Algebraic cycle

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 ${\displaystyle \operatorname {CH} ^{2}(S)}$ of rational equivalence classes of codimension two cycles in S is false.^[1] The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group ${\displaystyle H^{2}(S)}$ contains transcendental information, and in effect Mumford's theorem implies that, despite ${\displaystyle \operatorname {CH} ^{2}(S)}$ having a purely algebraic definition, it shares transcendental information with ${\displaystyle H^{2}(S)}$. Mumford's theorem has since been greatly generalized.^[2]

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.