Classical motives

A. J. Scholl

Introduction

This paper is based on a talk given on the first day of the conference, entitled “Examples”. Its purpose was to give an elementary survey of some examples of classical motives from a purely geometric standpoint (that is, without recourse to any cohomological methods). This report has the same aims and consequent shortcomings. There are four main parts:

(i) An account of the definitions and basic properties of motives. This is included for completeness; it can all be found, in a somewhat different form but in greater detail, in the definitive accounts of Kleiman [12] and Manin [19].

(ii) The relation between the motive of a curve and its Jacobian variety, due in essence to Weil.

(iii) The motives h^{1}(X) and h^{2d−1}(X) for a variety X of dimension d. Here we follow Murre [23], with minor modifications.

(iv) An elementary proof of the canonical decomposition of the motive of an abelian variety, inspired by the work of Deninger and Murre [6] and Künnemann [15, 16].

There is therefore little which is original contained in these pages. I have given a more or less complete proof of Murre’s result in §4, so as to make the comparison between the different decompositions in 5.3. Also included is a proof (3.5) of the unsurprising fact that the category of motives constructed using rational equivalence of cycles is in general not an abelian category. Otherwise proofs have been sketched or omitted.

A word about the notion of motive used in this paper is appropriate. Grothendieck’s definition of a motive involves replacing the category of varieties by a category with the same objects, but whose morphisms are correspondences, modulo a suitable equivalence relation. Depending on the equivalence relation chosen, one gets rather different theories (see 6.2 below for a discussion).

It is usual to take numerical or homological equivalence, obtaining motive categories M_{k}^{num} and M_{k}^{hom}, but in this report we concentrate more on M_{k}^{rat}, the category of motives for rational equivalence (sometimes called Chow motives). One reason to do this is that rational equivalence is the finest adequate equivalence on cycles, so that M_{k}^{rat} is in some sense universal.

Another reason is that many interesting cohomology-like functors factor through M_{k}^{rat} but not in general through M_{k}^{num} or M_{k}^{hom}. Examples of such functors include absolute ℓ-adic cohomology H^{∗}(X_{/k},Q_{ℓ}(n)), Deligne-Beilinson (absolute Hodge) cohomology, and motivic cohomology (see Nekovář’s paper in these Proceedings for details of both of these). In particular, to formulate Beilinson’s conjectures for motives it is at present necessary to work in M_{k}^{rat}.

The obvious disadvantage of using rational equivalence is that of not being in an abelian (or even conjecturally abelian) category. Arguments that are trivial when one uses numerical or homological equivalence can become cumbersome. Philosophically, one has to study M_{k}^{rat} as well as M_{k}^{hom} for the same reasons that in cohomology it is often necessary to work with complexes (i.e., in the derived category) rather than simply with cohomology groups. Unfortunately we do not yet have the whole derived category of motives to play with.

The author is grateful to J. Murre for pointing out an error in an earlier version of section 4 of this paper.

1. Formal properties of motives

1.1. We fix a base field k. Let V_{k} denote the category of smooth and projective k-schemes. We refer to the objects of V_{k} as simply varieties—note that they are not necessarily irreducible or even equidimensional. However we will only consider connected varieties from §3 onwards. If ϕ:Y → X is a morphism in V_{k}, we denote by Γ_{ϕ} ⊂ X ×Y its graph. 1.2. For a variety X and an integer d, the cycle group Z^{d}(X) is the free abelian group generated by irreducible subvarieties of X of codimension d. Central to the definition of motives is the choice of an adequate equivalence relation ∼ on cycles. For a precise definition of what that entails we refer to [12]; there are three important examples of adequate equivalence relations:

(i) rational equivalence; (ii) homological equivalence, with respect to a (fixed) Weil cohomology theory H^{∗}; (iii) numerical equivalence.

We write (with apologies for the notation) A^{d}(X) = Z^{d}(X)⊗Q/ ∼, where ∼ is a fixed adequate equivalence relation. If Z is a cycle on X we write [Z] for its class in A^{d}(X). The definition of adequate implies that the groups A^{d}(X) enjoy a number of functorial properties:—

• For a morphism ϕ: X → Y there are pullback and push-forward maps ϕ^{∗}: A^{∗}(Y) → A^{∗}(X), ϕ_{∗}:A^{∗}(X) → A^{∗+dim Y −dim X}(Y).

• There is a product structure A^{d}(X)⊗A^{e}(X) → A^{d+e}(X) given by intersection theory. In this report we will be mostly concerned with the case when ∼ is rational equivalence, in which case A^{d}(X) is the usual codimension d Chow group tensored with Q. The motives arising from this choice are sometimes called Chow motives. For further comments on the effect of choosing a different equivalence relation, see §§3.5, 6.2 below. 1.3. Let X, Y be in V_{k}. Define Corr^{r}(X, Y), the group of correspondences of degree r from X to Y, as follows. If X is purely d-dimensional, then

Corr^{r}(X,Y) = A^{d+r}(X ×Y).

In general, let X = ∐X_{i} where each X_{i} is a connected variety, and set

Corr^{r}(X,Y) = ⨁Corr^{r}(X_{i},Y) ⊂ A^{∗}(X ×Y).

For varieties X, Y and Z there is a composition of correspondences

Corr^{r}(X,Y)⊗Corr^{s}(Y,Z) → Corr^{r+s}(X,Z)

defined by

f ⊗g ↦ g ∘f = p_{13∗}(p_{13}^{∗}f ·p_{23}^{∗}g)

where p_{ij} are the projections of X ×Y ×Z onto products of factors. 1.4. The category M_{k} of k-motives is now defined as follows: an object of M_{k} is a triple (X,p,m) where X is a k-variety, m is an integer and p = p^{2} ∈ Corr^{0}(X,X) is an idempotent. If (X,p,m) and (Y,q,n) are motives, then

Hom_{M_{k}}((X,p,m),(Y,q,n)) = pCorr^{n−m}(X,Y)q ⊂ Corr^{∗}(X,Y)

and composition is given by composition of correspondences.

This is the definition of the category of motives as given, for example, in [9]. It is equivalent to the definition found in other places (for example [19] and [12]) because of the following elementary fact:

Lemma 1.5. Let p, q be commuting endomorphisms of an abelian group B, which are idempotents. Then the map x ↦ pqx gives an isomorphism

\frac{ker(p−q)}{kerp∩kerq} \underrightarrow{∼} pqB.

Theorem 1.6. M_{k} is an additive, Q-linear category, which is pseudoabelian.

If (X,p,m) and (Y,q,n) are motives with m = n then their direct sum is defined to be

(X,p,m)⊕(Y,q,m) =^{def}(X ∐Y,p⊕q,m).

It is immediate that this satisfies the necessary properties. The general construction of the direct sum is given in §1.14 below; the reader can check that the proof is not circular.

Now recall that an additive category such as M_{k} is pseudoabelian if every projector f ∈ EndM has an image, and the canonical map Im(f)⊕Im(1−f) → M is an isomorphism. In this case it follows formally from the definition that there is a decomposition

M = (X,pfp,m)⊕(X,p−pfp,m)

if M = (X,p,m) is a motive and f = pfp ∈ EndM is a projector. 1.7. Remark. One should bear in mind that the category M_{k} is in general not abelian (see 3.5 below), and some caution must therefore be exercised when discussing kernels of arbitrary morphisms. In view of this it is worth making the following trivial observation:

If f:M → N is a morphism in M_{k} (or any pseudoabelian category) which has a left inverse, then it has an image, which is (non-canonically!) a direct factor of N, and M → Imf is an isomorphism.

In fact if g is any such retract of f, then fg is an idempotent, so fgN ⊂ N exists, and f:M \underrightarrow{∼} fgN. Dually, if f has a section, then N is canonically a quotient object of M, and non-canonically a direct factor of M. 1.8. There is a functor

h:V_{k}^{opp} → M_{k}

which on objects is given by h(X) = (X,id,0), and on morphisms ϕ:Y → X by

h(ϕ) = [Γ_{ϕ}] ∈ Corr^{0}(X,Y) = Hom(h(X),h(Y))

(usually one writes ϕ^{∗} for h(ϕ)). 1.9. There is a tensor product on M_{k}, defined on objects by

(X,p,m)⊗(Y,q,n) = (X ×Y,p⊗q,m+n)

and on morphisms by

\begin{matrix}p_{1}f_{1}q_{1} ⊗p_{2}f_{2}q_{2} = & (p_{1} ⊗p_{2})(f_{1} ⊗f_{2})(q_{1} ⊗q_{2}) ∈ Corr^{∗}(X_{1} ×X_{2},Y_{1} ×Y_{2}) \\ if p_{i}f_{i}q_{i}:(X_{i},p_{i},m_{i}) → (Y_{i},q_{i},n_{i}). \end{matrix}

One writes 11 = (Spec k, id, 0) (the unit motive) and 𝕃 = (Spec k, id, −1) (the Lefschetz motive). Then 11 is the identity for the tensor product, and every motive is a direct factor of h(X)⊗𝕃^{⊗n} for suitable X and n; in fact, if p ∈ Corr^{0}(X,X) = Endh(X) is a projector, then

(X,p,m) = ph(X)⊗𝕃^{⊗−m} ⊂ h(X)⊗𝕃^{⊗−m}.

