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Cycle Map for Strictly Decomposable Cycles

We introduce a class of cycles, called nondegenerate, strictly decomposable cycles, and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed Hodge structures does not vanish. This class contains certain cycles in the kernel of the Abel-Jacobi map. The construction gives a refinement of Nori's argument in the case of a self-product of a curve. As an application, we show that a higher cycle which is not annihilated by the reduced higher Abel-Jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1-form.


Introduction
Let X be a smooth projective complex variety, and let CH p (X) be the Chow group of codimension p cycles modulo rational equivalence. Consider the subgroup CH p hom (X) consisting of cycles homologically equivalent to zero. We have the Abel-Jacobi map to the intermediate Jacobian [24]. If p > 1, it is known that this map is neither surjective nor injective in general. D. Mumford [30] showed that the kernel can be very large, see also [7]. However it is not necessarily easy to construct a nonzero element in the kernel, because the rational equivalence is rather complicated for explicit calculations. An argument constructing such an example was found by M. Nori in the case of a selfproduct of a curve, see [38]. His argument involves the transcendental part of the second cohomology, and requires some more calculations to verify the hypothesis on cycles. In this paper, we simplify and generalize it by introducing the notion of a nondegenerate, strictly decomposable cycle.
Let X 1 , X 2 be smooth projective varieties over C, and set X = X 1 × X 2 . Take divisors ζ i ∈ CH 1 (X i ) which are homologically equivalent to zero. We assume that they are not torsion. Consider ζ := ζ 1 × ζ 2 in CH 2 (X) Q . This belongs to the kernel of the Abel-Jacobi map. It may be zero, for example, if X 1 and X 2 are the same elliptic curve and ζ 1 and ζ 2 are the same cycle of the form [P ] − [Q], see (2.9.2). So we are interested in the following question: When is ζ not rationally equivalent to zero (with rational coefficients)? This is nontrivial even in the case of a self-product of an elliptic curve. In this paper we give a partial answer as follows.
0.1. Theorem. Assume there exists an algebraically closed subfield k of C such that X 1 , X 2 and ζ 1 are defined over k, but ζ 2 is not. Then the cycle ζ does not vanish in CH 2 (X) Q . This is an example of an exterior product of cycles [39] although the varieties here are all defined over k. In the curve case, Theorem (0.1) implies (by replacing k) that does not vanish in CH 2 (X) Q if there exists an algebraically closed subfield k of C such that X 1 , X 2 , Q 1 , Q 2 are defined over k, but P 1 , P 2 are not, and if the field of definition k(P 1 ) of P 1 is algebraically independent of k(P 2 ) (i.e. if they generate a subfield of transcendence degree 2). Here k(P i ) is the image of the morphism k(X i ) → C defined by P i , see (2.9).
The proof of (0.1) uses a refined cycle map and the associated Leray filtration on the Chow group, which follow from the theory of (arithmetic) mixed sheaves [34], [36], [37] (see also [1]) as conjectured in [4], [7]. This construction was originally considered in order to define an analogue of "spreading out" of algebraic cycles [7] for mixed Hodge Modules (see [34], 1.9), and was greatly inspired by earlier work of M. Green [23], C. Voisin [40], [41] and others. We remark that our Leray filtration seems to coincide with a filtration which has been studied recently by M. Green in the case the variety is defined over k as explained in his talk at Azumino, July 2000. We can use also Green's formulation for the proof of Theorem (0.1), see Remark (ii) of (1.5). The theory of mixed Hodge modules is essential in the case the variety is not defined over k.
Note that ζ 1 in Theorem (0.1) can be a cycle of any codimension provided that its image by the Abel-Jacobi map is not torsion. Indeed, we have a more general assertion (2.6) whose proof is reduced to a simple lemma on Hodge structures (2.7). We can extend Theorem (2.6) to the case of higher Chow groups CH p (X, m) [9] as in (3.3), which implies 0.2. Theorem. Let ζ = ζ 1 × ζ 2 ∈ CH p (X, m) Q with ζ 1 ∈ CH p 1 (X 1 ) Q and ζ 2 ∈ CH p 2 (X 2 , m) Q . Then ζ does not vanish if there exists an algebraically closed subfield k of C such that X 1 , X 2 are defined over k and the following condition is satisfied for i = 1 or 2 : (i) ζ i is defined over k, but ζ 3−i is not, (ii) the image of ζ i by the cycle map (3.1.2) does not vanish, (iii) p 3−i = 1.
