on V that is represented by a linear combination of subvarieties of V. Therefore the algebraic cycles on V are the part of the algebraic topology of V that is directly accessible in algebraic geometry. With the formulation of some fundamental conjectures in the 1950s and 1960s, the study of algebraic cycles became one of the main objectives of the algebraic geometry of general varieties.

The nature of the difficulties is quite plain: the existence of algebraic cycles is easy to predict, but the current methods of constructing them are deficient. The major conjectures on algebraic cycles include the Hodge conjecture and the Tate conjecture. In the search for a proof of the Weil conjectures, Alexander Grothendieck and Enrico Bombieri formulated what are now known as the standard conjectures of algebraic cycle theory.

Algebraic cycles have also been shown to be closely connected with algebraic K-theory.

For the purposes of a well-working intersection theory, one uses various equivalence relations on algebraic cycles. Particularly important is the so-called rational equivalence. Cycles up to rational equivalence form a graded ring, the Chow ring, whose multiplication is given by the intersection product. Further fundamental relations include algebraic equivalence, numerical equivalence, and homological equivalence. They have (partly conjectural) applications in the theory of motives.