In future we write simply 𝕃^{n} for 𝕃^{⊗n}, and for a morphism f:M → N of motives we will write f also for the tensor product M ⊗𝕃^{⊗n} → N ⊗𝕃^{⊗n}.

The diagonal Δ: X → X × X defines a product structure on h(X), given by the composite

m_{X}:h(X)⊗h(X) = h(X ×X)⟶^{Δ^{∗}}h(X).

1.10. If ϕ:Y → X and X and Y are purely d- and e-dimensional, respectively, then the transpose [^{t}Γ_{ϕ}] ∈ A^{d}(Y ×X) is a correspondence of degree d−e from Y to X, and so defines a morphism

ϕ_{∗}:h(Y) → h(X)⊗𝕃^{e−d}.

Suppose that d = e and that ϕ is generically finite, of degree r. Then the composite ϕ_{∗} ∘ϕ^{∗} ∈ Endh(X) is multiplication by r. In fact,

ϕ_{∗} ∘ϕ^{∗} = p_{13∗}(p_{13}^{∗}[Γ_{ϕ}]·p_{24}^{∗}[^{t}Γ_{ϕ}]) = p_{13∗}(ϕ,id,ϕ)_{∗}[Y] = r[Δ_{X}]

where Δ_{X} ⊂ X ×X is the diagonal.

If f ∈ Corr^{0}(X,Y) = Hom(h(X),h(Y)) and ϕ:X → X′, ψ:Y → Y′ are morphisms in V_{k} then the formula for the composite map of motives is

ψ_{∗} ∘f ∘ϕ^{∗} = (ϕ×ψ)_{∗}f.

Similarly if ϕ:X′ → X and ψ:Y′ → Y then

ψ^{∗} ∘f ∘ϕ_{∗} = (ϕ×ψ)^{∗}f.

Because of the inherent confusion in formulae of this type, we will attempt to distinguish between operations on cycles (direct and inverse image and intersection) and on correspondences by frequent use of the usual symbols · and ∘. In particular, the notation c^{2} will generally denote c·c and not c∘c. 1.11. Suppose that X is irreducible of dimension d, and that there is a k-rational point x ∈ X(k). Denote by α:X → Speck the structural morphism. Then x^{∗} ∘α^{∗} = id, so by the remark in 1.7, the map

α^{∗}:11 → h(X)

is a subobject of h(X). Similarly, as α_{∗} ∘x_{∗} = id, the map

α_{∗}:h(X) → 𝕃^{d}

is a quotient object.

More generally, let X be irreducible, and write k′ = Γ(X,O_{X}), and α:X → Speck′ for the structural morphism. Let k′′/k′ be a finite separable extension such that there exists x ∈ X(k′′). Write γ:Speck′′ → Speck′ for the natural map. Then by 1.10, the composite

h(Speck′)⟶^{α^{∗}}h(X)⟶^{x^{∗}}h(Speck′′)⟶^{γ_{∗}}h(Spec(k′)

is multiplication by [k′′:k′]. Therefore

α^{∗}:h(Speck′) → h(X)

defines a subobject of h(X), denoted h^{0}(X). It is well-defined as a subobject up to unique isomorphism.

We denote the quotient of h(X) by h^{0}(X) as h^{≥1}(X). The quotient exists because h^{0}(X) is (non-canonically) a direct factor of h(X). The choice of a point x determines a splitting h(X) = h^{0}(X)⊕h^{≥1}(X). Remark: When ∼ is homological or numerical equivalence, the class of Γ_{x} in A^{∗}(X ⊗_{k′} k′′) is independent of x, and so the splitting is in this case canonical.

In particular, if X is absolutely irreducible, then h^{0}(X) = 11 is a direct summand of h(X). We can use this to eliminate the nuisance of dealing with varieties with components of different dimension as follows:

Proposition 1.12. Any motive M can be expressed as a direct factor of some h(X′)⊗𝕃^{n}, with X′ equidimensional.

Proof. It is enough to show this for M = h(X). Let X = ∐X_{i} be the decomposition of X into its components. Choose integers d_{i} ≥ 0 such that dimX_{i} + d_{i} does not depend on i. We then have

h(X) = ⨁h(X_{i}) = ⨁(h(X_{i})⊗h^{0}(P^{d_{i}}))

and we have seen that this is a direct factor of ⨁h(X_{i} ⊗P^{d_{i}}) = h(∐X_{i} ×P^{d_{i}}). 1.13. Continuing with the assumptions of 1.11, let d = dimX. Then the composite

h(Speck′)⊗𝕃^{d}⟶^{x_{∗}γ^{∗}}h(X)⟶^{α_{∗}}h(Speck′)⊗𝕃^{d}

is multiplication by [k′′:k′], and so

α_{∗}:h(X) → h(Speck′)⊗𝕃^{d}

is a quotient object of h(X), denoted h^{2d}(X).

If X is irreducible and has a rational point x ∈ X(k) then

\begin{aligned}h^{0}(X) & ≃ (X,\{x\}×X,0) ≃ 11 \\ h^{2d}(X) & ≃ (X,X ×\{x\},0) ≃ 𝕃^{d} \end{aligned}

Assuming only that X is irreducible, let Z be any zero-cycle on X whose degree d is positive. Then p_{0} = (1/d)[Z×X] ∈ A^{d}(X ×X) is an idempotent, and gives rise to canonical isomorphisms

h^{0}(X) ≃ (X,p_{0},0), h^{2d}(X) ≃ (X,p_{2d},0) where p_{2d} =^{t}p_{0}.

For example, consider X = P^{1}. Since the cycles P^{1} × {x}, {x} × P^{1} on P^{1} × P^{1} do not depend on the choice of a rational point x ∈ P^{1}(k), and since their sum is rationally equivalent to the diagonal, we have canonically

h(P^{1}) = h^{0}(P^{1})⊕h^{2}(P^{1}) = 11⊕𝕃.

In most treatments this is taken as the definition of 𝕃. 1.14. We can now construct arbitrary direct sums in M_{k}. Let M = (X, p, m) and N = (Y,q,n) be motives. Assume that m < n. Then

M = (X,p,n)⊗L^{n−m} = (X,p,n)⊗h^{2}(P^{1})^{n−m} = (X ×(P^{1})^{n−m},p′,n)

for a suitable projector p′, and the direct sum of M and N is then

(X ×(P^{1})^{n−m} ∐Y,p′ ⊕q,n)

as in 1.6 above. 1.15. There is an involution ^{∨}:M_{k}^{opp} → M_{k}, defined on objects by

(X,p,m)^{∨} = (X,^{t}p,d−m) if X is purely d-dimensional

and on morphisms as the transpose of correspondences. In particular h(X)^{∨} = h(X)⊗𝕃^{−d} (“Poincaré duality”). Clearly M^{∨}^{∨} = M for every M, and the standard formula

Hom(M ⊗N,P) = Hom(M,N^{∨} ⊗P)

is trivially seen to hold. Then one can define an internal Hom in M_{k} by the formula Hom(M, N) = M^{∨} ⊗ N. These constructions give M_{k} the structure of a rigid additive tensor category, once commutativity and associativity constraints are defined. (This tensor structure gives what in the Tannakian setting is usually called the false category of motives—see [5], p.200ff.) 1.16. It is sometimes convenient to construct a decomposition of motives over an extension field (this is needed in §4, for instance). Because of what was said in 1.10, the groups A^{∗} satisfy Galois descent: if k′/k is a finite Galois extension then A^{∗}(X) is the subspace of invariants of A^{∗}(X ⊗ k′) under Gal(k′/k). Therefore Galois-invariant decompositions of motives descend to the ground field: more precisely, we have the following easy result.

Lemma 1.17. Let X ∈ V_{k} be purely d-dimensional, and let k′/k be a finite Galois extension of degree m. Write X′ = X ⊗_{k} k′, and denote by β the canonical map X′ ×_{k′} X′ → X ×_{k} X. Suppose that p′_{1},..., p′_{r} ∈ A^{d}(X′ ×_{k′} X′) is a complete system of orthogonal idempotents, which are invariant under Gal(k′/k). Then the correspondences p_{i} = (1/m)β_{∗}(p′_{i}) form a complete system of orthogonal idempotents in A^{d}(X ×_{k} X).

2. Cycles and Manin’s identity principle

2.1. An immediate consequence of the definition of motives is that the cycle class groups A^{∗} can be interpreted in terms of M_{k}: in fact

A^{d}(X) = Hom(𝕃^{d},h(X)).

If ξ ∈ A^{d}(X), then we write ξ_{∗}:𝕃^{d} → h(X) for the corresponding mapping of motives, and ξ^{∗}:h(X) → 𝕃^{dim X−d} for its transpose. (If ξ is the class of a rational point x ∈ X(k), this then agrees with the notations x_{∗} and x^{∗}.) There is then defined a morphism ξ̄:h(X)⊗𝕃^{d} → h(X) (“cup-product with ξ”) by

h(X)⊗𝕃^{d}⟶^{id ⊗ξ_{∗}}h(X)⊗h(X)⟶^{m_{X}}h(X).

The morphism ξ̄ is represented by the cycle class

Δ_{∗}(ξ) ∈ Corr^{d}(X,X) ⊂ A^{∗}(X ×X).

If η is another cycle class then

ξ̄∘η̄= \overline{ξ ·η} and ξ̄∘η_{∗} = (ξ ·η)_{∗}.  (2.1.1)

In possible conflict with the convention of 1.10 the notation ξ̄^{i} will mean the correspondence induced by ξ^{i}, which is the i-th iterate of ξ̄ and not the i-fold intersection. We define for any motive M and d ∈ Z the cycle groups of M by

A^{d}(M) =^{def} Hom(𝕃^{d},M).

Then A^{∗}(−) is a Z-graded additive functor from M_{k} to Vect_{Q}, the category of Q-vector spaces. For M = h(X) this therefore agrees with the previous notation.

2.2. If M and N ∈ M_{k} then

Hom(N,M) = Hom(11,M ⊗N^{∨}) = A^{0}(M ⊗N^{∨}).

Therefore by the Yoneda lemma the functor which attaches to M ∈ M_{k} the functor A^{0}(M ⊗−):M_{k} → Vect_{Q} is fully faithful.

Now any N ∈ M_{k} is a direct factor of some h(Y) ⊗ 𝕃^{n} with Y ∈ V_{k} and n ∈ Z, and A^{0}(M ⊗ h(Y) ⊗ 𝕃^{n}) = A^{−n}(M ⊗ h(Y)). Therefore if we denote by ω_{M}:V_{k}^{opp} → Vect_{Q} the functor

ω_{M}(Y) = A^{∗}(M ⊗h(Y))

then M ↦ ω_{M} is fully faithful. From this one can deduce (using for example [20], II.7.1):

2.3 (Manin’s identity principle). (i) Let f, g: M → N be morphisms of motives. Then f is an isomorphism if and only if the induced map

ω_{f}(Y):A^{∗}(M ⊗h(Y)) → A^{∗}(N ⊗h(Y))

is an isomorphism for every Y ∈ V_{k}; and f = g if and only if ω_{f}(Y) = ω_{g}(Y) for every Y.

(ii) A sequence 0⟶M′⟶^{f}M⟶^{g}M′′⟶0 in M_{k} is exact if and only if, for every Y ∈ V_{k}, the sequence

0⟶A^{∗}(M′ ⊗h(Y)) ⟶^{ω_{f} (Y)}A^{∗}(M ⊗h(Y)) ⟶^{ω_{g}(Y)}A^{∗}(M′′ ⊗h(Y))⟶0

is exact.

2.4. As a first example we calculate the motive of a projective bundle following Manin [19]. Let S ∈ V_{k} and let E be a locally free sheaf on S of constant rank r + 1 ≥ 1. Let X = P_{S}[E]⟶^{π}S be the associated projective bundle, and ξ = c_{1}(O_{X}(1)) ∈ A^{1}(X) be the divisor class of the tautological line bundle on X.

Recall that A^{∗}(X) is a free module over A^{∗}(S) (via π^{∗}) with basis 1, ξ,...,ξ^{r}, and that the multiplication is given by

ξ^{r+1} = ∑_{j=0}^{r}(−1)^{r−j}c_{r−j+1}(E)ξ^{j}

where c_{i}(E) ∈ A^{i}(S) are the Chern classes of E. For any integer n ≥ 0, write ξ^{n} = ∑_{j=0}^{r} θ_{n,j}ξ^{j}. Then θ_{n,j} ∈ A^{n−j}(S) are given by certain universal polynomials in the Chern classes.

Theorem 2.5. The map

∑_{i=0}^{r} ξ̄^{i} ∘π^{∗}:⨁_{i=0}^{r}h(S)⊗𝕃^{i} ⟶ h(X)

is an isomorphism of motives. In terms of this isomorphism, the product structure on h(X) is given by

(h(S)⊗𝕃^{i})⊗(h(S)⊗𝕃^{n−i})⟶^{m_{S}}h(S)⊗𝕃^{n}⟶^{(θ̄_{n,j})} ⨁_{j=0}^{r}h(S)⊗𝕃^{j}

Proof. Manin’s identity principle implies that it is equivalent to know that the corresponding mapping at the level of cycle class groups

\begin{matrix}⨁_{i=0}^{r}A^{∗−i}(S ×Y) & ⟶ A^{∗}(X ×Y) \\ (z_{i}) & ↦ ∑_{i=0}^{r}ξ^{i} ·(π ×id_{Y})^{∗}(z_{i}) \end{matrix}

is a ring isomorphism for every Y ∈ V_{k}. But since the Chern classes of pr_{1}^{∗}E on S ×Y are simply pr_{1}^{∗}c_{i}(E), this follows from the facts recalled in 2.4 (replacing X/S by X×Y/S×Y). 2.6. Similar results hold for other varieties which admit cellular decompositions relative to a base (grassmanians, etc...). Suppose that π:X → S is a flat morphism of pure relative dimension n, and that X admits a filtration by closed subschemes X = X_{0} ⊃ X_{1} ⊃ ... such that X_{i} − X_{i+1} is S-isomorphic to the affine space A_{S}^{n−d_{i}}, for some d_{i} ∈ Z. Then there is an isomorphism

⨁_{i}A^{∗−d_{i}}(S) ⟶ A^{∗}(X)  (2.6.1)

which is functorial with respect to cartesian squares

\begin{matrix}X′ & ⟶ & X \\ ↓_{π′} & & ↓^{π} \\ S′ & ⟶ & S \end{matrix}

and so by Manin’s identity principle h(X) ≃ ⨁ h(S) ⊗ 𝕃^{d_{i}}. See [14], particularly the appendix, for a proof of (2.6.1) and further examples. Calculations for many of the classical varieties of this type can be found already in SGA6. 2.7. The identity principle can also be used to calculate the motive of a blowup. Let Y ⊂ X be a non-singular subvariety which is purely of codimension r +1 > 1, and let

\begin{matrix}Y′ & ⟶^{i′} & X′ \\ ↓_{π′} & & ↓^{π} \\ Y & ⟶^{i} & X \end{matrix}

be the blowup of X along Y. Then Y′ → Y is a projective bundle; let ξ ∈ A^{1}(Y′) be the class of the tautological line bundle. Consider the sequence of motives:

0 → h(Y)⊗𝕃^{r+1} ⟶^{(i_{∗},ξ̄^{r}·π^{∗})}h(X)⊕(h(Y)⊗𝕃) ⟶^{π′^{∗}⊕(−i′_{∗})}h(X′) → 0.  (1)

There is a retract of the first arrow given by 0⊕π_{∗}:h(X)⊕(h(Y)⊗𝕃) → h(Y)⊗𝕃^{r+1}.

Theorem 2.8 (Manin). The sequence (1) is split exact.

The proof relies on the identity principle and the behaviour of A^{∗} under blowups. For details we refer the reader to [19]. This result was used there by Manin to prove the Weil conjectures for unirational varieties of dimension three.

3. Curves and abelian varieties (I)

3.1. One reason for Grothendieck’s introduction of motives was to serve as analogues of the Jacobian of a curve in higher dimensions. Here we explain the precise relationship between the motive of a curve and its Jacobian. For the rest of the paper all varieties will be assumed connected. 3.2. Let X ∈ V_{k} be a curve, with field of constants k′. As explained in §1.11–1.13 there is a submotive h^{0}(X) ≃ h(Spec k′) and a quotient motive h^{2}(X) ≃ h(Spec k′) ⊗ 𝕃 of h(X). The choice of a zero-cycle Z on X of positive degree determines projectors p_{0}, p_{2} ∈ Corr^{0}(X,X) = Endh(X) with h^{i}(X) ≃ (X,p_{i},0) for i = 0, 2 as in 1.13.

