Algebraic Cycles and Fibrations

Charles Vial

Received: March 23, 2012 Revised: July 19, 2013

Communicated by Alexander Merkurjev

Abstract. Let f : X → B be a projective surjective morphism between quasi-projective varieties. The goal of this paper is the study of the Chow groups of X in terms of the Chow groups of B and of the fibres of f. One of the applications concerns quadric bundles. When X and B are smooth projective and when f is a flat quadric fibration, we show that the Chow motive of X is “built” from the motives of varieties of dimension less than the dimension of B.

2010 Mathematics Subject Classification: 14C15, 14C25, 14C05, 14D99 Keywords and Phrases: Algebraic cycles, Chow groups, quadric bundles, motives, Chow–Künneth decomposition.

Contents

1. Surjective morphisms and motives 1526 2. On the Chow groups of the fibres 1530 3. A generalisation of the projective bundle formula 1532 4. On the motive of quadric bundles 1534 5. On the motive of a smooth blow-up 1536 6. Chow groups of varieties fibred by varieties

with small Chow groups 1540 7. Applications 1547 References 1551 For a scheme X over a field k, CH_{i}(X) denotes the rational Chow group of idimensional cycles on X modulo rational equivalence. Throughout, f : X → B will be a projective surjective morphism defined over k from a quasi-projective variety X of dimension d_{X} to an irreducible quasi-projective variety B of dimension d_{B}, with various extra assumptions which will be explicitly stated. Let h be the class of a hyperplane section in the Picard group of X. Intersecting with h induces an action CH_{i}(X) → CH_{i−1}(X) still denoted h. Our first observation is Proposition 1.6: when B is smooth, the map

(1)  ⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(B) ⟶ CH_{l}(X)

is injective for all l and a left-inverse can be expressed as a combination of the proper pushforward f_{∗}, the refined pullback f^{∗} and intersection with h. It is then not too surprising that, when both B and X are smooth projective, the morphism of Chow motives

⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) ⟶ 𝔥(X)

is split injective; see Theorem 1.4. By taking a cohomological realisation, for instance by taking Betti cohomology if k ⊆ C, we thus obtain that the map

⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} H^{n−2i}(B, Q) ⟶ H^{n}(X, Q)

is split injective for all n and thus realises the left-hand side group as a sub-Hodge structure of the right-hand side group. This observation can be considered as a natural generalisation of the elementary fact that a smooth projective variety X of Picard rank 1 does not admit a non-constant dominant map to a smooth projective variety of smaller dimension. Let now Ω be a universal domain containing k, that is an algebraically closed field containing k which has infinite transcendence degree over its prime subfield. Let us assume that there is an integer n such that the fibres X_{b} of f over Ω-points b of B satisfy CH_{l}(X_{b}) = Q for all l < n. If f is flat, then Theorem 3.2 shows that (1) is surjective for all l < n. When X and B are both smooth projective, we deduce in Theorem 4.2 a direct sum decomposition of the Chow motive of X as

(2)  𝔥(X) ≅ ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) ⊕ M(n),

where M is isomorphic to a direct summand of the motive of some smooth projective variety Z of dimension d_{X} − 2n. This notably applies when X is a projective bundle over a smooth projective variety B to give the well-known isomorphism ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) ⟶^{≃} 𝔥(X). Such a morphism is usually shown to be an isomorphism by an existence principle, namely Manin’s identity principle. Here, we actually exhibit an explicit inverse to that isomorphism. The same arguments are used in Theorem 5.3 to provide an explicit inverse to the smooth blow-up formula for Chow groups. More interesting is the case when a smooth projective variety X is fibred by complete intersections of low degree. For instance, the decomposition (2) makes it possible in Corollary 4.4 to construct a Murre decomposition (see Definition 4.3) for smooth projective varieties fibred by quadrics over a surface, thereby generalising a result of del Angel and Müller–Stach [5] where a Murre decomposition was constructed for 3-folds fibred by conics over a surface, and also generalising a previous result [27] where, in particular, a Murre decomposition was constructed for 4-folds fibred by quadric surfaces over a surface. Another consequence of the decomposition (2) is that rational and numerical equivalence agree on smooth projective varieties X fibred by quadrics over a curve or a surface defined over a finite field; see Corollary 4.8. It should be mentioned that our approach bypasses the technique of Gordon–Hanamura–Murre [8], where Chow–Künneth decompositions are constructed from relative Chow–Künneth decompositions. In our case, we do not require the existence of a relative Chow–Künneth decomposition, nor do we require f to be smooth away from finitely many points as is the case in [8]. Finally, it should be noted, for instance if X is complex smooth projective fibred by quadrics, that (2) actually computes some of the Hodge numbers of X without going through a detailed analysis of the Leray–Serre spectral sequence.

