Algebraic Cycles

Anand Sawant

Abstract Algebraic cycles arose in the study of for algebraic varieties. This note, based on a a lecture in the Mathematics Students’ Seminar at TIFR on September 7, 2012 is meant to give an intorduction to algebraic cycles and various adequate equivalence relations on them. We then state the Standard Conjecture D and state some of its consequences including the conjectural theory of motives.

1 Intersection theory and algebraic cycles

Intersection theory deals with the problem of studying intersection of two closed subschemes of a . Perhaps the first result in intersection theory was Bezout’s theorem, which says that given 2 two plane curves of degree m and n in PC having no component in common, their intersection consists of at most mn points (exactly mn points if counted with multiplicity). A systematic study of intersection theory involves the study of the group of algebraic cycles on a scheme X. In this section, we give a brief account of basic definitions and properties of algebraic cycles; for details, the reader is referred to Fulton’s book [1]. For simplicity, all the schemes considered in this note will be of finite type over a field F.

Definition 1.1. An algebraic cycle on a scheme X is an element of the free abelian group on the set of all the closed integral subschemes of X. We shall denote the group of algebraic cycles by Z(X).

Since closed integral subschemes of X are determined uniquely by their generic points, we have

Z(X) = ⊕ Z. ζ∈X

We usually grade this group by dimension, by letting Zp(X) denote the algberaic cycles of dimension p. If X(p) denotes the set of points of X of dimension p, then Zp(X):= ⊕ Z. A point x ∈ X is x∈X(p) said to have codimension p, if the local ring OX,x has Krull dimension p. We may grade Z(X) by codimension as well: writing X(p) for the set of points of X of codimension p, we set Z p(X):= ⊕ Z. If X is equidimensional of dimension n, we have x∈X(p)

p Zn−p(X) = Z (X).

If Z ,→ X is a closed integral subscheme, we denote its class in Z(X) by [Z], and call it a prime cycle.

1 Definition 1.2. An algebraic cycle associated to a closed subscheme Y ,→ X is defined to be

[Y]:= ⊕ `(OY,Yi )[Yi], i

where the sum is taken over all the irreducible components Yi of Y and `(OY,Yi ) denotes the length

of OY,Yi as module over itself. Note that the length is finite, OY,Yi being an artinian ring. For a well-working theory of algebraic cycles, we need to have a notion of intersection product of the cycles associated to two subvarieties Y and Z of a scheme X of dimension n, which will closely reflect the geometry of their intersection Y ∩ Z. Just taking the set-theoretic intersection of subvarieties in question will not work, because subvarieties may occur in bad position - for instance, two parallel lines in a plane or the case of intersecting a subvariety with itself. In case the cycles intersect properly, that is, if dim Y + dim Z − dim Y ∩ Z = n, we can define

[Y] · [Z] = [Y ∩ Z].

In order to take care of the bad cases, we must have an adequate equivalence relation on Z(X) which is broad enough to ensure that given any two cycles on X we can find equivalent cycles intersecting properly such that their intersection is equivalent to the original intersection. The most important such equivalence relation which is essential for the purposes of intersection theory on smooth varieties is rational equivalence.

Definition 1.3 (Rational equivalence). An i-dimensional cycle α on a scheme X is said to be ratio- nally equivalent to zero and we write α ∼rat 0 if there exist (i+1)-dimensional subvarieties V1,... Vr 1 1 of X × P , such that the projection maps π j : V j → P are dominant and we have

Xr α = [V j(0)] − [V j(∞)], j=1

−1 1 1 where [V j(P)] = p∗[π j (P)] for a point P ∈ P . Here p denotes the projection X × P → X. The group of i-dimensional cycles that are rationally equivalent to zero is denoted by Rati(X). We use the notation Rati(X), if the cycles are graded by codimension.

Thus, in other words, an i-dimensional cycle α on a scheme X is rationally equivalent to zero if there is an (i + 1)-dimensional cycle β on X × P1 such that

α = [β(0)] − [β(∞)].