Let p_{1} = 1−p_{0} −p_{2} ∈ Corr^{0}(X,X), and write h^{1}(X) = (X,p_{1},0) ∈ M_{k}. Then there is a direct sum decomposition

h(X) = h^{0}(X)⊕h^{1}(X)⊕h^{2}(X)

which in general depends on the choice of Z (more specifically, its class in A^{1}(X)). However h^{1}(X) is well-defined up to unique isomorphism; in fact, it is the kernel of the composite map h^{≥1}(X) → h(X) → h^{2}(X), so is well-defined as a subquotient of h(X). The theory of the motives h^{1}(X) is essentially that of Jacobian varieties:

Proposition 3.3. If X, X′ are curves with Jacobian varieties J, J′, then

Hom(h^{1}(X),h^{1}(X′)) = Hom(J,J′)⊗Q

and

Hom(L,h^{1}(X)) = \begin{cases}0 & if ∼ is numerical (or homological) equivalence \\ J(k)⊗Q & if ∼ is rational equivalence. \end{cases}

(Some explanation for the dichotomy will be given at the end of the paper.) Proof. By [27] we have A^{1}(X × X′) = A^{1}(X) ⊕ A^{1}(X′) ⊕ Hom(J,J′) ⊗ Q giving the first formula. For the second, we have Hom(L, h(X)) = A^{1}(X) and Hom(L, h^{0}(X)) = 0, while Hom(L, h^{2}(X)) = Q is generated by the class of a closed point in A^{1}(X). Hence Hom(L,h^{1}(X)) = ker(deg:A^{1}(X) → Q), giving the second formula.

Corollary 3.4. Let C_{k} be the full subcategory of M_{k} whose objects are direct summands of motives of the form h^{1}(X), where X is of dimension one. Then C_{k} is equivalent to the category of abelian varieties over k up to isogeny.

This follows from the fact that every abelian variety is an abelian subvariety of a Jacobian, and Poincaré’s complete reducibility theorem. (Note that the result is independent of the particular equivalence relation chosen to define M_{k}.)

Corollary 3.5. Assume that k is not contained in the algebraic closure of a finite field, and that ∼ is rational equivalence. Then M_{k}^{rat} is not an abelian category.

Proof. . Under the hypothesis on k, one can find an elliptic curve E/k and a point P ∈ E(k) of infinite order. Let ξ ∈ A^{1}(E) be the class of the divisor (P) − (0). Then the morphism ξ_{∗}:𝕃 → h^{1}(E) is non-zero by 3.3. The composite ξ_{∗} ∘ξ^{∗}:h^{1}(E)⊗𝕃 → h^{1}(E) is represented by the class of the zero-cycle η(P, P) + (0, 0) − (P, 0) − (0, P) on E × E. If P = 2Q for Q ∈ E(k̄), then in CH^{2}(E ×E_{/k̄}) we can write

η = [(P,P)+(0,0)−2(Q,Q)]+[2(Q,Q)−(P,0)−(0,P)]

and this is rationally equivalent to zero. Therefore η is a torsion class in CH^{2}(E × E), whence ξ_{∗} ∘ ξ^{∗} = 0. Thus ξ_{∗} is not a monomorphism. If M_{k}^{rat} were abelian then ker ξ_{∗} would be a proper subobject of 𝕃. Tensoring by 𝕃^{−1} this would imply that 11 had a nontrivial subobject. But the unit object in a rigid abelian tensor category is completely decomposable, by [5] Prop. 1.17, so as End11 = Q this is impossible. 3.6. In the next section we will give a generalisation of the above to higher dimensional varieties. First we recall the bare essentials of the theory of Albanese and Picard varieties (see for example [17]). 3.7. Let X be a variety over k. The Albanese variety J_{X} is an abelian variety over k equipped with a morphism ν: X × X → J_{X}, satisfying the following universal property: any morphism ϕ:X ×X → A to an abelian variety such that ϕ(x,y)+ϕ(y,z) = ϕ(x,z) (or equivalently, ϕ(x,x) = 0) factors uniquely as ϕ = β ∘ν for some homomorphism β:J_{X} → A.

If X has a k-rational point x_{0} and γ:X → J_{X} is the morphism γ(x) = ν(x,x_{0}), then γ satisfies the usual universal property for morphisms from X to abelian varieties. If ψ:X → Y is a morphism, then the universal property defines a homomorphism J_{ψ}:J_{X} → J_{Y}, so that J is a covariant functor. 3.8. Let P_{X} be the functor on varieties given by

P_{X}(S) = \frac{\left\{{isomorphism classes of line bundles L on X ×S such that \atop L|_{X×s} is algebraically equivalent to 0 for all s ∈ S(k̄)}\right\}}{\{pr_{2}^{∗}G for G ∈ PicS\}}

Recall that the Picard variety P_{X} is an abelian variety over k such that there are functorial injections

P_{X}(S) ⟶ P_{X}(S)

for varieties S, which are bijections whenever X(S) is non-empty. (See for example §5(d) in chapter 0 of [22].) P_{X} is contravariant in X, and is (functorially) isomorphic to the dual abelian variety of J_{X}.

Theorem 3.9. Let X and Y ∈ V_{k} be varieties of dimensions d and e. Then:

(i)  Hom(J_{X},P_{Y})⊗Q ≃ \frac{A^{1}(X ×Y)}{pr_{1}^{∗}A^{1}(X)+pr_{2}^{∗}A^{1}(Y)}

(ii) Let ξ ∈ A^{d}(X), η ∈ A^{e}(Y) be 0-cycles of positive degree. Then there is an isomorphism

Ω:Hom(J_{X},P_{Y})⊗Q \underrightarrow{∼} \{c ∈ A^{1}(X ×Y) | c∘ζ_{∗} = 0 and η^{∗} ∘c = 0.\}

Recall the proof: first assume that X(k) and Y (k) are nonempty, and let x_{0} ∈ X(k). Then by the universal properties we have

Hom(J_{X},P_{Y}) = {ϕ:X → P_{Y} | ϕ(x_{0}) = 0}

= \frac{\left\{{isomorphism classes of line bundles L on \atop X ×Y such that L|_{\{x_{0}\}×Y} ≃ O_{Y}}\right\}}{\{pr_{1}^{∗}G for G ∈ PicX\}}

= \frac{PicX ×Y}{pr_{1}^{∗} PicX +pr_{2}^{∗} PicY}

by the seesaw theorem. In general this will hold when k is replaced by a finite Galois extension k′/k. Tensoring with Q and taking invariants under Gal(k′/k) then yields (i). The isomorphism (ii) is obtained by combining (i) and seesaw.

The following easy proposition gives the functorial behaviour of Ω:

Proposition 3.10. Let ϕ: X′ → X, ψ: Y′ → Y be morphisms of varieties. Choose positive zero-cycles ξ′, η′ on X′, Y′ with direct images ξ, η on X and Y. If β:J_{X} → P_{Y} is a homomorphism, then

Ω(P_{ψ} ∘β) = ψ^{∗} ∘Ω(β) and Ω(β ∘J_{ϕ}) = Ω(β)∘ϕ_{∗}

(where Ω denotes the isomorphism of 3.9(ii) with respect to the chosen 0-cycles.)

4. The motives h^{1}(X) and h^{2d−1}(X)

4.1. Let X/k be a variety of dimension d, and fix a projective embedding of X of degree m. Let ξ ∈ A^{1}(X) be the class of a hyperplane section. Following Murre [23] we will define projectors p_{1}, p_{2d−1} ∈ A^{d}(X × X) such that the corresponding motives h^{i}(X) = (X,p_{i},0) satisfy the analogue of the hard Lefschetz theorem:

ξ̄^{d−1}:h^{1}(X) \underrightarrow{∼} h^{2d−1}(X)⊗𝕃^{1−d}

and are closely related to the Picard and Albanese varieties of X. This generalises work of Grothendieck, Kleiman and Lieberman, who proved this when ∼ is homological equivalence (see [11], Appendix to §2). 4.2. Assume from now on that there is a 1-dimensional linear section C of X which is a smooth (and connected) curve. This does not involve loss of generality, since one can always find such a section after a finite base extension k′/k. Then the projectors p_{i} may be constructed over k′ and descended to k by Lemma 1.17.

Let Z be a 0-cycle on C which is cut out by a hyperplane, and let ζ ∈ A^{d}(X) be the class of Z. Write i:C ⟶ X for the embedding. Then

ξ̄^{d−1} = i_{∗} ∘i^{∗}:h(X) ⟶ h(X)⊗𝕃^{1−d}.