More generally, we are interested in computing, in some sense, the Chow groups of X in terms of the Chow groups of B and of the fibres of f. Let us first clarify what is meant by “fibres”. We observed in [27, Theorem 1.3] that if B is smooth and if a general fibre of f has trivial Chow group of zero-cycles (i.e. if it is spanned by the class of a point), then f_{∗} : CH_{0}(X) → CH_{0}(B) is an isomorphism with inverse a rational multiple of h^{d_{X}−d_{B}} ∘ f^{∗}. We thus see that, as far as zero-cycles on the fibres are concerned, it is enough to consider only the general fibre. For that matter, we show in Proposition 2.4 that, provided the ground field is a universal domain, it is actually enough that a very general fibre have trivial Chow group of zero-cycles. However, if one is willing to deal with positive-dimensional cycles, it is no longer possible to ignore the Chow groups of some of the fibres. For instance, if X̃_{Y} → X is a smooth blow-up along a smooth center Y ⊆ X, then CH_{1}(X̃_{Y}) is isomorphic to CH_{1}(X) ⊕ CH_{0}(Y), although a general fibre of X̃_{Y} → X is reduced to a point and hence has trivial CH_{1}. One may argue that a smooth blow-up is not flat. Let us however consider as in [1] a complex flat conic fibration f : X → P^{2}, where X is smooth projective. All fibres F of f satisfy CH_{0}(F) = Q. A smooth fibre F of f is isomorphic to P^{1} and hence satisfies CH_{1}(F) = Q. A singular fibre F of f is either a double line, or the union of two lines meeting at a point. In the latter case CH_{1}(F) = Q⊕Q. This reflects in CH_{1}(X) and, as shown in [1], CH_{1}(X)_{hom} is isomorphic to the Prym variety attached to the discriminant curve of f. This suggests that a careful analysis of the degenerations of f is required in order to derive some precise information on the Chow groups of X. On another perspective, the following examples show what kind of limitation is to be expected when dealing with fibres with non-trivial Chow groups. For instance, consider a complex Enriques surface S and let T → S be the 2-covering by a K3-surface T. We know that CH_{0}(S) = Q, and the fibres T_{s} are disjoint union of two points and hence satisfy CH_{0}(T_{s}) = Q ⊕ Q. However, we cannot expect CH_{0}(T) to be correlated in some way to the Chow groups of S and of the fibres, because a theorem of Mumford [17] says that CH_{0}(T) is infinite-dimensional in a precise sense. Another example is given by taking a pencil of high-degree hypersurfaces in P^{n}. Assume that the base locus Z is smooth. By blowing up Z, we get a morphism P̃_{Z}^{n} → P^{1}. In that case, CH_{0}(P^{1}) = Q and the CH_{0} of the fibres is infinite-dimensional, but CH_{0}(P̃_{Z}^{n}) = Q. Going back to the case where the general fibre of f : X → B has trivial Chow group of zero-cycles, we see that CH_{0}(X) is supported on a linear section of dimension d_{B}. We say that CH_{0}(X) has niveau d_{B}. More generally, Laterveer [15] defines a notion of niveau on Chow groups as follows. For a variety X, the group CH_{i}(X) is said to have niveau ≤ n if there exists a closed subscheme Z of X of dimension ≤ i + n such that CH_{i}(Z) → CH_{i}(X) is surjective, in other words if the i-cycles on X are supported in dimension i + n. It can be proved [27, Theorem 1.7] that if a general fibre F of f : X → B is such that CH_{0}(F) has niveau ≤ 1, then CH_{0}(X) has niveau ≤ d_{B} + 1. In that context, a somewhat more precise question is: what can be said about the niveau of the Chow groups of X in terms of the niveau of the Chow groups of the fibres of f : X → S? A statement one would hope for is the following: if the fibres X_{b} of f : X → B are such that CH_{∗}(X_{b}) has niveau ≤ n for all Ω-points b ∈ B, then CH_{∗}(X) has niveau ≤ n + d_{B}. We cannot prove such a general statement but we prove it when some of the Chow groups of the fibres of f are either spanned by linear sections or have niveau 0, i.e. when they are finitedimensional Q-vector spaces. Precisely, if f : X → B is a complex projective surjective morphism onto a smooth quasi-projective variety B, we show that CH_{l}(X) has niveau ≤ d_{B} in the following cases:

• CH_{i}(X_{b}) = Q for all i ≤ l and all b ∈ B(C) (Theorem 6.10);

• d_{B} = 1 and CH_{i}(X_{b}) is finitely generated for all i ≤ l and all b ∈ B(C) (Theorem 6.12);

• f is smooth away from finitely many points, CH_{i}(X_{b}) = Q for all i < l and CH_{l}(X_{b}) is finitely generated, for all b ∈ B(C) (Theorem 6.13).

These results, which are presented in Section 6, complement the generalisation of the projective bundle formula of Theorem 3.2 by dropping the flatness condition on f and by requiring in some cases that the Chow groups of the fibres be finitely generated instead of one-dimensional. Their proofs use standard techniques such as localisation for Chow groups (for that matter, information on the Chow groups of the fibres of f is extracted from information on the Chow groups of the closed fibres of f in Section 2), relative Hilbert schemes and a Baire category argument. Let us mention that the assumption of Theorem 6.13 on the singular locus of f being finite is also required in [8] where the construction of relative Chow–Künneth decompositions is considered. Finally, Theorem 7.1 gathers known results about smooth projective varieties whose Chow groups have small niveau. Together with the results above, in Section 7, we prove some conjectures on algebraic cycles (such as Kimura’s finite-dimensionality conjecture [12], Murre’s conjectures [18], Grothendieck’s standard conjectures [13], the Hodge conjecture) for some smooth projective varieties fibred by very low degree complete intersection, or by cellular varieties over surfaces. For instance, we show the existence of a Murre decomposition for smooth projective varieties fibred by cellular varieties over a curve (Proposition 7.7) and for 6-folds fibred by cubics over a curve (Proposition 7.4), and the standard conjectures for varieties fibred by smooth cellular varieties of dimension ≤ 4 (Proposition 7.7) or by quadrics (Proposition 7.3) over a surface.