Definition 1.4. The of dimension i-cycles on a scheme X is defined to be

CHi(X):= Zi(X)/Rati(X).

We will henceforth assume that all the schemes considered here are equidimensional and will i write CH (X):= CHn−i(X), where n is the dimension of X. Theorem 1.5. If X is a smooth scheme of dimension n over a field F, then the intersection product n makes CH∗(X) = ⊕ CHi(X) into a commutative ring. i=1

2 This theorem was first proved in the Chevalley seminar for smooth quasiprojective varieties using the moving lemma and was later generalized by Fulton by replacing the moving lemma by using the techniques of reduction to the diagonal and of deformation to the normal cone. We summarize the imporatant properties of Chow groups in the next proposition: Proposition 1.6. (1) If f : Y → X is proper, there is a pushforward map

f∗ : CHp(Y) → CHp(X), for all p. (2) If g : Y → X is flat of relative dimension d, then there is a pullback map

∗ g : CHp(Y) → CHp+d(X), for all p. (3) Let i : Z ,→ X be a closed subscheme and let U = X − Z. Let j : U ,→ X be the inclusion of U into X. Then the sequence

∗ i∗ j CHp(Z) −→ CHp(X) −→ CHp(U) → 0 is exact for all p. (4) If π : E → X is an affine bundle of rank d, then the flat pullback

∗ π : CHp(X) → CHp+d(E), is an isomorphism for all p.

2 Examples of adequate equivalence relations

We now see examples of some other adequate equivalence relations on algebraic cycles on smooth projective varieties over an algebraically closed field k. We shall denote the category of smooth projectivve varieties by SmProj/k.

Definition 2.1 (Algebraic equivalence). We say that α ∈ Zi(X) is algebraically equivalent to zero and write α ∼alg 0 if there exists a smooth curve C, points c1, c2 ∈ C and a cycle β ∈ Zi+1(X × C) such that α = [β(c1)] − [β(c2)].

The subgroup of Zi(X) generated by cycles algebraically equivalent to zero is denoted by Algi(X). We use the notation Algi(X), if the cycles are graded by codimension.

Clearly, it can be seen that Rati(X) ⊆ Algi(X). The inclusion is proper in general. We next introduce homological equivalence of algebraic cycles. This depends on the notion of a Weil theory, which we define next. Definition 2.2. A Weil cohomology theory is a contravariant functor

H∗ : SmProj/k → Graded K-algebras,

∗ i for a field K such that if X ∈ SmProj/k is of dimension n, then H (X) = ⊕iH (X) satisfies the following properties:

3 1. Hi(X) is a finite dimensional K- for each i. 2. Hi(X) = 0 for i < 0 and for i > 2n. 3. H2n ' K. 4. (Poincare´ duality) There exists a nondegenerate pairing Hi(X) × H2n−i(X) → H2n(X) ' K, for each i. 5. (Kunneth¨ isomorphism) There is a canonical isomorphism ∼ H∗(X) ⊗ H∗(Y) −→ H∗(X × Y), induced by the projection maps. 6. (Cycle map) There exists a group homomorphism ∗ γX : Z(X) → H (X) that is functorial with proper pushforwards and flat pullbacks, respects products and direct sums and such that it coincides with the canonical map Z → K in the case X = Spec k. 7. (Weak Lefschetz property) For any smooth hyperplane section j : W ,→ X (that is, W = X∩H for some hyperplane H), j∗ : Hi(X) → Hi(W) is an isomorphism for i ≤ n − 2 and a monomorphism for i ≤ n − 1.

8. (Hard Lefschetz property) For any smooth hyperplane section j : W ,→ X, let w = γX([W]) ∈ H2(X). Let L : Hi(X) → Hi+2(X) be the Lefschetz operator given by x 7→ x ∪ w. Then Li : Hn−i(X) → Hn+1(X) is an isomorphism for i = 1,..., n. The classical examples of Weil cohomology theories include the singular (Betti) cohomology, , l-adic cohomology, crystalline cohomology and Hodge cohomology. We are now set ot define the adequate equivalence relation of homological equivalence. Definition 2.3 (Homological equivalence). Fix a Weil cohomology theory H∗ on SmProj/k. A cycle α ∈ Zi(X) is said to be homologically equivalent to zero and we write α ∼hom 0 if γX(α) = 0. The subgroup of Zi(X) generated by cycles homologiically equivalent to zero is denoted by Homi(X). We use the notation Homi(X), if the cycles are graded by codimension.

