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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Chow groups (named after
Wei-Liang Chow Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician and stamp collector born in Shanghai, known for his work in algebraic geometry. Biography Chow was a student in the US, graduating from the ...
by ) of 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. Mo ...
over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
are algebro-geometric analogs of the
homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...
of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
. The elements of the Chow group are formed out of subvarieties (so-called
algebraic cycles 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 alg ...
) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
, the Chow groups can be interpreted as cohomology groups (compare
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
) and have a multiplication called the
intersection product In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general.


Rational equivalence and Chow groups

For what follows, define a variety over a field k to be an
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
coefficients. (Here and below, subvarieties are understood to be closed in X, unless stated otherwise.) For a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
i, the group Z_i(X) of i-dimensional cycles (or i-cycles, for short) on X is the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
on the set of i-dimensional subvarieties of X. For a variety W of dimension i+1 and any
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
f on W which is not identically zero, the
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 ...
of f is the i-cycle :(f) = \sum_Z \operatorname_Z (f) Z, where the sum runs over all i-dimensional subvarieties Z of W and the integer \operatorname_Z(f) denotes the order of vanishing of f along Z. (Thus \operatorname_Z(f) is negative if f has a pole along Z.) The definition of the order of vanishing requires some care for W singular. For a scheme X of finite type over k, the group of i-cycles rationally equivalent to zero is the subgroup of Z_i(X) generated by the cycles (f) for all (i+1)-dimensional subvarieties W of X and all nonzero rational functions f on W. The Chow group CH_i(X) of i-dimensional cycles on X is the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
of Z_i(X) by the subgroup of cycles rationally equivalent to zero. Sometimes one writes /math> for the class of a subvariety Z in the Chow group, and if two subvarieties Z and W have = /math>, then Z and W are said to be rationally equivalent. For example, when X is a variety of dimension n, the Chow group CH_(X) is the
divisor class group In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumfo ...
of X. When X is smooth over k (or more generally, a locally Noetherian normal scheme ), this is isomorphic to the
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
s on X.


Examples of Rational Equivalence


Rational Equivalence on Projective Space

Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree d, so f,g \in H^0(\mathbb^n, \mathcal O(d)), we can construct a family of hypersurfaces defined as the vanishing locus of sf + tg. Schematically, this can be constructed as : X = \text\left( \frac\right) \hookrightarrow \mathbb^1 \times \mathbb^n using the projection \pi_1: X \to \mathbb^1 we can see the fiber over a point _0:t_0/math> is the projective hypersurface defined by s_0 f + t_0 g. This can be used to show that the cycle class of every hypersurface of degree d is rationally equivalent to d mathbb^/math>, since sf + tx_0^d can be used to establish a rational equivalence. Notice that the locus of x_0^d=0 is \mathbb^ and it has multiplicity d, which is the coefficient of its cycle class.


Rational Equivalence of Cycles on a Curve

If we take two distinct line bundles L, L' \in\operatorname(C) of a smooth projective curve C, then the vanishing loci of a generic section of both line bundles defines non-equivalent cycle classes in CH(C). This is because \operatorname(C) \cong \operatorname(C) for smooth varieties, so the divisor classes of s \in H^0(C, L) and s' \in H^0(C, L') define inequivalent classes.


The Chow ring

When the scheme X is smooth over a field k, the Chow groups form a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, not just a graded abelian group. Namely, when X is smooth over k, define CH^i(X) to be the Chow group of
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the ...
-i cycles on X. (When X is a variety of dimension n, this just means that CH^i(X) = CH_(X).) Then the groups CH^*(X) form a commutative
graded ring In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
with the product: :CH^i(X) \times CH^j(X) \rightarrow CH^(X). The product arises from intersecting algebraic cycles. For example, if Y and Z are smooth subvarieties of X of codimension i and j respectively, and if Y and Z intersect transversely, then the product /math> in CH^(X) is the sum of the irreducible components of the intersection Y\cap Z, which all have codimension i+j. More generally, in various cases,
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
constructs an explicit cycle that represents the product /math> in the Chow ring. For example, if Y and Z are subvarieties of complementary dimension (meaning that their dimensions sum to the dimension of X) whose intersection has dimension zero, then /math> is equal to the sum of the points of the intersection with coefficients called
intersection number In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
s. For any subvarieties Y and Z of a smooth scheme X over k, with no assumption on the dimension of the intersection, William Fulton and Robert MacPherson's intersection theory constructs a canonical element of the Chow groups of Y\cap Z whose image in the Chow groups of X is the product /math>.


