View source for Algebraic cycle

''on an algebraic variety''

An element of the [[Free Abelian group|free Abelian group]] the set of free generators of which is constituted by all closed irreducible subvarieties of the given algebraic variety. The subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114601.png" /> of algebraic cycles on a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114602.png" /> generated by a subvariety of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114603.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114604.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114605.png" /> can be represented as the direct sum

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The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114607.png" /> is identical with the group of Weil divisors (cf. [[Divisor|Divisor]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114608.png" />.

In what follows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a0114609.png" /> will denote a non-singular projective algebraic variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146010.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146012.png" /> is the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146013.png" />, then each algebraic cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146014.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146015.png" />-dimensional homology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146016.png" /> and, in accordance with Poincaré duality, a cohomology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146017.png" />. The homology (or, respectively, cohomology) classes of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146018.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146019.png" />) are called algebraic homology (respectively, cohomology) classes. (Hodge's conjecture) Each analytic cycle is homologous with an algebraic cycle. It is believed that an integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146020.png" />-dimensional cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146022.png" /> is homologous with an algebraic cycle if and only if the integrals of all closed differential forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146024.png" />, over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146025.png" /> are equal to zero. This conjecture has only been proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146026.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146027.png" /> [[#References|[6]]], and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146028.png" /> [[#References|[7]]]), for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146029.png" />, and for isolated classes of varieties [[#References|[4]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146030.png" /> is an algebraic cycle on the product of two varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146031.png" />, then the set of cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146032.png" /> of the form

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is known as a family of algebraic cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146034.png" /> parametrized by the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146035.png" />. The usual requirement in this connection is that the projection of each subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146037.png" /> be a flat morphism. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146038.png" /> is defined by an irreducible subvariety, the corresponding family of algebraic cycles on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146039.png" /> is called a family of algebraic subvarieties. In particular, for any flat morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146040.png" /> of algebraic varieties its fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146041.png" /> form a family of algebraic subvarieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146042.png" /> parametrized by the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146043.png" />. A second particular case of this concept is that of a [[Linear system|linear system]]. All members of a family of algebraic subvarieties (or, respectively, algebraic cycles) of a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146044.png" />, parametrized by a connected base, have the same [[Hilbert polynomial|Hilbert polynomial]] (respectively, virtual [[Arithmetic genus|arithmetic genus]]).

Two algebraic cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146046.png" /> on a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146047.png" /> are algebraically equivalent (which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146048.png" />) if they belong to the same family, parametrized by a connected base. Intuitively, equivalence of algebraic cycles means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146049.png" /> may be algebraically deformed into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146050.png" />. If this definition includes the condition that the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146051.png" /> is a rational variety, the algebraic cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146053.png" /> are called rationally equivalent (which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146054.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146055.png" />, the concept of rational equivalence reduces to the concept of linear equivalence of divisors. The subgroup of algebraic cycles rationally (or, respectively, algebraically) equivalent to zero, is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146056.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146057.png" />). Each of these groups is a direct sum of its components

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The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146060.png" /> is finitely generated and is called as the Neron–Severi group of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146061.png" />. The problem of the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146062.png" /> being finitely generated for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146063.png" /> remains open at the time of writing (1977). The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146064.png" /> has the structure of an Abelian variety (cf. [[Picard scheme|Picard scheme]]). The operation of intersection of cycles makes it possible to define a multiplication in the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146065.png" />, converting it into a commutative ring, called the Chow ring of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146066.png" /> (cf. [[Intersection theory|Intersection theory]]).

For any Weil cohomology theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146067.png" /> there exists a uniquely defined homomorphism of groups

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Two algebraic cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146070.png" /> are called homologically equivalent (which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146071.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146072.png" />. The subgroup of algebraic cycles that are homologically equivalent with zero is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146073.png" />. The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146074.png" /> is valid. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146075.png" /> is finitely generated, and is a subring in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146076.png" />, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146077.png" /> and is known as the ring of algebraic Weil cohomology classes. It is not known (1986) whether or not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146078.png" /> depends on the Weil cohomology theory that has been chosen.