4.3. The functoriality of the Picard and Albanese varieties defines a composite homomorphism

α:P_{X} ⟶^{P_{i}}P_{C} = J_{C} ⟶^{J_{i}}J_{X}.

The construction relies upon the theorem of Weil ([28], cor. 1 to thm. 7) that α is an isogeny, and does not depend on the choice of section C. Choose n ≥ 1 and β:J_{X} → P_{X} such that α ∘ β = [×n]. Since duality interchanges the functors P and J, one has α̂ = α and β̂= β.

By 3.9, β corresponds to a cycle β̃:= Ω(β) ∈ A^{1}(X × X) = Hom(h(X) ⊗ 𝕃^{1−d},h(X)) satisfying β̃∘ ζ_{∗} = 0 and ζ^{∗} ∘ β̃ = 0. Since β̂ = β we have ^{t}β̃ = β̃. Normalise the projectors p_{0}, p_{2d} of section 1.13 above by p_{0} = \frac{1}{m}[Z ×X] =^{t}p_{2d}, and define

\begin{matrix}p_{1}^{?} = \frac{1}{n}β̃∘ξ̄^{d−1} = \frac{1}{n}β̃·[C ×X] = \frac{1}{n}β̃∘i_{∗} ∘i^{∗}, and \\ p_{2d−1}^{?} =^{t}p_{1}^{?} = \frac{1}{n}i_{∗} ∘i^{∗} ∘β̃. \end{matrix}

Define p_{1} = p_{1}^{?} ∘(1− \frac{1}{2}p_{2d−1}^{?}) and p_{2d−1} =^{t}p_{1}. (We will show that p_{1} = p_{1}^{?} if d > 2; see the proof of (i) below.) Let h^{i}(X) = (X,p_{i},0) ∈ M_{k}.

Theorem 4.4. (i) p_{0}, p_{1}, p_{2d−1}, p_{2d} are orthogonal idempotents. (ii) The composite morphism

h^{1}(X) ⟶ h(X) ⟶^{ξ̄^{d−1}} h(X)⊗𝕃^{1−d} ⟶h^{2d−1}(X)⊗𝕃^{1−d}

is an isomorphism. (iii) Assume ∼ is rational equivalence. Let A^{r}(X)^{0} ⊂ A^{r}(X) be the subgroup of cycle

classes numerically equivalent to zero, and let Alb:A^{d}(X)^{0} → J_{X}(k) ⊗ Q be the Albanese map. The cycle class groups of h^{i}(X) are given as follows:

A^{r}(h^{1}(X)) = \begin{cases}0 & if r ≠ 1 \\ A^{1}(X)^{0} & if r = 1; \end{cases}
A^{r}(h^{2d−1}(X)) = \begin{cases}0 & if r ≠ d \\ A^{d}(X)^{0}/ker(Alb) & if r = d. \end{cases}

Proof. (i) We first consider the p_{i}^{?}. We have

n^{2}p_{1}^{?} ∘p_{1}^{?} = β̃∘i_{∗} ∘i^{∗} ∘β̃∘i_{∗} ∘i^{∗}.

Now if f = i^{∗} ∘β̃∘i_{∗} ∈ A^{1}(C ×C) and g = β̃∘i_{∗} ∈ A^{1}(C ×X) then Ω^{−1}(f) = P_{i} ∘β ∘J_{i} and Ω^{−1}(g) = β ∘J_{i}, by 3.10. Therefore

Ω^{−1}(g ∘f) = β ∘J_{i} ∘P_{i} ∘β ∘J_{i} = [×n]∘β ∘J_{i} ∈ Hom(J_{C},P_{X}).

Since ζ^{∗} ∘ f = ζ^{∗} ∘ g ∘ f = 0 and f ∘ ζ_{∗} = g ∘ f ∘ ζ_{∗} = 0, from 3.9 we deduce that g ∘ f = ng. Therefore

p_{1}^{?} ∘p_{1}^{?} = p_{1}^{?}.  (4.4.1)

Next, β̃∘ζ_{∗} = 0 = ζ^{∗} ∘β̃ implies pr_{2∗}(β̃·[Z ×X]) = pr_{1∗}(β̃·[X ×Z]) = 0, and so we get

p_{1}^{?} ∘p_{2d} = 0 = p_{0} ∘p_{1}^{?}.  (4.4.2)

Also p_{1}^{?}∘p_{0} = (1/mn)[Z]×pr_{2∗}(p_{1}^{?}) = ap_{0} for some a ∈ Q. Squaring gives a^{2}p_{0} = (p_{1}^{?}∘p_{0})^{2} = 0, so a = 0, and

p_{1}^{?} ∘p_{0} = 0.  (4.4.3)

A similar argument shows that

p_{2d} ∘p_{1}^{?} = 0  (4.4.4)

and by transposition

p_{2d−1}^{?} ∘p_{i} = p_{i} ∘p_{2d−1}^{?} = 0 (i = 0, 2d)  (4.4.5)

Now consider

n^{2}p_{2d−1}^{?} ∘p_{1}^{?} = i_{∗} ∘i^{∗} ∘β̃∘β̃∘i_{∗} ∘i^{∗}.

We have β̃ ∘ β̃ ∈ A^{2−d}(X × X). So if d > 2 then β̃ ∘ β̃ vanishes, and if d = 2 then β̃∘ β̃ = a[X × X] for some a ∈ Q. Now β̃∘ ζ_{∗} = 0, whereas [X × X] ∘ ζ_{∗} = m[X], so in fact a = 0. Thus in every case β̃∘β̃= 0, and

p_{2d−1}^{?} ∘p_{1}^{?} = 0  (4.4.6)

Combining (4.4.1–4.4.6) with the definition of p_{1} and p_{2d−1} we get all the required orthogonalities after a series of trivial computations.

For completeness we finally calculate p_{1}^{?} ∘p_{2d−1}^{?}. We have
n^{2}p_{1}^{?} ∘p_{2d−1}^{?} = β̃∘(i_{∗} ∘i^{∗})∘(i_{∗} ∘i^{∗})∘β̃.

If d > 2 then (i_{∗} ∘i^{∗})∘(i_{∗} ∘i^{∗}) = 0 since [C]^{2} = 0 in A^{∗}(X). But if d = 2 then we have

(i_{∗} ∘i^{∗})∘(i_{∗} ∘i^{∗}) = [Δ_{∗}(Z)].

If it happens that

n[Δ_{∗}(Z)] = [Z ×Z] ∈ A^{4}(X ×X)

then we will have p_{1}^{?} ∘ p_{3}^{?} = n^{−3}β̃∘ ζ_{∗} ∘ ζ^{∗} ∘ β̃ = 0. However in general it appears that this need not hold (the situation is rather similar to Remark (iv) below), and the correcting terms in the definition of p_{1} and p_{3} are needed.

(ii) Consider the morphisms

h^{1}(X)\begin{matrix}p_{2d−1} ∘ξ̄^{d−1} ∘p_{1} \\ ⟶ \\ ⟵ \\ p_{1} ∘β̃∘p_{2d−1} \end{matrix}h^{2d−1}(X)⊗𝕃^{1−d}.

Now

p_{1} ∘β̃∘p_{2d−1} = p_{1}^{?} ∘(1− \frac{1}{2}p_{2d−1}^{?})∘β̃∘(1− \frac{1}{2}p_{1}^{?})∘p_{2d−1}^{?}.

As β̃∘β̃= 0 (cf. the proof of (i) above) we get p_{2d−1}^{?} ∘β̃= 0 = β̃∘p_{1}^{?} and so

p_{1} ∘β̃∘p_{2d−1} = p_{1}^{?} ∘β̃∘p_{2d−1}^{?} = p_{1}^{?} ∘β̃= β̃∘p_{2d−1}^{?}.

Therefore

p_{1} ∘β̃∘p_{2d−1} ∘ξ̄^{d−1} ∘p_{1} = p_{1}^{?} ∘β̃∘ξ̄^{d−1} ∘p_{1} = np_{1}.

Similarly,

p_{2d−1} ∘ξ̄^{d−1} ∘p_{1} ∘β̃∘p_{2d−1} = p_{2d−1} ∘ξ̄^{d−1} ∘β̃∘p_{2d−1}^{?} = np_{2d−1}

and thus the arrows are isomorphisms.

(iii) See [23] for details of this part.

Proposition 4.5. Let X, X′ ∈ V_{k} with chosen projective embeddings. Then

Hom(h^{1}(X),h^{1}(X′)) = Hom(P_{X},P_{X′})⊗Q

and Hom(h^{2d−1}(X),h^{2d−1}(X′)) = Hom(J_{X′},J_{X})⊗Q.