Examples


Projective space

The Chow ring of
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
\mathbb P^n over any field k is the ring : CH^*(\mathbb P^n) \cong \mathbf Z (H^), where H is the class of a hyperplane (the zero locus of a single linear function). Furthermore, any subvariety Y of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
d and codimension a in projective space is rationally equivalent to dH^a. It follows that for any two subvarieties Y and Z of complementary dimension in \mathbb P^n and degrees a, b, respectively, their product in the Chow ring is simply : \cdot = a\, b\, H^n where H^n is the class of a k-rational point in \mathbb P^n. For example, if Y and Z intersect transversely, it follows that Y\cap Z is a zero-cycle of degree ab. If the base field k is
algebraically closed In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
, this means that there are exactly ab points of intersection; this is a version of
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
, a classic result of
enumerative geometry In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory. History The problem of Apollonius is one of the earliest examp ...
.


Projective bundle formula

Given a vector bundle E \to X of rank r over a smooth proper scheme X over a field, the Chow ring of the associated projective bundle \mathbb(E) can be computed using the Chow ring of X and the Chern classes of E. If we let \zeta = c_1(\mathcal O_(1)) and c_1,\ldots, c_r the Chern classes of E, then there is an isomorphism of rings : CH^\bullet(\mathbb(E)) \cong \frac


Hirzebruch surfaces

For example, the Chow ring of a
Hirzebruch surface In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by . Definition The Hirzebruch surface \Sigma_n is the \mathbb^1-bundle, called a Projective bundle, over \mathbb^1 associated to the sheaf\mathca ...
can be readily computed using the projective bundle formula. Recall that it is constructed as F_a = \mathbb(\mathcal\oplus\mathcal(a)) over \mathbb^1. Then, the only non-trivial Chern class of this vector bundle is c_1 = aH. This implies that the Chow ring is isomorphic to : CH^\bullet(F_a) \cong \frac \cong \frac


Remarks

For other algebraic varieties, Chow groups can have richer behavior. For example, let X be an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
over a field k. Then the Chow group of zero-cycles on X fits into an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
: 0 \rightarrow X(k) \rightarrow CH_0(X) \rightarrow \mathbf \rightarrow 0. Thus the Chow group of an elliptic curve X is closely related to the group X(k) of k-
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
s of X. When k is a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
, X(k) is called the
Mordell–Weil group In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A defined over a number field K, it is an arithmetic invariant of the Abelian variety. It is simply the group of K-points of A, so A(K) is the Mo ...
of X, and some of the deepest problems in number theory are attempts to understand this group. When k is the complex numbers, the example of an elliptic curve shows that Chow groups can be
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
abelian groups.


Functoriality

For a
proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field '' ...
f: X\to Y of schemes over k, there is a pushforward homomorphism f_*: CH_i(X)\to CH_i(Y) for each integer i. For example, for a
proper scheme In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field ...
X over k, this gives a homomorphism CH_0(X)\to \mathbf Z, which takes a closed point in X to its degree over k. (A closed point in X has the form \operatorname(E) for a finite extension field E of k, and its degree means the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
of the field E over k.) For a
flat morphism In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e., :f_P\colon \mathcal_ \t ...
f: X\to Y of schemes over k with fibers of dimension r (possibly empty), there is a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
f^*: CH_i(Y)\to CH_(X). A key computational tool for Chow groups is the localization sequence, as follows. For a scheme X over a field k and a closed subscheme Z of X, there is an
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...
:CH_i(Z) \rightarrow CH_i(X) \rightarrow CH_i(X-Z) \rightarrow 0, where the first homomorphism is the pushforward associated to the proper morphism Z\to X, and the second homomorphism is pullback with respect to the flat morphism X - Z \to X. The localization sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore) motivic homology groups, also known as higher Chow groups. For any morphism f: X\to Y of smooth schemes over k, there is a pullback homomorphism f^*: CH^i(Y)\to CH^i(X), which is in fact a ring homomorphism CH^*(Y)\to CH^*(X).


Examples of flat pullbacks

Note that non-examples can be constructed using blowups; for example, if we take the blowup of the origin in \mathbb^2 then the fiber over the origin is isomorphic to \mathbb^1.