Here CH 1 (X 2 , m) = 0 for m > 1 and CH 1 (X 2 , 1) = C by loc. cit. So we may assume X 2 = pt if the above hypothesis is satisfied for i = 1 and m = 1.
Originally this theory was considered in the case when both ξ 1 1 and ξ 1 2 are nonzero with the notation of (2.1) in order to show the nonvanishing of some extension classes. In the study of Nori's construction [38], a special case of (2.6) was obtained for a self-product of a curve of certain type [32]. Then these were generalized to (2.6).
As an application, we have the following. For a smooth complex projective variety X, let CH p ind (X, 1) Q denote the group of indecomposable higher cycles with rational coefficients, see (4.1). We have the reduced higher Abel-Jacobi map (4.1.2) which is analogous to Griffiths' Abel-Jacobi map. 0.3. Theorem. Let X = X 1 ×X 2 as above. If Γ(X 1 , Ω 1 X 1 ) = 0 and the reduced higher Abel-Jacobi map (4.1.2) for X 2 is not zero, then CH p+1 ind (X, 1) Q is uncountable. More precisely, let ζ 2 be an element of CH p (X 2 , 1) such that its image by (4.1.2) does not vanish. Let P denote the Picard variety of X 1 . Assume X i , ζ 2 and P are defined over k. Then for any ζ 1 ∈ P(C) \ P(k), the image of ζ 1 × ζ 2 in CH p+1 ind (X, 1) Q is nonzero. This does not contradicts Voisin's conjecture [42] on the countability of CH 2 ind (X, 1) Q , because CH 1 ind (X 2 , 1) Q = 0. It is known that CH p+1 ind (X, 1) Q is uncountable for certain varieties X, see [13]. The assumption on the nonvanishing of the image of ζ 2 by (4.1.2) is satisfied for some examples, see [14], [35].
The paper is organized as follows. In Sect. 1 we recall the notion of arithmetic mixed Hodge structures. In Sect. 2 we prove Theorem (0.1). Then Sect. 3 treats the case of higher cycles, and the application to indecomposable higher cycles is given in Sect. 4.
In this paper, a variety means a separated scheme of finite type over a field. For a complex variety X and a ring A, we denote H j (X an , A) by H j (X, A) to simplify the notation.
1. Arithmetic mixed Hodge structure 1.1. Mixed Hodge structure. For a subfield k of C, let MHS k denote the category of graded-polarizable mixed Q-Hodge structure with k-structure, see [17], [18], [26], etc. (In this paper, l-adic cohomology is not used, because k is assumed to be algebraically closed and the embedding of k into C is fixed.) An object H of MHS k consists of a bifiltered k-vector space (H k ; F, W ) and a filtered Q-vector space (H Q , W ) together with an filtered isomorphism α : (H k , W )⊗ k C = (H Q , W )⊗ Q C, and they define a graded-polarizable mixed Q-Hodge structure in the sense of [16] after tensoring H k with C. Here the polarizations on the graded pieces of W are assumed to be defined over k so that the graded pieces are semisimple. Morphisms are pairs of morphisms of bifiltered or filtered vector spaces compatible with α. This is an abelian category such that every morphism is bistrictly compatible with (F, W ). We have naturally a constant object Q k . We can define the Tate twist (p) for p ∈ Z as in loc. cit.
For H, H ′ ∈ MHS k , we have by an argument similar to [12], see [36]. Let X k be a k-variety. Then we have naturally the cohomology H i (X k , Q) in MHS k by [16] together with the compatibility of de Rham cohomology with base change. Furthermore, we can define canonically K H (X k ) in the derived category D b MHS k such that its cohomology H i K H (X k ) is isomorphic to H i (X k , Q) in MHS k . This can be done by applying the construction of the direct image for perverse sheaves [5] to mixed Hodge Modules. In the case X k is smooth and quasiprojective, we may assume that the complement of X k in a smooth projective compactification of X k is a divisor. Then we can take two sets of general hyperplane sections {H k,i } and {H ′ k,j } such that ∩ i H k,i = ∩ j H ′ k,j = ∅, and use the Cech and co-Cech complexes associated to the affine coverings {U k,i = X k \ H k,i } and {U ′ k,j = X k \ H ′ k,j } together with the generalization of the weak Lefschetz theorem in [6], so that we get a complex whose p-th component is

Deligne cohomology.