Proof. Use ′ to denote the corresponding objects in the construction above when applied to X′. By 4.4(ii) we have

\begin{aligned}Hom(h^{1}(X),h^{1}(X′)) & ≃ Hom(h^{2d−1}(X)⊗𝕃^{1−d},h^{1}(X′)) \\ & = \{c ∈ A^{1}(X ×X′) | p′_{1} ∘c∘p_{2d−1} = c\} \end{aligned}

and the isogeny β gives an isomorphism

Hom(P_{X},P_{X′})⊗Q ≃ Hom(J_{X},P_{X′})⊗Q ≃ {c ∈ A^{1}(X ×X′) | ζ′^{∗} ∘c = 0 = c∘ζ_{∗}}.

SInce ζ′^{∗} ∘p′_{1} = 0 = p_{2d−1} ∘ζ_{∗} by 4.4(i) the first subspace of A^{1}(X ×X′) is contained in the second. To get an inclusion in the other direction, suppose that ζ′^{∗} ∘c = 0 = c∘ζ_{∗}, so that c = Ω(ν) for some ν ∈ Hom(J_{X},P_{X′})⊗Q. Then

c∘p_{2d−1}^{?} = \frac{1}{n}Ω(ν)∘i_{∗} ∘i^{∗} ∘Ω(β)

= \frac{1}{n}Ω(ν ∘J_{i})∘Ω(P_{i} ∘β)

= \frac{1}{n}Ω(ν ∘J_{i} ∘P_{i} ∘β) = \frac{1}{n}Ω([×n]∘ν) = c

and by transposition p_{1}^{?′} ∘c = c also. This settles the case d > 2 completely. If d = 2, then consider c ∘ p_{1}^{?} = c ∘ β̃∘ i_{∗} ∘ i^{∗}. We have c ∘ β̃∘ i_{∗} ∈ A^{0}(C × X′), so c ∘ β̃∘ i_{∗} = a[C × X′] for some a ∈ Q. Since ζ′^{∗} ∘ c = 0 we have a = 0. Thus c ∘ p_{1}^{?} = 0, hence c ∘ p_{3} = c ∘ p_{3}^{?} = c and likewise p_{1} ∘c = c.

The second equality in the proposition follows from the first by duality. 4.6. By 4.4(i) we can write

h(X) = h^{0}(X)⊕h^{1}(X)⊕M ⊕h^{2d−1}(X)⊕h^{2d}(X)

for some M. Suppose now that X has dimension 2. We can then define p_{2} = 1−p_{0} −p_{1} − p_{3} −p_{4} and h^{2}(X) = (X,p_{2},0) = M, and there is a decomposition

h(X) = ⨁_{i=0}^{4}h^{i}(X).

When ∼ is rational equivalence the cycle groups A^{j}(h^{i}(X)) are given by the following table:

M =  h^{0}(X)  h^{1}(X)  h^{2}(X)  h^{3}(X)  h^{4}(X)
A^{0}(M) =  A^{0}(X)  0  0  0  0
A^{1}(M) =  0  A^{1}(X)^{0}  NS(X)⊗Q  0  0
A^{2}(M) =  0  0  ker(Alb)  A^{2}(X)^{0}/ker(Alb)  Q

Remarks. (i) Murre calls the motives h^{1}(X) and h^{2d−1}(X) the Picard and Albanese motives of X, respectively, in view of the previous result. Observe that p_{1} factors through h(C), and in fact h^{1}(X) is a direct summand of h^{1}(C). (ii) The minor differences between the construction we give and that in [23] are in slightly different normalisations of the projectors p_{i}; we use the zero-cycle ζ rather than an auxiliary rational point to normalise p_{0} and p_{2d}, and in the case d = 2 we have made a different choice of p_{1} and p_{3} to preserve Poincaré duality and the analogue of the hard Lefschetz theorem. (iii) To complete the picture in 4.5 one would like to know for surfaces X, X′ the nature of the group Hom(h^{2}(X),h^{2}(X′)), and in particular that it did not depend on the choice of equivalence relation on cycles. This would be answered by enough knowledge about the conjectural filtration on Chow groups; see §6.2 and Jannsen’s paper in these proceedings [10] for more information. (iv) The reader is warned against reading too much into 4.4(ii). In particular, in M_{k}^{rat} it will not generally be the case that h(X) has a decomposition into primitive pieces which satisfy a naïve analogue of the hard Lefschetz theorem. The analogous situation in the derived category of Q_{ℓ}-sheaves is considered in [3] and especially [4]; here we give a simple example.

Consider a curve X of genus g over an algebraically closed field embedded in projective space by a multiple of P + Q, where P, Q are points on X. We take ∼ to be rational equivalence. Then cup-product with the hyperplane section ξ does of course give an isomorphism

h^{0}(X) \underrightarrow{∼} h^{2}(X)⊗𝕃^{−1}

where the projectors are taken to be

p_{0} = \frac{1}{2}[(P +Q)×X] =^{t}p_{2}, p_{1} = 1−p_{0} −p_{2}.

However it also induces a morphism

h^{1}(X) ⟶ h^{1}(X)⊗𝕃^{−1}

which is represented by a non-zero multiple of the cycle

η = (P,P)+(Q,Q)−(P,Q)−(Q,P).

Using arguments of Bloch, Mumford and Roitman one sees that for g > 1 and P, Q sufficiently generic, η does not vanish. (This can be deduced fairly easily from Theorem 3.1(a) of [2], for instance.) So h^{1}(X) is not killed by ξ̄ in general.

5. Abelian varieties (II)

5.1. In this section we will consider an abelian variety X over k of dimension g. For an integer n, write [×n]:X → X for multiplication by n. Also let μ:X ×X → X be the group law, ε ∈ X(k) the identity element and σ:X → X multiplication by −1.

Theorem 5.2.

(i) There is a unique decomposition in M_{k}

h(X) = ⨁_{i=0}^{2g}h^{i}(X)^{can}

which is stable under [×n]^{∗}, and such that [×n]^{∗}|_{h^{i}(X)} is multiplication by the scalar n^{i}, for every n ∈ Z.

(ii) The iterated product maps

h(X)⊗...⊗h(X) = h(X ×...×X) ⟶^{diag^{∗}}h(X)

induce for every i ≥ 0 isomorphisms

⋀^{i} h^{1}(X)^{can} \underrightarrow{∼} h^{i}(X)^{can}.

(iii) Let ξ ∈ A^{1}(X) be the class of an ample symmetric line bundle on X. Then there is a commutative diagram

h^{i}(X)  ⟶  h(X)
↓^{≀}  ↓ξ̄^{g−i}
h^{2g−i}(X)⊗𝕃^{i−g} ⟶ h(X)⊗𝕃^{i−g}

in which the horizontal arrows are the obvious inclusions.

(Recall that ξ ∈ A^{∗}(X) is symmetric if σ^{∗}ξ = ξ.) There is also a relation between (i) and the decomposition of the previous section. Let p_{i}^{can} ∈ Endh(X) be the projectors for which h^{i}(X)^{can} = (X,p_{i}^{can},0), and for i = 0, 1, 2g − 1, 2g let p_{i} be the projectors defined in §4, using the class ξ of a very ample line bundle on X.

Theorem 5.3. If ξ is symmetric, then p_{i}^{can} = p_{i} for i = 0, 1, 2g −1 and 2g.

5.4. For numerical (or homological) equivalence, Theorem 5.2 was proved by Grothendieck, Kleiman and Lieberman; see the appendix to §2 in [11]. For rational equivalence, the existence of a decomposition h(X) = ⊕h^{i}(X) in M_{k}^{rat} satisfying (ii) was first proved by Manin and Shermenev [26], using Jacobians. An elegant proof of (i) was recently found by Deninger and Murre [6], who used the Fourier transform on Chow groups [1]. Künnemann extended their ideas to prove (ii); see his paper [15] in these proceedings, where he gives an elegant explicit formula for the projectors p_{i}^{can} using the Pontryagin product on the Chow groups of X. The work of Deninger-Murre and Künnemann applies more generally to abelian schemes over any smooth variety. Finally, in his 1992 Ph.D. thesis Künnemann not only proves (iii), but also obtains a complete Lefschetz decomposition of h(X), just as one has in cohomology. For details see [16].

Rather than reproduce any of these arguments here, we will give an elementary proof of 5.2(i) from which 5.2(iii) and 5.3 will be easy consequences, and which uses Fourier theory at just one point (5.7 below). 5.5. We first introduce some simple notations. If i ∈ Z then write A^{∗}(X)^{(i)} for the subspace comprising all c ∈ A^{∗}(X) such

[×n]^{∗}(c) = n^{i}c for every n ∈ Z.