Branched coverings of curves

Consider the branched covering of curves :f: \operatorname\left( \frac \right) \to \mathbb^1_x Since the morphism ramifies whenever f(\alpha) = 0 we get a factorization :g(\alpha,y) = (y - a_1)^\cdots(y-a_k)^ where one of the e_i>1. This implies that the points \ = f^(\alpha) have multiplicities e_1,\ldots,e_k respectively. The flat pullback of the point \alpha is then :f^*
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , whic ...
= e_1
alpha Alpha (uppercase , lowercase ; grc, ἄλφα, ''álpha'', or ell, άλφα, álfa) is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph , whic ...
+ \cdots + e_k alpha_k/math>


Flat family of varieties

Consider a flat family of varieties :X \to S and a subvariety S' \subset S. Then, using the cartesian square : \begin S'\times_ X & \to & X \\ \downarrow & & \downarrow \\ S' & \to & S \end we see that the image of S'\times_ X is a subvariety of X. Therefore, we have :f^* '= '\times_S X/math>


Cycle maps

There are several homomorphisms (known as cycle maps) from Chow groups to more computable theories. First, for a scheme ''X'' over the complex numbers, there is a homomorphism from Chow groups to Borel–Moore homology: :\mathit_i(X) \rightarrow H_^(X,\mathbf). The factor of 2 appears because an ''i''-dimensional subvariety of ''X'' has real dimension 2''i''. When ''X'' is smooth over the complex numbers, this cycle map can be rewritten using
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
as a homomorphism :\mathit^j(X) \rightarrow H^(X,\mathbf). In this case (''X'' smooth over C), these homomorphisms form a ring homomorphism from the Chow ring to the cohomology ring. Intuitively, this is because the products in both the Chow ring and the cohomology ring describe the intersection of cycles. For a smooth complex
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory,
Deligne cohomology In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordin ...
. This incorporates the
Abel–Jacobi map In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name der ...
from cycles homologically equivalent to zero to the
intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by put ...
. The
exponential sequence In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaf (mathematics), sheaves used in complex geometry. Let ''M'' be a complex manifold, and write ''O'M'' for the sheaf of holomorphic functions on ''M''. Le ...
shows that ''CH''1(''X'') maps isomorphically to Deligne cohomology, but that fails for ''CH''''j''(''X'') with ''j'' > 1. For a scheme ''X'' over an arbitrary field ''k'', there is an analogous cycle map from Chow groups to (Borel–Moore) etale homology. When ''X'' is smooth over ''k'', this homomorphism can be identified with a ring homomorphism from the Chow ring to etale cohomology.


Relation to K-theory

An (algebraic)
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
''E'' on a smooth scheme ''X'' over a field has
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
es ''c''''i''(''E'') in ''CH''''i''(''X''), with the same formal properties as in topology. The Chern classes give a close connection between vector bundles and Chow groups. Namely, let ''K''0(''X'') be the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic i ...
of vector bundles on ''X''. As part of the
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is it ...
, Grothendieck showed that the
Chern character In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
gives an isomorphism :K_0(X)\otimes_\mathbf \cong \prod_i \mathit^i(X)\otimes_\mathbf. This isomorphism shows the importance of rational equivalence, compared to any other
adequate equivalence relation In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined i ...
on algebraic cycles.


Conjectures

Some of the deepest conjectures in algebraic geometry and number theory are attempts to understand Chow groups. For example: *The Mordell–Weil theorem implies that the divisor class group ''CH''''n''-1(''X'') is finitely generated for any variety ''X'' of dimension ''n'' over a number field. It is an open problem whether all Chow groups are finitely generated for every variety over a number field. The
Bloch Bloch is a surname of German origin. Notable people with this surname include: A–F * (1859-1914), French rabbi *Adele Bloch-Bauer (1881-1925), Austrian entrepreneur *Albert Bloch (1882–1961), American painter * (born 1972), German motor journal ...
Kato conjecture on values of L-functions predicts that these groups are finitely generated. Moreover, the rank of the group of cycles modulo homological equivalence, and also of the group of cycles homologically equivalent to zero, should be equal to the order of vanishing of an L-function of the given variety at certain integer points. Finiteness of these ranks would also follow from the Bass conjecture in algebraic K-theory. *For a smooth complex projective variety ''X'', the
Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectu ...
predicts the image ( tensored with the rationals Q) of the cycle map from the Chow groups to singular cohomology. For a smooth projective variety over a finitely generated field (such as a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
or number field), the
Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The c ...
predicts the image (tensored with Q''l'') of the cycle map from Chow groups to
l-adic cohomology In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
. *For a smooth projective variety ''X'' over any field, the
Bloch Bloch is a surname of German origin. Notable people with this surname include: A–F * (1859-1914), French rabbi *Adele Bloch-Bauer (1881-1925), Austrian entrepreneur *Albert Bloch (1882–1961), American painter * (born 1972), German motor journal ...
Beilinson conjecture predicts a filtration on the Chow groups of ''X'' (tensored with the rationals) with strong properties. The conjecture would imply a tight connection between the singular or etale cohomology of ''X'' and the Chow groups of ''X''. :For example, let ''X'' be a smooth complex projective surface. The Chow group of zero-cycles on ''X'' maps onto the integers by the degree homomorphism; let ''K'' be the kernel. If the
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 complex m ...
''h''0(''X'', Ω2) is not zero, Mumford showed that ''K'' is "infinite-dimensional" (not the image of any finite-dimensional family of zero-cycles on ''X''). The Bloch–Beilinson conjecture would imply a satisfying converse, Bloch's conjecture on zero-cycles: for a smooth complex projective surface ''X'' with geometric genus zero, ''K'' should be finite-dimensional; more precisely, it should map isomorphically to the group of complex points of the
Albanese variety In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. Precise statement The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to ...
of ''X''.


