# Algebraic cycle

*on an algebraic variety*

An element of the 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 $ C(X) $ of algebraic cycles on a variety $ X $ generated by a subvariety of codimension $ p $ is denoted by $ C ^ {p} (X) $. The group $ C(X) $ can be represented as the direct sum

$$ C (X) = \oplus _ { p } C ^ {p} (X) . $$

The subgroup $ C ^ {1} (X) $ is identical with the group of Weil divisors (cf. Divisor) on $ X $.

In what follows $ X $ will denote a non-singular projective algebraic variety of dimension $ n $ over an algebraically closed field $ k $. If $ k $ is the field of complex numbers $ \mathbf C $, then each algebraic cycle $ Z \in C ^ {p} (X) $ defines a $ (2n - 2p) $- dimensional homology class $ [ Z ] \in H _ {2n-2p} (X, \mathbf Z ) $ and, in accordance with Poincaré duality, a cohomology class $ \gamma (Z) \in H ^ {2p} (X, \mathbf Z ) $. The homology (or, respectively, cohomology) classes of type $ [ Z ] $( or $ \gamma (Z) $) 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 $ (2n - 2p) $- dimensional cycle $ \Gamma $ on $ X $ is homologous with an algebraic cycle if and only if the integrals of all closed differential forms of type $ ( 2p - q, q) $, $ q \neq p $, over $ \Gamma $ are equal to zero. This conjecture has only been proved for $ p = 1 $( for $ n = 2 $[6], and for all $ n $[7]), for $ p = n - 1 $, and for isolated classes of varieties [4].

If $ W = \sum n _ {i} W _ {i} $ is an algebraic cycle on the product of two varieties $ X \times T $, then the set of cycles on $ X $ of the form

$$ \sum n _ {i} W _ {i} \cap ( X \times \{ t \} ) $$

is known as a family of algebraic cycles on $ X $ parametrized by the base $ T $. The usual requirement in this connection is that the projection of each subvariety $ W _ {i} $ on $ T $ be a flat morphism. If $ W = W _ {i} $ is defined by an irreducible subvariety, the corresponding family of algebraic cycles on $ X $ is called a family of algebraic subvarieties. In particular, for any flat morphism $ f: X \rightarrow Y $ of algebraic varieties its fibres $ X _ {y} $ form a family of algebraic subvarieties of $ X $ parametrized by the base $ Y $. A second particular case of this concept is that of a linear system. All members of a family of algebraic subvarieties (or, respectively, algebraic cycles) of a projective variety $ X $, parametrized by a connected base, have the same Hilbert polynomial (respectively, virtual arithmetic genus).

Two algebraic cycles $ Z $ and $ Z ^ \prime $ on a variety $ X $ are algebraically equivalent (which is denoted by $ Z \sim _ { \mathop{\rm alg} } Z ^ \prime $) if they belong to the same family, parametrized by a connected base. Intuitively, equivalence of algebraic cycles means that $ Z $ may be algebraically deformed into $ Z ^ \prime $. If this definition includes the condition that the base $ T $ is a rational variety, the algebraic cycles $ Z $ and $ Z ^ \prime $ are called rationally equivalent (which is denoted by $ Z \sim _ { \mathop{\rm rat} } Z ^ \prime $). If $ Z, Z ^ \prime \in C ^ {1} (X) $, 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 $ C _ { \mathop{\rm rat} } (X) $( respectively, $ C _ { \mathop{\rm alg} } (X) $). Each of these groups is a direct sum of its components

$$ C _ { \mathop{\rm rat} } ^ {p} (X) = C _ { \mathop{\rm rat} } (X) \cap C ^ {p} (X) , $$

$$ C _ { \mathop{\rm alg} } ^ {p} (X) = C _ { \mathop{\rm alg} } (X) \cap C ^ {p} (X) . $$

The quotient group $ C ^ {1} (X) / C _ { \mathop{\rm alg} } ^ {1} (X) $ is finitely generated and is called as the Neron–Severi group of the variety $ X $. The problem of the quotient group $ C ^ {p} (X) / C _ { \mathop{\rm alg} } ^ {p} (X) $ being finitely generated for $ p > 1 $ remains open at the time of writing (1977). The quotient group $ C _ { \mathop{\rm alg} } ^ {1} (X) / C _ { \mathop{\rm rat} } ^ {1} (X) $ has the structure of an Abelian variety (cf. Picard scheme). The operation of intersection of cycles makes it possible to define a multiplication in the quotient group $ C(X) / C _ { \mathop{\rm rat} } (X) $, converting it into a commutative ring, called the Chow ring of the variety $ X $( cf. Intersection theory).

For any Weil cohomology theory $ H ^ {*} (X) $ there exists a uniquely defined homomorphism of groups

$$ \gamma : C ^ {p} (X) \rightarrow H ^ {2p} (X) . $$

Two algebraic cycles $ Z $ and $ Z ^ \prime $ are called homologically equivalent (which is denoted by $ Z \sim _ { \mathop{\rm hom} } Z ^ \prime $) if $ \gamma ( Z ) = \gamma ( Z ^ \prime ) $. The subgroup of algebraic cycles that are homologically equivalent with zero is denoted by $ C _ { \mathop{\rm hom} } (X) $. The imbedding $ C _ { \mathop{\rm alg} } ( X ) \subset C _ { \mathop{\rm hom} } ( X) $ is valid. The quotient group $ C( X ) / C _ { \mathop{\rm hom} } ( X ) $ is finitely generated, and is a subring in the ring $ H ^ {*} ( X ) $, which is denoted by $ A ^ {*} ( X ) $ and is known as the ring of algebraic Weil cohomology classes. It is not known (1986) whether or not $ A ^ {*} ( X ) $ depends on the Weil cohomology theory that has been chosen.

Two algebraic cycles $ Z $ and $ Z ^ \prime $ are called $ \tau $- equivalent (which is denoted by $ Z \sim _ \tau Z ^ \prime $) if there exists an $ m \geq 1 $ such that $ m Z \sim _ { \mathop{\rm alg} } m Z ^ \prime $. The subgroup of algebraic cycles that are $ \tau $- equivalent to zero, is denoted by $ C _ \tau (X) $. Two algebraic cycles $ Z $ and $ Z ^ \prime $ from $ C ^ {p} (X) $ are called numerically equivalent (which is denoted by $ Z \sim _ { \mathop{\rm num} } Z ^ \prime $) if the equality $ WZ = W Z ^ \prime $ is valid for any $ W \in C ^ {n-p} ( X ) $, provided both sides of the equality are defined. The subgroup of algebraic cycles numerically equivalent with zero is denoted by $ C _ { \mathop{\rm num} } ( X ) $. The imbeddings

$$ C _ \tau (X) \subset C _ { \mathop{\rm hom} } (X) \subset C _ { \mathop{\rm num} } (X) $$

are valid. For divisors the groups $ C _ \tau ( X ) \cap C ^ {1} ( X ) $, $ C _ { \mathop{\rm hom} } ( X ) \cap C ^ {1} ( X ) $ and $ C _ { \mathop{\rm num} } ( X ) \cap C ^ {1} ( X ) $ are identical [6]. However, in accordance with the counterexample in [5] for the case $ k = \mathbf C $

$$ C _ \tau (X) \neq C _ { \mathop{\rm hom} } (X) , $$

where $ C _ { \mathop{\rm hom} } ( X ) $ is considered with respect to the ordinary cohomology theory with rational coefficients. A similar counterexample was established for a field $ k $ of arbitrary characteristic and for the $ l $- adic theory of Weil cohomology. The question as to the equality of the groups $ C _ { \mathop{\rm hom} } ( X ) $ and $ C _ { \mathop{\rm num} } ( X ) $ has been solved [9].

Let $ X $ be imbedded in a projective space and let $ L _ {X} $ be the cohomology class of a hyperplane section. An algebraic cohomology class

$$ x \in A ^ {p} (X) = A ^ {*} (X) \cap H ^ {2p} (X) $$

is called primitive if $ x L _ {X} ^ {n-p} = 0 $. In such a case, if $ k $ is the field of complex numbers $ \mathbf C $, the bilinear form

$$ ( a , b ) \rightarrow ( - 1 ) ^ {n} L _ {X} ^ {n-2p} ab $$

is positive definite on the subspace of primitive classes in $ A ^ {p} ( X ) $. A similar proposition for arbitrary $ k $, which is closely connected with the Weil conjectures on the zeta-function of an algebraic variety, has been proved for $ n \leq 2 $ only.

If a variety $ X $ is defined over a field $ k $ that is not algebraically closed, the Galois group $ G ( \overline{k}\; / k) $ of the separable algebraic closure of the field $ k $ acts on the Weil cohomology $ H ^ {*} ( \overline{X}\; ) $, where $ \overline{X}\; = X \otimes _ {\overline{k}\; } \overline{k}\; $. Each element of $ A ^ {*} (X) $ is invariant with respect to some subgroup of finite index of the group $ G ( \overline{k}\; /k) $. It is believed (Tate's conjecture on algebraic cycles) that the converse proposition is also true if $ k $ is finitely generated over its prime subfield. Many conjectures on the zeta-function of algebraic varieties are based on this assumption [2].

#### References

[1] | M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902 |

[2] | J.T. Tate, "Algebraic cohomology classes" , Summer school of algebraic geometry Woods Hole, 1964 Zbl 0213.22901 |

[3] | 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 |

[4] | 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 & Masson (1968) pp. 359–386 MR0292838 |

[5] | P.A. Griffiths, "On the periods of certain rational integrals II" Ann. of Math. (2) , 90 : 3 (1969) pp. 496–541 Zbl 0215.08103 |

[6] | S. Lefschetz, "L'analysis situs et la géométrie algébrique" , Gauthier-Villars (1924) MR0033557 MR1520618 |

[7] | W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952) MR0051571 |

[8] | "Groupes de monodromie en géométrie algébrique" M. Raynaud (ed.) D.S. Rim (ed.) A. Grothendieck (ed.) , Sem. Geom. Alg. , 7 , Springer (1972–1973) MR0354656 |

[9] | P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–308 MR0340258 Zbl 0314.14007 Zbl 0287.14001 |

#### Comments

In 1983 H. Clemens proved that $ C ^ {p} (X) / C _ { \mathop{\rm alg} } ^ {p} (X) $ is not finitely generated [a1]. He also proved that $ C _ { \mathop{\rm hom} } (X) / C _ \tau (X) $ is not finitely generated, even after tensoring with the field of rational numbers [a1].

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

Much of the recent progress of the theory of algebraic cycles is related to algebraic $ K $- theory, see [a4].

#### References

[a1] | H. Clemens, "Homological equivalence modulo algebraic equivalence is not finitely generated" Publ. Math. IHES , 58 (1983) pp. 19–38 MR720930 Zbl 0529.14002 |

[a2] | T. Shiado, "What is known about the Hodge conjecture" , North-Holland & Kinokuniya (1983) |

[a3] | M.F. Atiyah, F. Hirzebruch, "Analytic cycles on complex manifolds" Topology , 1 (1961) pp. 25–45 MR0145560 Zbl 0108.36401 |

[a4] | S. Bloch, "Lectures on algebraic cycles" , IV , Dept. Math. Duke Univ. (1980) MR0558224 Zbl 0436.14003 |

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Algebraic cycle.

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