Notations. We work over a field k and Ω denotes a universal domain that contains k. A variety over k is a reduced scheme of finite type over k. Throughout, f : X → B denotes a projective surjective morphism defined over k from a quasi-projective variety X of dimension d_{X} to an irreducible quasi-projective variety B of dimension d_{B}. Given a scheme X over k, the group CH_{i}(X) is the Q-vector space with basis the i-dimensional irreducible reduced subschemes of X modulo rational equivalence. By definition, we set CH_{j}(X) = 0 for j < 0 and we say that CH_{i}(X) is finitely generated if it is finitely generated as a Q-vector space, i.e. if it is a finite-dimensional Q-vector space. If Z is an irreducible closed subscheme of X, we write [Z] for the class of Z in CH_{∗}(X). If α is the class of a cycle in CH_{∗}(X), we write |α| for the support in X of a cycle representing α. If Y is a scheme over k and if β is a cycle in CH_{∗}(X × Y), we define its transpose ^{t}β ∈ CH_{∗}(Y × X) to be the proper pushforward of β under the obvious map τ : X × Y → Y × X. If X and Y are smooth projective, a cycle γ ∈ CH_{∗}(X × Y) is called a correspondence. The correspondence γ acts both on CH_{∗}(X) and CH_{∗}(Y) in the following way. Let p_{X} : X × Y → X and p_{Y} : X × Y → Y be the first and second projections, respectively. These are proper and flat and we may define, for α ∈ CH_{∗}(X), γ_{∗}α := (p_{Y})_{∗}(γ · p_{X}^{∗}α). Here “·” is the intersection product on non-singular varieties as defined in [7, §8]. We then define, for β ∈ CH_{∗}(Y), γ^{∗}β := (^{t}γ)_{∗}β. Given another smooth projective variety Z and a correspondence γ′ ∈ CH_{∗}(Y × Z), the composite γ′ ∘ γ ∈ CH_{∗}(X × Z) is defined to be (p_{XZ})_{∗}(p_{XY}^{∗} γ · p_{Y Z}^{∗}γ′), where p_{XY} : X × Y × Z → X × Y is the projection and likewise for p_{XZ} and p_{Y Z}. The composition of correspondence is compatible with the action of correspondences on Chow groups [7, §16]. Motives are defined in a covariant setting and the notations are those of [27]. Briefly, a Chow motive (or motive, for short) M is a triple (X, p, n) where X is a variety of pure dimension d_{X}, p ∈ CH_{d}(X × X) is an idempotent (p ∘ p = p) and n is an integer. The motive of X is denoted 𝔥(X) and, by definition, is the motive (X, Δ_{X}, 0) where Δ_{X} is the class in CH_{d_{X}} (X × X) of the diagonal in X × X. We write 𝟙 for the unit motive (Spec k, Δ_{Spec k}, 0) = 𝔥(Spec k). With our covariant setting, we have 𝔥(P^{1}) = 𝟙 ⊕ 𝟙(1). A morphism between two motives (X, p, n) and (Y, q, m) is a correspondence in q∘CH_{d_{X}+n−m}(X ×Y)∘p. If f : X → Y is a morphism, Γ_{f} denotes the graph of f in X ×Y. By abuse, we also write Γ_{f} ∈ CH_{d_{X}} (X × Y) for the class of the graph of f. It defines a morphism Γ_{f} : 𝔥(X) → 𝔥(Y). By definition we have CH_{i}(X, p, n) = p_{∗} CH_{i−n}(X) and H_{i}(X, p, n) = p_{∗}H_{i−2n}(X), where we write H_{i}(X) := H^{2d−i}(X(C), Q) for singular homology when k ⊆ C, or H_{i}(X) := H^{2d−i}(X\overline{k}, Q_{ℓ}) for ℓ-adic homology (ℓ ≠ char k) otherwise. Given an irreducible scheme Y over k, η_{Y} denotes the generic point of Y. If f : X → B and if Y is a closed irreducible subscheme of B, X_{η_{Y}} denotes the fibre of f over the generic point of Y and X_{\overline{η}_{Y}} denotes the fibre of f over a geometric generic point of Y.