It is a fact that Algi(X) ⊆ Homi(X) for each i. The inclusion can be proper, as was shown first by Griffiths and later by Clemens and others. (Griffiths showed that Hom2(X)/ Alg2(X) is nonzero 4 and actually has an element of infinite order, if X is a generic hypersurface of degree 5 in PC.) We now define the coarsest of the adequate equivalence relations. Recall that the degree map h·i : Z(X) → Z is defined by

 P P n  m j, if ξ = m j x j ∈ Z (X); hξi =  j j  0, otherwise.

4 Definition 2.4 (Numerical equivalence). Let dim X = n. We say that α ∈ Zi(X) is numerically n−i i equivalent to zero and we write α ∼num 0, if hα · βi = 0, for all β ∈ Z (X). The subgroup of Z (X) generated by cycles numerically equivalent to zero is denoted by Numi(X). If X is equidimensional, n−i we write Numi(X) = Num (X). Note that for each i, we have Homi(X) ⊆ Numi(X); this follows from the fact that the cycle class ∗ ∗ map γX induces a ring homomorphism CH (X) → H (X). To see this, let ∆X : X → X × X be the diagonal; then for any α ∈ Zi(X) and any β ∈ Zn−i(X), we have

∗ ∗ γX(α · β) = γX∆X(α × β) = ∆X(γX(α) ⊗ γX(β)) = γX(α) · γX(β), as desired. Grothendieck’s Standard Conjecture D says that up to torsion, homological and numerical equivalence coincide.

i i Standard Conjecture D: For each i, we have Hom (X) ⊗Z Q = Num (X) ⊗Z Q.

Note that the Standard Conjecture D implies that homological equivalence is independent of the choice of a Weil cohomology theory!

Remark 2.5. The standard conjecture is known to hold for divisors, that is, the case i = 1. If k = C, then the Standard Conjecture D is implied by the . Thus, by virtue of this, the Standard Conjecture D is true for abelian varieties in characteristic 0.

Remark 2.6. The standard conjectures on algebraic cycles were first stated by Grothendieck in 1968 (see [2]). Grothendieck’s motivation for them was that the celebrated were a consequence of the standard conjectures. For an account of these, see the articles [6] and [4] by Kleiman. For an account of motives, see [5].

3 Motives

Let X be a smooth complex projective variety. We then have the following relationship between the classical Weil cohomology theories:

i i HdR(X, C) ' Hsing(X, Z) ⊗ C, and i i i Hsing(X, Z) ⊗ Ql ' Hsing(X, Ql) ' H´et(X, Ql). This suggests that there should exist a universal cohomology theory of which these cohomology theories are realizations. This is the basis of Grothendieck’s conjectural theory of motives. The category of pure motives M (k), according to Grothendieck, is supposed to be a Q-linear admitting a functor from SmProj/k such that any Weil cohomology theory factors through M (k). SmProj/k / M (k) QQ QQQ QQQ ∃ ! H∗ QQQ QQ(  Graded K-algebras

5 The category SmProj/k itself is not additive as it does not have enough morphisms. An attempt to linearize it by adding a few more morphisms can be made by regarding a morphism f : X → Y as a cycle on X × Y of dimension dim(X) given by the graph Γ f of f . Definition 3.1. A correspondence of degree r from X to Y is defined to be a cycle on X × Y of dimension dim(X) + r. We write

Corrr(X, Y) = Zdim(X)+r(X × Y).

0 0 For a morphism f : X → Y, its graph Γ f ∈ Corr (X, Y). Any α ∈ Corr (X, Y) defines a homomorphism H∗(Y) → H∗(X) by

∗ y 7→ pX∗(pY y ∪ γX×Y (α)). Let ∼ be an adequate equivalence relation and define

0 0 Corr∼(X, Y):= Corr (X, Y)/ ∼ .

0 0 Let Corr∼(X, Y)Q := Corr∼(X, Y) ⊗ Q. We can now define a category Corr(k) by taking an ob- 0 jects h(X) for each X ∈ SmProj/k, and defining HomCorr(k)(h(X), h(Y)) := Corr∼(X, Y)Q. The cor- 0 0 respondences are composed as follows: if α ∈ Corr∼(X, Y) and β ∈ Corr∼(Y, Z), then β ◦ α = ∗ ∗ pXZ∗ (pXY (α) · pYZ(β)). This gives us a preadditive category. e f f We now take the pseudo-abelian envelope of this. Define a category M∼ (k) by taking for 0 objects h(X, e), where X ∈ SmProj(k) and e ∈ Corr∼(X, X) is an idempotent, that is, e ◦ e = e. Define the morphisms by setting

0 Hom e f f (h(X, e), h(Y, f )) := f ◦ Corr (X, Y)Q ◦ e. M∼ (k) ∼

e f f This makes M∼ (k) a pseudo-abelian category. e f f Definition 3.2. M∼ (k) is called the category of effective motives for ∼ over k. The category of pure motives for ∼ is now obtained by adding twists. This will make every object in the category to have a dual.

Definition 3.3 (Pure motives). The category M∼(k) whose objects are triples h(X, e, m), where h(X, e) is as above, m ∈ Z, and the morphisms are defined by

n−m HomM∼(k)(h(X, e, m), h(Y, f, n)) := f ◦ Corr∼ (X, Y)Q ◦ e. is a rigid pseudo-abelian category. It is called the category of pure motives for ∼ over k.

We end by enlisting some of the known properties of M∼(k).

• M∼(k) has direct sums and tensor products.

• If ∼= Num, then Hom-sets in M∼(k) ae finite dimensional Q-vector spaces.

• A remarkable theorem Jannsen asserts that M∼(k) is a semisimple abelian category if and only if ∼= Num. (Thus, M∼(k) is the only possible candidate for motives. But there are no realization functors if the Standard Conjecture D does not hold!)

6 We end by indicating how one can get the realization functors if ∼ is finer than or equal to homological equivalence. Fix a Weil cohomology theory H∗. We indicate how to construct a functor

M∼(k) → Graded K-algebras.

m dim(X)+m Let M = (X, p, m) ∈ M∼(k) be a pure motive. Note that p ∈ Corr∼ (X, X) = CH (X × X) is an idempotent. Its cycle class γX(p) is an element of the group

H2 dim(X)+2m(X × X) ' ⊕ H2 dim(X)−i(X) ⊗ Hi+2m(X) i ' ⊕ Hi+2m(X)∗ ⊗ Hi+2m(X) i ' ⊕ Hom(Hi+2m(X), Hi+2m(X)) i by Poincare´ duality and the Kunneth¨ isomorphism. We thus obtain the induced maps

p : Hi+2m(X) → Hi+2m(X).

Define Hi(M) to be the image of p inside Hi+2m(X) and set

H∗(M):= ⊕ Hi(M). i

References

[1] Fulton, W.: Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 2. Springer-Verlag, Berlin, 1998.

[2] Grothendieck, A.: Standard conjectures on algebraic cycles. Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press, p. 193 – 199.

[3] Jannsen, U.: Motives, numerical equivalence, and semi-simplicity. Invent. Math. 197(1992), p. 447 – 452.

[4] Kleiman, S.: Algebraic cycles and the Weil conjectures. In: Dix exposes´ sur la cohomologie des schemas,´ North-Holland 1968, p. 359 – 386.

[5] Kleiman, S.: Motives. Algebraic Geometry, Oslo 1970 (edited by F. Oort), Walters-Noordhoff, Groningen, 1972, p. 53 – 82.

[6] Kleiman, S.: The standard conjectures. Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics 55, American Mathematical Society, p. 3 – 20.

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