We define an analogue of Deligne cohomology by If k = C, this coincides with the absolute p-Hodge cohomology of Beilinson [3] which is denoted by H i D (X, Q(j)) in this paper, see also [20], [21], [25], [33]. In general, if we put X = X k ⊗ k C, then the forgetful functor induces a natural morphism More generally, for H ∈ MHS k , we define By (1.1.1) we have a natural short exact sequence Let CH p (X k ) Q be the Chow group of codimension p cycles with rational coefficients modulo rational equivalence. Assume X k is smooth. Then we have a cycle map (see e.g.
This is compatible with the cycle map for X = X k ⊗ k C via (1.2.1). The cycle map (1.2.4) induces Griffiths' Abel-Jacobi map [24] tensored with Q: where CH p hom (X) denotes the subgroup consisting of homologically equivalent to zero cycles, and the last isomorphism is due to [12].
1.3. Leray filtration. Let S k be a k-variety, and put Y k = X k × k S k . Then we have a canonical isomorphism There is an increasing Leray filtration Ker d for i = r, and 0 for i > r .) We have a canonical isomorphism Since the filtration F L splits by (1.

1.2), it induces a decreasing Leray filtration
Here the filtration F L is shifted as in [16]. We have a natural short exact sequence The filtration F L and this short exact sequence are compatible with the pull-back induced by the base change under a morphism of k-varieties S ′ k → S k . Assume X, S are smooth. By the cycle map (1.2.3) for Y k , we get the induced Leray filtration F L on CH p (Y ) Q such that is injective.
1.4. Arithmetic mixed Hodge structure. Let k be a subfield of C. We assume k is algebraically closed for simplicity (hence l-adic sheaves are not used in this paper). Then the category of arithmetic mixed Hodge structures M k is defined to be the inductive limit of the categories of mixed Hodge Modules on S k = Spec R (i.e. mixed Hodge Modules on S := S k ⊗ k C whose underlying bifiltered D-Modules and polarizations are defined over S k ), where R runs over the finitely generated smooth k-subalgebra of C. Although we can also define it to be the inductive limit of the categories of admissible variations of mixed Hodge structures on S k , we need mixed Hodge Modules in order to calculate higher extension groups, because the adjunction relation between direct images and pullbacks does not hold for the derived category of admissible variations. There is a canonical pull-back functor induced by the projection S k = Spec R → Spec k for any finitely generated smooth ksubalgebra R of C. Let Q k = π * Q k . We have also the forgetful functor by restricting to the geometric generic point of S k defined by the inclusion R → C. For a complex algebraic variety X, the cohomology H i (X, Q k ) is naturally defined in M k by taking an R-scheme X R such that X R ⊗ R C = X, because the direct image of the constant sheaf by X R → Spec R is defined by the theory of mixed Hodge Modules. Furthermore, it can be lifted to a complex K k (X) ∈ D b M k having a canonical isomorphism We define an analogue of Deligne cohomology by From now on, we assume X = X k ⊗ k C for a smooth k-variety X k . Then the argument is very much simplified, and we have natural isomorphisms .
which induces the (decreasing) Leray filtration F L on CH p (X) Q such that is injective. This cycle map is compatible with (1.2.4).
1.5. Remarks. (i) By the compatibility of the cycle maps (1.2.4) and (1.4.5), F 1 L CH p (X) Q coincides with the subgroup CH p hom (X) Q of homologically equivalent to zero cycles, and F 2 L CH p (X) Q is contained in the kernel of the Abel-Jacobi map CH p AJ (X) Q . Here we have equality if p = dim X (see e.g. [36], 3.6). This follows from Murre's result on Albanese motives [31] together with the compatibility of the action of a correspondence. Restricting to the subgroup CH p alg (X) Q of algebraically equivalent to zero cycles, we have more generally the equality F 2 L CH p alg (X) Q = CH p AJ (X) Q ∩ CH p alg (X) Q for any p, see [37], 3.9. We can show Gr r F L CH p (X) Q = 0 for r > p. In the case k = Q, it is expected that the filtration F L gives the conjectural filtration of Bloch [7] and Beilinson [4], see also [27].
The problem is the injectivity of the cycle map (1.4.5). In the case p = 2 and m = 0, this can be reduced to a conjecture of Beilinson [4] and Bloch on the injectivity of Abel-Jacobi map for codimension 2 cycles defined over number fields (see [11] for a special case where the conjecture is verified).