1. Surjective morphisms and motives

Let us start by recalling a few facts about intersection theory. Let f : X → Y be a morphism of schemes defined over k and let l be an integer. If f is proper, then there is a well-defined proper pushforward map f_{∗} : CH_{l}(X) → CH_{l}(Y); see [7, §1.4]. If f is flat, then there is a well-defined flat pullback map f^{∗} : CH^{l}(Y) → CH^{l}(X); see [7, §1.7]. Pullbacks can also be defined in the following two situations. On the one hand, if D is a Cartier divisor with support ι : |D| ↪ X, there is a well-defined Gysin map ι^{∗} : CH_{l}(X) → CH_{l−1}(|D|) and the composite ι_{∗} ∘ ι^{∗} : CH_{l}(X) → CH_{l−1}(X) does not depend on the linear equivalence class of D, that is, there is a well-defined action of the Picard group Pic(X) on CH_{l}(X); see [7, §2]. For instance, if X is a quasi-projective variety given with a fixed embedding X ↪ P^{N}, then there is a well-defined map h : CH_{l}(X) → CH_{l−1}(X) given by intersecting with a hyperplane section of X. More generally, if τ : Y ↪ X is a locally complete intersection of codimension r, then there is a well-defined Gysin map τ^{∗} : CH_{l}(X) → CH_{l−r}(Y); see [7, §6]. For n ≥ 0, we write h^{0} = id : CH_{l}(X) → CH_{l}(X) and h^{n} for the nfold composite h ∘ … ∘ h : CH_{l}(X) → CH_{l−n}(X). By functoriality of Gysin maps [7, §6.5], if ι^{n} : H^{n} ↪ X denotes a linear section of codimension n, then the composite map h ∘ … ∘ h coincides with ι_{∗}^{n} ∘ (ι^{n})^{∗}. When X is smooth projective, we write Δ_{H^{n}} for the diagonal inside H^{n} × H^{n}, and the correspondence Γ_{ι^{n}} ∘ ^{t}Γ_{ι^{n}} = (ι^{n} × ι^{n})_{∗}[Δ_{H^{n}}] ∈ CH_{d_{X}−n}(X × X) induces a map CH_{l}(X) → CH_{l−n}(X) that coincides with the map h^{n}; see [7, §16]. By abuse, we also write h^{n} = Γ_{ι^{n}} ∘ ^{t}Γ_{ι^{n}} for n > 0 and h^{0} := [Δ_{X}]. On the other hand, if f : X → Y is a morphism to a non-singular variety Y and if x ∈ CH_{∗}(X) and y ∈ CH_{∗}(Y), then there is a well-defined refined intersection product x ·_{f} y ∈ CH_{∗}(|x| ∩ f^{−1}(|y|)), where “∩” denotes the scheme-theoretic intersection; see [7, §8]. The pullback f^{∗}y is then defined to be the proper pushforward of [X] ·_{f} y in CH_{∗}(X). Let us denote γ_{f} : X → X × Y the morphism x ↦ (x, f(x)). Because Y is non-singular, this morphism is a locally complete intersection morphism and the pullback f^{∗} is by definition γ_{f}^{∗} ∘ p_{Y}^{∗}, where p_{Y} : X × Y → Y is the projection and γ_{f}^{∗} is the Gysin map; see [7, §8].

Finally, if f is flat, then this pullback map coincides with flat pullback [7, Prop. 8.1.2].

We have the following basic lemma.

Lemma 1.1. Let f : X → B be a projective surjective morphism between two quasi-projective varieties. Let X′ ↪ X be a linear section of X of dimension ≥ d_{B}. Then f|_{X′} : X′ → B is surjective.

Proof. Let X ↪ P^{N} be an embedding of X in projective space and let H ↪ P^{N} be a linear subspace such that X′ is obtained as the pullback of X along H ↪ P^{N}. The linear subvariety H has codimension at most d_{X} − d_{B} in P^{N} while a geometric fibre of f has dimension at least d_{X} − d_{B}. Thus every geometric fibre of f meets H and hence X′. It follows that f|_{X′} is surjective. □

Lemma 1.2. Let f : X → B be a projective surjective morphism to a smooth quasi-projective variety B. Then there exists a positive integer n such that, for all i, f_{∗} ∘ h^{d_{X}−d_{B}} ∘ f^{∗} : CH_{i}(B) → CH_{i}(B) is multiplication by n. If moreover X is smooth and B is projective, then

Γ_{f} ∘ h^{d_{X}−d_{B}} ∘^{t}Γ_{f} = n · Δ_{B} ∈ CH_{d_{B}}(B × B).