Likewise if X′ is a second abelian variety, of dimension g′, write A^{∗}(X ×X′)^{(i,j)} for the set of all c ∈ A^{∗}(X ×X′) such that

([×m]×[×n])^{∗}(c) = m^{i}n^{j}c for all m, n ∈ Z.

From 1.10 it follows that c ∈ A^{∗}(X×X′)^{(i,j)} if and only if, for every n ∈ Z, one has identities of correspondences

[×n]_{X′}^{∗} ∘c = n^{j}c and c∘[×n]_{X}^{∗} = n^{2g−i}c.

Therefore if c ∈ A^{∗}(X ×X′)^{(i,j)} and d ∈ A^{∗}(X′×X′′)^{(r,s)} one has d∘c = 0 unless j = 2g′−r. 5.6. Recall ([21], §II.6) that if L is any line bundle on X then

[×n]^{∗}L ≃ L^{⊗n(n+1)/2} ⊗(σ^{∗}L)^{⊗n(n−1)/2}.

Therefore ξ ∈ A^{1}(X) is symmetric if and only if ξ ∈ A^{1}(X)^{(2)}, or equivalently [×n]^{∗} ∘ ξ̄= n^{2}ξ̄∘ [×n]^{∗}. Likewise, ξ is antisymmetric if and only if ξ ∈ A^{1}(X)^{(1)} (which holds if and only if ξ is algebraically equivalent to zero). 5.7. If ξ ∈ A^{1}(X) is the class of an ample line bundle we have the usual formula

deg(ξ^{g}) = g!d

where d^{2} is the degree of the polarisation determined by ξ. Using Fourier theory one can show that if ξ is symmetric then

ξ^{g} = g!d[ε] ∈ A^{g}(X)

(see for example [1], middle of page 249). 5.8. Now pick a symmetric ξ which is the class of an ample line bundle, and write λ = μ^{∗}ξ −pr_{1}^{∗}ξ −pr_{2}^{∗}ξ ∈ A^{1}(X ×X). Then

λ ∈ A^{1}(X ×X)^{(1,1)}.

Indeed, the restrictions of λ to the fibres of pr_{1} and pr_{2} are algebraically equivalent to zero, so ([×m]×[×n])^{∗}λ−mnλ has zero restriction to the fibres by 5.6, hence is zero.

Remark. If ϕ_{L}:X → X̂ is the usual homomorphism x → T_{x}^{∗}L⊗L^{∨} attached to a line bundle L and ξ = c_{1}(L), then λ is the pullback by id×ϕ_{L} of the class of the Poincaré bundle on X ×X̂. This is the link between the formulae given below and Fourier theory.

5.9. Proof of 5.2(i), (iii). Firstly, it is obvious that the decomposition (i) is unique if it exists. This shows in particular that the choice of ξ is unimportant.

If 0 ≤ i ≤ 2g define

\begin{aligned}f_{i} & = ∑_{max(0,i−g)≤j≤i/2} \frac{1}{j!(g −i+j)!(i−2j)!}pr_{1}^{∗}ξ^{j} ·pr_{2}^{∗}ξ^{j} ·λ^{i−2j} \\ q_{i} & = ∑_{max(0,i−g)≤j≤i/2} \frac{1}{j!(g −i+j)!(i−2j)!}pr_{1}^{∗}ξ^{g−i+j} ·pr_{2}^{∗}ξ^{j} ·λ^{i−2j}. \end{aligned}

If 0 ≤ i ≤ g then

q_{i} = pr_{1}^{∗}ξ^{g−i} ·f_{i} = f_{i} ∘ξ̄^{g−i} and q_{2g−i} = pr_{2}^{∗}ξ^{g−i} ·f_{i} = ξ̄^{g−i} ∘f_{i}.  (5.9.1)

Also we have

f_{i} ∈ A^{i}(X ×X)^{(i,i)} and q_{i} ∈ A^{g}(X ×X)^{(2g−i,i)}.

In particular q_{i} ∘q_{i′} = 0 if i ≠ i′. Now

∑_{i=0}^{2g}q_{i} = ∑_{{0≤i≤2g \atop max(0,i−g)≤j≤i/2}} \frac{1}{j!(g −i+j)!(i−2j)!}pr_{1}^{∗}ξ^{g−i+j} ·pr_{2}^{∗}ξ^{j} ·λ^{i−2j}

= \frac{1}{g!}(pr_{1}^{∗}ξ +pr_{2}^{∗}ξ +λ)^{g}

= \frac{1}{g!}μ^{∗}ξ^{g} = dμ^{∗}[ε] = d[Γ_{σ}].

Define p_{i}^{can} = \frac{1}{d}σ^{∗} ∘q_{i} = \frac{(−1)^{i}}{d} q_{i} ∈ A^{g}(X ×X)^{(2g−i,i)}. Then by the above,

∑p_{i}^{can} = 1 and ∑p_{i}^{can} ∘p_{i}^{can} = (∑p_{i}^{can})^{2} = 1.

This forces p_{i}^{can} ∘p_{i}^{can} = p_{i}^{can}, and then h^{i}(X)^{can} = (X,p_{i}^{can},0) satisfies (i).

By 5.6 and (i) the cup-product ξ̄^{g−i} maps h^{i}(X)^{can} into h^{2g−i}(X)^{can} ⊗ 𝕃^{i−g}. Then the formulae (5.9.1) show that it is an isomorphism, giving (iii).

Corollary 5.10. The natural map Hom(X,X′) ⊗ Q → Hom(h^{1}(X′)^{can},h^{1}(X)^{can}) is an isomorphism.

Proof. Composition with ξ̄^{g−1} gives

Hom(h^{1}(X′)^{can},h^{1}(X)^{can}) ≃ Hom(h^{2g−1}(X′)^{can} ⊗𝕃^{1−g},h^{1}(X)^{can})
= A^{1}(X′ ×X)^{(1,1)}
≃ A^{1}(X′ ×X)/pr_{1}^{∗}A^{1}(X′)+pr_{2}^{∗}A^{1}(X)
≃ NS(X′ ×X)⊗Q
≃ Hom(X′,X)⊗Q

the last isomorphism coming from the polarisation ξ. It is easily checked that this is inverse to the map of the corollary. 5.11. A formal consequence of 5.2(i) is the eigenspace decomposition of the Chow groups: we have

A^{d}(X) = Hom(𝕃^{d},h(X)) = ⨁_{i=0}^{2g}Hom(𝕃^{d},h^{i}(X)^{can}) = ⨁_{i=0}^{2g}A^{d}(X)^{(i)}.

Since composition with ξ̄^{g−i} gives an isomorphism A^{d}(X)^{(i)} \underrightarrow{∼} A^{g+d−i}(X)^{(2g−i)} one gets by dimensional considerations that A^{d}(X)^{(i)} = 0 unless d ≤ i ≤ g +d. Similarly, we have

A^{d}(X ×X′) = ⨁_{(i,j)}A^{d}(X ×X′)^{(i,j)}

where the sum is over pairs (i,j) ∈ Z^{2} such that d − g′ ≤ i ≤ d + g, d − g ≤ j ≤ d + g′ and d ≤ i + j ≤ d + g + g′. These decompositions are of course direct consequences of Fourier theory [1, 6], and it was using them that Deninger and Murre proved 5.2(i).

The interpretation of motives for rational equivalence as complexes (see 6.2 below) suggests that a stronger vanishing result

A^{d}(X)^{(i)} = Hom(𝕃^{d},h^{i}(X)^{can}) = 0 for i > 2d

holds. See [10], where the conjectural filtration on Chow groups (of which this vanishing is a part) is explained. Proof of 5.3. We have p_{2g−i}^{can} =^{t}p_{i}^{can} and p_{2g−i} =^{t}p_{i}, so it is enough to treat p_{0} and p_{1}. We

have p_{0} = \frac{1}{g!d} (ξ^{g} ×[X]), so by 5.7 this is the same as p_{0}^{can}.

For p_{1} we first observe that the difficulties arising in the previous section in dimension 2 disappear here; for by 5.7 we have ξ^{g} = d!g[ε] so that (4.4.7) obviously holds. So in every case we have p_{1} = p_{1}^{?} (in the notations of 4.3).

Lemma 5.12. [×n]^{∗} ∘p_{1} = np_{1} = p_{1} ∘[×n]^{∗}

Proof. By definition p_{1} = (1/m)β̃∘ξ̄^{g−1} for a certain isogeny β:X → X̂. We therefore have, using 3.10

[×n]^{∗} ∘β̃= Ω(P_{[×n]} ∘β) = Ω([×n]∘β) = nΩ(β).