(ii) It is also possible to consider a variant of Deligne cohomology by replacing X k × k S k with X × S in (1.4.3) (i.e. by forgetting k-structure). Then we get a filtration similar to the one studied recently by M. Green and also to the one treated in [28]. (However, the definition given in loc. cit. in terms of currents does not seem to be correct even in the product case, because the Hodge filtration F on some graded pieces of Dec L on the complex of currents is filtered quasi-isomorphic to the filtration σ on the localization of holomorphic differential forms, and not the pole order filtration of Griffiths [24] and Deligne [15].) In view of the above conjecture on the injectivity of Abel-Jacobi map, it may be expected that we get still the same filtration on the Chow group after the modification. Note that we can forget the k-structure for the proof of Theorems (0.1) and (0.2).
(iii) For a subfield K of C containing k, we can repeat the arguments in (1.4) by restricting R to those contained in K.

Strictly decomposable cycles
2.1. Let X 1 and X 2 be smooth complex projective varieties defined over an algebraically closed subfield k of C, i.e. there are smooth projective k-varieties X i,k such that X i = X i,k ⊗ k C. Put X k = X 1,k × k X 2,k , X = X k ⊗ k C as in the introduction. We say that a cycle ζ on X is strictly decomposable if there exist subfields K i of C finitely generated over k, together with cycles ζ i on X i,K i := X i ⊗ k K i for i = 1, 2 such that the canonical morphism K 1 ⊗ k K 2 → C is injective, and ζ coincides with the base change of the cycle by K 1 ⊗ k K 2 → C (replacing k with a finite extension if necessary). We say that a strictly decomposable cycle is of bicodimension (p 1 , p 2 ) if codim ζ i = p i . Put p = p 1 + p 2 . Let R i be a finitely generated smooth k-subalgebra of K i such that the fraction field of denote the Künneth components of the cycle class of If ξ 0 i = 0, then ζ i belongs to F 1 L CH p i (X i,k ⊗ k K i ) Q by (1.3) together with Remark (iii) in (1.5), and Gr 1 F L cl(ζ i ) gives by forgetting the k-structure, see (1.3.3).

2.2.
Proposition. With the above notation, ξ 1 i = 0 if and only if ξ 1 i comes from ξ ′ i ∈ Ext 1 MHS (Q, H i ) by the pull-back under a S i : S i → Spec C. In the case p i = 1, these conditions are further equivalent to that ζ i comes from CH 1 (X i,k ) Q .
Proof. This follows from the short exact sequence (1.2.2) for r = 1 and k = C, where the first term is isomorphic to Ext 1 MHS (Q, H i ), and the first morphism is induced by the pull-back functor for a S i . Indeed, ξ 1 i coincides with the image of ξ 1 i in the last term, and we get the first equivalence. Then the second is clear because the cycle map to Deligne cohomology is an isomorphism in the divisor case and the base change by k → C induces an injective morphism of Chow groups with rational coefficients.

Remark. By (2.2) the cohomology class ξ 1
i measures how the cycle ζ i ⊗ k C varies along the parameter space S i . If the first equivalent conditions of (2.2) are satisfied, we may assume R i = k as long as Gr 1 F L cl(ζ i ) is concerned, e.g. if p i = 1. (Indeed, we can restrict to a k-valued point of Spec R i .)

Definition. We say that
A strictly decomposable cycle ζ is called separately degenerate if one of ζ i is degenerate, and cohomologically degenerate if ξ 0 i = ξ 1 i = 0 for both i, and degenerate if it is separately degenerate or cohomologically degenerate. In other words, ζ is nondegenerate if one of ξ 0 i and ξ 1 i does not vanish for each i and one of ξ 0 1 , ξ 1 1 , ξ 0 2 , ξ 1 2 does not vanish. 2.5. Remark. Assume p i = 1. Then ζ i is degenerate if and only if it is zero in the Chow group with rational coefficients. If ζ is cohomologically degenerate, it comes from a cycle on X k by the pull-back under the projection to X k , see (2.2). If furthermore k is a number field, it is expected that ζ is rationally equivalent to zero according to the conjecture of Beilinson [2] and Bloch (see also [8], [26]). If ζ is nondegenerate, however, we can detect it by the refined cycle map due to the following: 2.6. Theorem. Let ζ be a nondegenerate, strictly decomposable cycle of codimension p, and ζ 1 , ζ 2 be as in (2.1). Then cl(ζ) = 0. More precisely, if r = #{i : and ξ is the external product of ξ 1 and ξ 2 using (1.3.1). Since Q(p)) and hence ζ ∈ F r L CH p (X) Q for r as above. Then the assertion is clear if one of ξ 0 i does not vanish. Indeed, it is reduced to the case R i = k (replacing k with a finite extension if necessary), and follows from the nonvanishing of ξ 0 Thus we may assume ξ 0 1 = ξ 0 2 = 0 and r = 2. By (1.3.1) we have the product in the notation of (2.1.3), and it is induced by Gr 2 F L ξ. We have to show that the pull-back of ξ 1 1 ⊗ ξ 1 2 by any dominant morphism S ′ → S does not vanish. It is enough to show that the restriction of ξ 1 1 ⊗ ξ 1 2 to any nonempty open subvariety U of S does not vanish by the same argument as in [36], 2.6. (Indeed, we can restrict to a subvariety of S ′ which is finité etale over an open subvariety of S, and then take the direct image.) If both ξ 1 1 and ξ 1 2 are nonzero, the assertion is easy, and follows from the same argument as in [36], 4.4. Indeed, if H 2 (S, Q) ∨ denotes the dual of H 2 (S, Q), then ξ 1 1 ⊗ ξ 1 2 corresponds to a morphism of mixed Hodge structures and its image has level 2 by hypothesis. (Here the level of a Hodge structure is the difference between the maximal and minimal numbers p such that Gr p F = 0.) This level does not change by replacing S with any nonempty open subvariety U of S, see [16]. So we get the assertion in this case.