Two algebraic cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146080.png" /> are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146082.png" />-equivalent (which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146083.png" />) if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146084.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146085.png" />. The subgroup of algebraic cycles that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146086.png" />-equivalent to zero, is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146087.png" />. Two algebraic cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146089.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146090.png" /> are called numerically equivalent (which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146091.png" />) if the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146092.png" /> is valid for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146093.png" />, provided both sides of the equality are defined. The subgroup of algebraic cycles numerically equivalent with zero is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146094.png" />. The imbeddings

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are valid. For divisors the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146098.png" /> are identical [[#References|[6]]]. However, in accordance with the counterexample in [[#References|[5]]] for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a01146099.png" />

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460101.png" /> is considered with respect to the ordinary cohomology theory with rational coefficients. A similar counterexample was established for a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460102.png" /> of arbitrary characteristic and for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460103.png" />-adic theory of Weil cohomology. The question as to the equality of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460105.png" /> has been solved [[#References|[9]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460106.png" /> be imbedded in a projective space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460107.png" /> be the cohomology class of a hyperplane section. An algebraic cohomology class

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is called primitive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460109.png" />. In such a case, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460110.png" /> is the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460111.png" />, the bilinear form

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is positive definite on the subspace of primitive classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460113.png" />. A similar proposition for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460114.png" />, which is closely connected with the Weil conjectures on the zeta-function of an algebraic variety, has been proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460115.png" /> only.

If a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460116.png" /> is defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460117.png" /> that is not algebraically closed, the Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460118.png" /> of the separable algebraic closure of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460119.png" /> acts on the Weil cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460120.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460121.png" />. Each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460122.png" /> is invariant with respect to some subgroup of finite index of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460123.png" />. It is believed (Tate's conjecture on algebraic cycles) that the converse proposition is also true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460124.png" /> is finitely generated over its prime subfield. Many conjectures on the zeta-function of algebraic varieties are based on this assumption [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Baldassarri,   "Algebraic varieties" , Springer  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.T. Tate,   "Algebraic cohomology classes" , ''Summer school of algebraic geometry Woods Hole, 1964''</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.V. Dolgachev,   V.A. Iskovskikh,   "Geometry of algebraic varieties"  ''J. Soviet Math.'' , '''5''' :  6  (1976)  pp. 803–864  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''12'''  (1974)  pp. 77–170</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.L. Kleiman,   "Algebraic cycles and the Weil conjecture"  A. Grothendieck (ed.)  J. Giraud (ed.)  et al. (ed.) , ''Dix exposés sur la cohomologie des schémas'' , North-Holland &amp; Masson  (1968)  pp. 359–386</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.A. Griffiths,   "On the periods of certain rational integrals II"  ''Ann. of Math. (2)'' , '''90''' :  3  (1969)  pp. 496–541</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Lefschetz,   "L'analysis situs et la géométrie algébrique" , Gauthier-Villars  (1924)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  W.V.D. Hodge,   "The theory and application of harmonic integrals" , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  "Groupes de monodromie en geometrie algebrique"  M. Raynaud (ed.)  D.S. Rim (ed.)  A. Grothendieck (ed.) , ''Sem. Geom. Alg.'' , '''7''' , Springer  (1972–1973)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P. Deligne,   "La conjecture de Weil I"  ''Publ. Math. IHES'' , '''43'''  (1974)  pp. 273–308</TD></TR></table>



====Comments====
In 1983 H. Clemens proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460125.png" /> is not finitely generated [[#References|[a1]]]. He also proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460126.png" /> is not finitely generated, even after tensoring with the field of rational numbers [[#References|[a1]]].

A state-of-the-art survey concerning the Hodge conjecture is in [[#References|[a2]]]. See also [[#References|[a3]]].

Much of the recent progress of the theory of algebraic cycles is related to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011460/a011460127.png" />-theory, see [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Clemens,   "Homological equivalence modulo algebraic equivalence is not finitely generated"  ''Publ. Math. IHES'' , '''58'''  (1983)  pp. 19–38</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Shiado,   "What is known about the Hodge conjecture" , North-Holland &amp; Kinokuniya  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.F. Atiyah,   F. Hirzebruch,   "Analytic cycles on complex manifolds"  ''Topology'' , '''1'''  (1961)  pp. 25–45</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Bloch,   "Lectures on algebraic cycles" , '''IV''' , Dept. Math. Duke Univ.  (1980)</TD></TR></table>