Proof. Let ι′ := ι^{d_{X}−d_{B}} : H′ ↪ X be a linear section of X of dimension d_{B}. We first check, for lack of reference, that (f ∘ ι′)^{∗} = (ι′)^{∗} ∘ f^{∗} on Chow groups. Here, f ∘ ι′ and f are morphisms to the non-singular variety B and as such the pullbacks (f ∘ ι′)^{∗} and (ι′)^{∗} are the ones of [7, §8], while ι′ is the inclusion of a locally complete intersection and as such the pullback (ι′)^{∗} is the Gysin pullback of [7, §6]. Let σ ∈ CH^{∗}(B), then (ι′)^{∗}f^{∗}σ = (ι′)^{∗}γ_{f}^{∗}([X]×σ) = (γ_{f} ∘ι′)^{∗}([X]×σ), where the second equality follows from the functoriality of Gysin maps [7, §6.5]. Since γ_{f} ∘ ι′ = (ι′ × id_{B}) ∘ γ_{f∘ι′}, we get by using functoriality of Gysin maps once more that (ι′)^{∗}f^{∗}σ = γ_{f∘ι′}^{∗}(ι′ × id_{B})^{∗}([X] × σ). Now, we have (ι′ × id_{B})^{∗}([X] × σ) = (ι′)^{∗}[X] × σ = [H′] × σ; see [7, Example 6.5.2]. We therefore obtain that (ι′)^{∗}f^{∗}σ = γ_{f∘ι′}^{∗}([H′] × σ) := (f ∘ ι′)^{∗}σ, as claimed. Thus, since in addition both f and ι′ are proper, we have by functoriality of proper pushforward (f ∘ ι′)_{∗}(f ∘ ι′)^{∗} = f_{∗}ι′_{∗}(ι′)^{∗}f^{∗} = f_{∗} ∘ h^{d_{X}−d_{B}} ∘ f^{∗}. By Lemma 1.1, the composite morphism g := f ∘ ι′ is generically finite, of degree n say. It follows from the projection formula [7, Prop. 8.1.1(c)] and from the definition of proper pushforward that, for all γ ∈ CH_{i}(B),

f_{∗} ∘ h^{d_{X}−d_{B}} ∘ f^{∗}γ = g_{∗}([H′] · g^{∗}γ) = g_{∗}([H′]) · γ = n[B] · γ = nγ.

Assume now that X and B are smooth projective. In that case, we have Γ_{f} ∘ h^{d_{X}−d_{B}} ∘ ^{t}Γ_{f} = Γ_{g} ∘ ^{t}Γ_{g} := (p_{1,3})_{∗}(p_{1,2}^{∗}^{t}Γ_{g} · p_{2,3}^{∗}Γ_{g}), where p_{i,j} denotes projection from B × H′ × B to the (i, j)-th factor. By refined intersection, we see that Γ_{g} ∘ ^{t}Γ_{g} is supported on (p_{1,3})([^{t}Γ_{g} × B] ∩ [B × Γ_{g}]), which itself is supported on the diagonal of B × B. Thus Γ_{f} ∘ h^{d_{X}−d_{B}} ∘^{t}Γ_{f} is a multiple of Δ_{B}. We have already showed that f_{∗} ∘ h^{d_{X}−d_{B}} ∘ f^{∗} = (Γ_{f} ∘ h^{d_{X}−d_{B}} ∘^{t}Γ_{f})_{∗} acts by multiplication by n on CH_{i}(B). Therefore, Γ_{f} ∘ h^{d_{X}−d_{B}} ∘^{t}Γ_{f} = n · Δ_{B}. □

The following lemma is reminiscent of [7, Prop. 3.1.(a)].

Lemma 1.3. Let f : X → B be a projective morphism to a smooth quasiprojective variety B. Then, for all i, f_{∗} ∘ h^{l} ∘ f^{∗} : CH_{i}(B) → CH_{i+d_{X}−d_{B}−l}(B) is the zero map for all l < d_{X}−d_{B}. If moreover X is smooth and B is projective, then

Γ_{f} ∘ h^{l} ∘^{t}Γ_{f} = 0 ∈ CH_{d_{X}−l}(B × B) for all l < d_{X} − d_{B}.

