In
mathematics, an algebraic cycle on an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers ...
''V'' is a formal linear combination of
subvarieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
of ''V''. These are the part of the
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...
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
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s 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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
, 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
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded ...
proved that, on a smooth complete complex algebraic surface ''S'' with positive
geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for comp ...
, 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.
[Voisin, Claire, ''Chow Rings, Decomposition of the Diagonal, and the Topology of Families'', Annals of Mathematics Studies 187, February 2014, .]
The behavior of algebraic cycles ranks among the most important open questions in modern mathematics. The
Hodge conjecture, one of the
Clay Mathematics Institute
The Clay Mathematics Institute (CMI) is a private, non-profit foundation dedicated to increasing and disseminating mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address is now in Denver, Colorado. CMI's sc ...
's
Millennium Prize Problems
The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
, 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
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
.
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
Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...
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.
Definition
Let ''X'' be a
scheme which is finite type over a field ''k''. An algebraic ''r''-cycle on ''X'' is a formal linear combination
: