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 geometrie algebrique" 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