This gives the first equality. For the second we have

p_{1} ∘[×n]^{∗} = \frac{1}{m}β̃∘ξ̄^{g−1} ∘[×n]^{∗} = \frac{n^{2−2g}}{m} β̃∘[×n]^{∗} ∘ξ̄^{g−1}.

Now [×n]^{∗} ∘ [×n]_{∗} = n^{2g} ∈ End h(X), so if we write β̃ ∘ [×n]^{∗} = Ω(β′) for some β′ ∈ Hom(X,X̂)⊗Q then

n^{2g}β̃= β̃∘[×n]^{∗} ∘[×n]_{∗} = Ω(β′ ∘J_{[×n]}) = Ω(β′ ∘[×n]) = nΩ(β′).

Therefore β̃∘[×n]^{∗} = n^{2g−1}β̃ and thus p_{1} ∘[×n]^{∗} = np_{1}, proving the lemma.

Now from 5.12 we see that p_{1}, p_{1}^{can} commute and that p_{1} ∘ p_{1}^{can} = p_{1}. Therefore h^{1}(X) = (X, p_{1}, 0) is a direct factor of h^{1}(X)^{can} = (X, p_{1}^{can}, 0). But by 4.5 and 5.10, Endh^{1}(X) = Endh^{1}(X)^{can} = EndX ⊗Q. Therefore h^{1}(X) = h^{1}(X)^{can}.

Remark. One can also check that β can be chosen to be the isogeny ϕ_{L}: X → X̂ determined by ξ = c_{1}(L). This gives an alternative verification of 5.3.

6. Further topics

6.1. (Relative motives.) Let S be a smooth (not necessarily projective) variety, and let V_{S} be the category of smooth projective S-schemes. Then one can introduce the notion of a relative correspondence between two objects X, Y of V_{S} as a cycle class of codimension dim(X/S) on X×_{S}Y. In [6] this is used to define the category M_{S} of motives over the base S. Deninger and Murre show there that if X/S is an abelian scheme of relative dimension d, then the relative motive h(X/S) has a canonical decomposition h(X/S) = ⨁_{i=0}^{2d} h^{i}(X/S).

If S itself is projective, then the functor V_{S} → V_{k} induces a functor h(S,−):M_{S} → M_{k}. In other words, relative motives should be thought of as (complexes of) sheaves on S. For an abelian scheme X/S one can then form various motives h(S,h^{i}(X/S)).

In an ideal world one would like a more general notion of relative motive: for example, if j:S ⟶ S′ is the smooth compactification of a curve S, and X/S ∈ V_{S}, then one would like to be able to give a meaning to j_{∗}h(X/S), and define motives such as h^{p}(S′,j_{∗}h^{q}(X/S)).

There is one simple case in which it is possible to do this. Let S/Q be the standard modular curve of level n ≥ 3 (which parameterises elliptic curves together with a chosen basis for the subgroup of n-division points), and let π: E → S be the universal elliptic curve. Let j:S → S′ be the smooth compactification of S. Write M = h^{1}(E/S), a relative motive over S, and let M_{ℓ} be the corresponding ℓ-adic sheaf R^{1}π_{∗}Q_{ℓ} on S. The parabolic cohomology groups

kW_{ℓ} = H^{1}(S′ ⊗\overline{Q},j_{∗} Sym^{k} M_{ℓ})

are representations of Gal(\overline{Q}/Q) which are pure of weight k + 1. Since Sym^{k} M is a submotive of h(X), where X is the k-fold fibre product of E over S, _{k}W_{ℓ} is in fact a subquotient of the cohomology of X′, a smooth compactification of X. One can in this case prove [25] by brutal construction that there is a submotive _{k}W ⊂ h(X′) (in the category of Chow motives) such that the ℓ-adic cohomology of _{k}W is the parabolic cohomology. It is tempting to write this motive as h^{1}(S′,j_{∗} Sym^{k} M), but there is as yet no way to make similar constructions in a general situation. We do not give any details as the proof uses arguments which are rather dissimilar to what has gone before. 6.2. In this final section we will try to give some vague hints of how the results described above fit in to a general (mainly conjectural) picture. To begin with we discuss the effect of choosing different equivalence relations on cycles.

• Numerical equivalence. This is the coarsest adequate equivalence relation. In [9] Jannsen proves, without using the standard conjectures, that the category of motives over a field is abelian and semisimple if and only if the equivalence relation used is numerical equivalence.

• Homological equivalence (with respect to a fixed Weil cohomology theory H^{∗}). According to the standard conjectures (see [11] and [13]) M_{k}^{num} and M_{k}^{hom} should coincide. At present, it is only for M_{k}^{hom} that we can define the realisation functors—but see also remark (4) in [9].

The standard conjectures also predict that in M_{k}^{hom} the motive h(X) has a direct sum decomposition

h(X) = ⨁_{i=0}^{2 dim X} h^{i}(X)  (6.2.1)

such that the cohomology functor H^{i} factors through the projection h(X) → h^{i}(X). Now in the usual cohomology theories any homomorphism H^{i}(X) → H^{j}(Y) induced by an algebraic cycle is zero unless i = j—in the ℓ-adic theory this is by Deligne’s proof of the Weil conjectures. Therefore the decomposition (6.2.1) is unique.

• Rational equivalence. In general M_{k}^{rat} is not abelian, by 3.5 above. One expects however (for reasons explained below) that every h(X) has in M_{k}^{rat} a direct sum decomposition (6.2.1) in which the motives h^{i}(X) are well defined up to unique isomorphism. Moreover the filtration on h(X) by subobjects

h^{≤i}(X) = ⨁_{j≤i}h^{j}(X)

should be uniquely determined and functorial with respect to inverse image and duality. The direct sum decomposition itself will not however be uniquely determined. The corresponding filtration on the Chow groups A^{d}(X) = Hom(𝕃^{d}, h(X)) by the subgroups Hom(𝕃^{d},h^{≤i}(X)) would be the conjectural filtration discussed in [10]. See [24], where the idea of using such a “Chow-Künneth” decomposition to study the filtration is introduced and elaborated in detail.

If ξ ∈ A^{1}(X) is the class of an ample divisor, then the hard Lefschetz theorem should hold in the following sense: ξ̄^{i}:h(X)⊗𝕃^{i} → h(X) respects the filtration up to a shift by 2i, and induces an isomorphism between the subquotients h^{dim X−i}(X)⊗𝕃^{i} and h^{dim X+i}(X).

In (say) ℓ-adic cohomology the ring of correspondences Corr^{0}(X,X) acts not only on H^{∗}(X̄, Q_{ℓ}) but also on the object RΓ(X̄, Q_{ℓ}) in the derived category D^{b}(Spec k, Q_{ℓ}) of complexes of ℓ-adic representations of Gal(k̄/k). In this category there is the canonical filtration by truncation. Moreover a theorem of Deligne ([3], Prop. 2.4; see also [4]) states that there is an isomorphism (which is not unique) in D^{b}(Speck,Q_{ℓ})

RΓ(X̄,Q_{ℓ}) \underrightarrow{∼} ⨁_{i=0}^{2 dim X} H^{i}(X̄,Q_{ℓ})[−i].

One should therefore think of the object h^{i}(X) in M_{k}^{rat} as a complex concentrated in degree i, and of Hom(h^{i}(X),h^{j}(Y)) as an analogue of

Hom(H^{i}(X̄,Q_{ℓ})[−i],H^{j}(Ȳ ,Q_{ℓ})[−j]) = Ext^{i−j}(H^{i}(X̄,Q_{ℓ}),H^{j}(Ȳ ,Q_{ℓ})).

The image of the Lefschetz motive 𝕃 in the ℓ-adic setting is Q_{ℓ}(−1)[−2], so A^{d}(h^{i}(X)) = Hom(𝕃^{d},h^{i}(X)) should be analogous to

Hom(Q_{ℓ}(−d)[−2d],H^{i}(X̄,Q_{ℓ})[−i]) = Ext^{2d−i}(Q_{ℓ}(−d),H^{i}(X̄,Q_{ℓ})).

This fits in well with the formalism of mixed motives, and suggests that there exists a “derived category of mixed motives”. It would contain M_{k}^{num} as the (abelian, semisimple!) subcategory formed of direct sums of “pure complexes” (of any weight) concentrated in degree zero, and would contain M_{k}^{rat} as the subcategory of “pure complexes of weight zero”. In particular this would imply that Hom(h^{i}(X),h^{i}(Y)) was the same in M_{k}^{num} and M_{k}^{rat}. For further indications along these lines we refer to the papers of Jannsen [10] and Levine [18], as well as other papers in these Proceedings.

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Department of Mathematical Sciences Science Laboratories University of Durham Durham DH1 3LE England e-mail: a.j.scholl@durham.ac.uk