Now it remains to consider the case ξ 1 1 = 0 and ξ 1 2 = 0 (hence ξ ′ 2 = 0). By (2.2), ξ 1 1 ⊗ ξ 1 2 is the pull-back of ξ 1 1 ⊗ ξ ′ 2 by S → S 1 , and we may replace R 2 with k. So the assertion is reduced to the case R 2 = k. Then by (1.2.2) it is sufficient to show the nonvanishing of Assume ξ 1 1 ⊗ ξ ′ 2 = 0. Using the long exact sequence associated with (2.6.2) We have to show that this implies Thus, by using a direct sum decomposition of H ′ 0 ⊗ H 1 ⊗ H 2 compatible with (Im u) ⊗ H 2 and H ′ 0 ⊗ ( j Im v j ), the assertion is reduced to the following: 2.7. Lemma. Let H i be a polarizable Hodge structure of weight 1 for i = 0, 1, 2. Let u ∈ Hom HS (Q(−1), H 0 ⊗ H 1 ), and v j ∈ Hom HS (Q(−1), H 1 ⊗ H 2 ). Then Proof. It is sufficient to show the assertion for the underlying R-Hodge structure. Using the projections to the direct factors, we may assume that H 0 , H 1 , H 2 are simple (i.e. 2-dimensional), and furthermore H 0 is isomorphic to H 1 , H 2 (because u = 0 or v j = 0 otherwise). Let {e, e} be a basis of H 0 = H 1 = H 2 over C such that e generates F 1 , and e is the complex conjugate of e. Then u = a e ⊗ e + a e ⊗ e and v j = b j e ⊗ e + b j e ⊗ e for a, b j ∈ C. In particular, H 0 ⊗ ( j Im v j ) is contained in the subspace generated by H 0 ⊗ e ⊗ e and H 0 ⊗ e ⊗ e. Thus we get (2.7.1). This completes the proof of Theorem (2.6).

2.9.
Examples. (i) Let C 1 , C 2 be smooth proper curves over C, and assume they are defined over k (i.e. C i = C i,k ⊗ k C for a curve C i,k over k). Let P i , Q i ∈ C i (C), and assume that Q 1 , Q 2 and P 2 are defined over k, but P 1 is not, and that [P 2 ] − [Q 2 ] is not torsion in the Jacobian. Then Theorem (2.6) implies Note that the condition on [P 2 ] − [Q 2 ] is satisfied if there exists an algebraically closed subfield k 0 of k such that C 2 , Q 2 are defined over k 0 , but not P 2 (using the embedding of C 2 in the Jacobian). Replacing k, (2.9.1) holds if Q 1 , Q 2 are defined over k, and if the field of definition of P 1 (i.e. the image of k(C 1 ) to C by the morphism defined by P 1 ) and that of P 2 are algebraically independent (i.e. if there exists an algebraically closed subfield k ′ of C which contains k and such that P 2 is defined over k ′ , but P 1 is not). See [1] for the case C 1 = C 2 , P 1 = P 2 and Q 1 = Q 2 , where it is assumed that the rank of the Néron-Severi group of C 1 × C 2 is 3 and div(K C 1 ) − (2g − 2)[Q 1 ] is not torsion in the Jacobian of C 1 (here K C 1 is the canonical line bundle). Note that the assertion does not hold for an elliptic curve by the following.