Variants


Bivariant theory

Fulton Fulton may refer to: People * Robert Fulton (1765–1815), American engineer and inventor who developed the first commercially successful steam-powered ship * Fulton (surname) Given name * Fulton Allem (born 1957), South African golfer * Fult ...
and MacPherson extended the Chow ring to singular varieties by defining the "
operational Chow ring In mathematics, a bivariant theory was introduced by Fulton and MacPherson , in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring. On technical levels, a bivariant theory is ...
" and more generally a bivariant theory associated to any morphism of schemes. A bivariant theory is a pair of covariant and contravariant
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s that assign to a map a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
and a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
respectively. It generalizes a
cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
, which is a contravariant functor that assigns to a space a ring, namely a
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually und ...
. The name "bivariant" refers to the fact that the theory contains both covariant and contravariant functors. This is in a sense the most elementary extension of the Chow ring to singular varieties; other theories such as
motivic cohomology Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geo ...
map to the operational Chow ring.


Other variants

Arithmetic Chow group In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. Background The main motivation behind Arakelov geometry is t ...
s are an amalgamation of Chow groups of varieties over Q together with a component encoding Arakelov-theoretical information, that is,
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on the associated complex manifold. The theory of Chow groups of schemes of finite type over a field extends easily to that of
algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, w ...
s. The key advantage of this extension is that it is easier to form quotients in the latter category and thus it is more natural to consider
equivariant Chow group In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = /G/math>, the Chow group of ''X'' is the same as the ''G''-equivariant Chow group of ''Y''. A key di ...
s of algebraic spaces. A much more formidable extension is that of
Chow group of a stack In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = /G/math>, the Chow group of ''X'' is the same as the ''G''-equivariant Chow group of ''Y''. A key di ...
, which has been constructed only in some special case and which is needed in particular to make sense of a
virtual fundamental class In mathematics, specifically enumerative geometry, the virtual fundamental class \text_ of a space X is a replacement of the classical fundamental class \in A^*(X) in its chow ring which has better behavior with respect to the enumerative probl ...
.


History

Rational equivalence of divisors (known as
linear equivalence In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier (mathematician), Pierre Cartier ...
) was studied in various forms during the 19th century, leading to the
ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
in number theory and the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...
in the theory of algebraic curves. For higher-codimension cycles, rational equivalence was introduced by
Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algeb ...
in the 1930s. In 1956,
Wei-Liang Chow Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician and stamp collector born in Shanghai, known for his work in algebraic geometry. Biography Chow was a student in the US, graduating from the ...
gave an influential proof that the intersection product is well-defined on cycles modulo rational equivalence for a smooth quasi-projective variety, using
Chow's moving lemma In algebraic geometry, Chow's moving lemma, proved by , states: given algebraic cycles ''Y'', ''Z'' on a nonsingular quasi-projective variety ''X'', there is another algebraic cycle ''Z' '' on ''X'' such that ''Z' '' is rationally equivalent to '' ...
. Starting in the 1970s,
Fulton Fulton may refer to: People * Robert Fulton (1765–1815), American engineer and inventor who developed the first commercially successful steam-powered ship * Fulton (surname) Given name * Fulton Allem (born 1957), South African golfer * Fult ...
and MacPherson gave the current standard foundation for Chow groups, working with singular varieties wherever possible. In their theory, the intersection product for smooth varieties is constructed by deformation to the normal cone.Fulton, Intersection Theory, Chapters 5, 6, 8.


See also

*
Intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...
*
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is it ...
*
Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectu ...
*
Motive (algebraic geometry) In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomo ...


References


Citations


Introductory

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Advanced

* * * * * * * * *{{Citation , author1-first=Claire , author1-last=Voisin , author1-link=Claire Voisin , title=Hodge Theory and Complex Algebraic Geometry (2 vols.) , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, year=2002 , isbn=978-0-521-71801-1 , mr=1997577 Algebraic geometry Intersection theory Topological methods of algebraic geometry Zhou, Weiliang