(ii) Let E be an elliptic curve over C with the origin O. Then for any P ∈ E(C) This can be verified by taking Q such that 2Q = P , and using the diagonal and antidiagonal curves passing (Q, Q) (which is viewed as a new origin).
3. The higher cycle case 3.1. For a smooth complex variety X and a positive integer m, let CH p (X, m) Q denote the higher Chow group of X with rational coefficients [9]. By [36] we have the refined cycle map which is compatible with the cycle map in [35] (3. The latter is expected to be compatible with the one in [10], [19]. (For m = 1, it is easy to verify this by reducing to the case p = 1.) We assume X = X k ⊗ k C and X k is smooth proper. Then CH p (X, m) Q and H 2p−m D (X, Q k (p)) have the Leray filtration F L as before, and (3.1.1) induces because H 2p−m (X, Q k ) is pure of weight 2p − m. This is the inductive limit of Let X i , X i,k (i = 1, 2) and X, X k be as in (2.1). In this paper we define the notion of a strictly decomposable higher cycle (in the strong sense) by allowing a higher cycle for ζ 2 in the notation of (2.1). More precisely, we say that a higher cycle ζ ∈ CH p (X, m) Q is strictly decomposable if there exist subfields K 1 , K 2 of C and cycles ζ 1 ∈ CH p 1 (X 1,k ⊗ k K 1 ) Q , ζ 2 ∈ CH p 2 (X 2,k ⊗ k K 2 , m) Q such that the K i satisfy the conditions in (2.1), p 1 + p 2 = p, and ζ is the base change of ζ 1 × ζ 2 as before.
Proof. The argument is similar to (2.6). We can identify ξ 1 2 with a morphism of mixed Hodge structures where the source is the dual of H 1 (S 2 , Q), and the target is pure of weight −m − 1. Therefore the image of (3.3.1) is a direct sum of Q(1) if m = 1, and vanishes otherwise. For a nonempty open subvariety S ′ 2 of S 2 , we have the injectivity of the restriction morphism Then, taking the tensor with H 2p 1 (X 1 , Q)(p 1 )⊗H 2 , we get the assertion in the case ξ 0 1 = 0 and ξ 1 2 = 0 (where we may assume R 1 = k). If ξ 0 1 = 0 and ξ 1 2 = 0, we may assume R 1 = R 2 = k, and the assertion is clear. Thus the assertion is reduced to the case ξ 0 1 = 0. The argument is similar if both ξ 1 1 and ξ 1 2 are nonzero (in particular, m = 1). For any nonempty open subset S ′ of S := S 1 × S 2 , we have the injectivity of the restriction morphism Thus we get the assertion in this case, and we may assume either ξ 1 1 or ξ 1 2 vanishes. If ξ 1 1 = 0 and ξ 1 2 = 0 (hence ξ ′ 2 = 0), then the assertion follows immediately from the long exact sequence associated to (2.6.2), because m > 0. If ξ 1 1 = 0 and ξ 1 2 = 0 (hence ξ ′ 1 = 0 and m = 1), then we may assume R 1 = k and H 2 is a direct sum of Q(1) because the image of (3.3.1) is a direct factor of the target (and the similar assertion holds over k). Furthermore, it is sufficient to show the nonvanishing of the composition of ξ ′ 1 ⊗ ξ 1 2 with the morphism induced by the projection H 1 (S 2 , Q) → Gr W 2 H 1 (S 2 , Q) after replacing S 2 with any nonempty open subvariety. Thus the assertion follows from the injectivity of (3.3.2). This finishes the proof of (3.3). 3.5. Remarks. (i) Theorems (2.6) and (3.3) hold also for a variety X over a subfield K of C, where the K i are assumed to be subfields of K, see Remark (iii) of (1.5).
(ii) Let C be a curve of positive genus, and assume X 1 = X 2 = C, K 1 = k(X 1,k ), and K 2 = k. Then (2.6) simplifies Nori's construction explained in [38] (because (2.6) does not assume the condition on the relation to the transcendental part of the second cohomology). So Theorems (2.6) and (3.3) may be viewed as a refinement of Nori's construction.

4.5.
Remark. It is known that CH p+1 ind (X, 1) Q is uncountable in some cases, see e.g. [13] for the case of the Jacobian of a curve, and also [14], [35] for examples such that the hypothesis of (0.3) is satisfied. (We can show that a higher cycle on a product of elliptic curves in [22] is decomposable by restricting to the generic fiber of the first projection.)