Proof. By refined intersection [7, §8], the pullback f^{∗}α is represented by a welldefined class in CH_{i+d_{X}−d_{B}}(f^{−1}(|α|)) for any cycle α ∈ CH_{i}(B). It follows that h^{l} ∘ f^{∗}α is represented by a well-defined class in CH_{i+d_{X}−d_{B}−l}(f^{−1}(|α|)). Since f|_{f^{−1}(|α|)} : f^{−1}(|α|) → |α| is proper, we see by proper pushforward that f_{∗} ∘ h^{l} ∘ f^{∗}α is represented by a well-defined cycle β ∈ CH_{i+d_{X}−d_{B}−l}(|α|). But then, dim |α| = i so that if l < d_{X} − d_{B}, then CH_{i+d_{X}−d_{B}−l}(|α|) = 0. Let us now assume that X and B are smooth projective. Let ι^{l} : H^{l} ↪ X be a linear section of X of codimension l, and let h^{l} be the class of (ι^{l} × ι^{l})(Δ_{H^{l}}) in CH_{d_{X}−l}(X × X). By definition we have Γ_{f} ∘ h^{l} ∘ ^{t}Γ_{f} = (p_{1,4})_{∗}(p_{1,2}^{∗}^{t}Γ_{f} · p_{2,3}^{∗}h^{l} · p_{3,4}^{∗}Γ_{f}), where p_{i,j} denotes projection from B × X × X × B to the (i, j)- th factor. These projections are flat morphisms, therefore by flat pullback we have p_{1,2}^{∗}^{t}Γ_{f} = [^{t}Γ_{f} ×X ×B], p_{2,3}^{∗}h^{l} = [B×Δ_{H^{l}} ×B] and p_{3,4}^{∗}Γ_{f} = [B×X ×Γ_{f}]. By refined intersection, the intersection of the closed subschemes ^{t}Γ_{f} × X × B, B × Δ_{H^{l}} × B and B × X × Γ_{f} of B × X × X × B defines a (d_{X} − l)-dimensional class supported on their scheme-theoretic intersection {(f(h), h, h, f(h)) : h ∈ H^{l}} ⊂ B × X × X × B. Since f is projective, this is a closed subset of dimension d_{X} − l. Also its image under the projection p_{1,4} has dimension at most d_{B}, which is strictly less than d_{X} − l by the assumption made on l. The projection p_{1,4} is a proper map and hence, by proper pushforward, we get that (p_{1,4})_{∗}[{(f(h), h, h, f(h)) ∈ B × X × X × B : h ∈ H^{l}}] = 0. □

Theorem 1.4. Let f : X → B be a surjective morphism of smooth projective varieties over k. Consider the following two morphisms of motives

Φ := ⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘^{t}Γ_{f} : ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) ⟶ 𝔥(X)

and

Ψ := ⨁_{i=0}^{d_{X}−d_{B}} Γ_{f} ∘ h^{i} : 𝔥(X) ⟶ ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i).

Then Ψ ∘ Φ is an automorphism.

Proof. The endomorphism Ψ ∘ Φ : ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) → ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) can be represented by the (d_{X} − d_{B} + 1) × (d_{X} − d_{B} + 1)-matrix whose (i, j)^{th}-entries are the morphisms

(Ψ ∘ Φ)_{i,j} = Γ_{f} ∘ h^{d_{X}−d_{B}−(j−i)} ∘^{t}Γ_{f} : 𝔥(B)(j − 1) → 𝔥(B)(i − 1).

By Lemma 1.2, there is a non-zero integer n such that the diagonal entries satisfy (Ψ ∘ Φ)_{i,i} = n · id_{𝔥(B)(i−1)}. By Lemma 1.3, (Ψ ∘ Φ)_{i,j} = 0 as soon as j > i. Therefore

N := id − \frac{1}{n} · Ψ ∘ Φ

is a nilpotent endomorphism of ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) with N^{d_{X}−d_{B}+1} = 0. Let us define

Ξ := (n · id − n · N)^{−1}
= \frac{1}{n} · (id + N + N^{2} + · + N^{d_{X}−d_{B}}).

It then follows that Ξ is the inverse of Ψ ∘ Φ. □

In the situation of Theorem 1.4, the morphism

Θ := Ξ ∘ Ψ

then defines a left-inverse to Φ and the endomorphism

p := Φ ∘ Θ = Φ ∘ Ξ ∘ Ψ ∈ End(𝔥(X))

is an idempotent.

Proposition 1.5. With the notations above, the idempotent p ∈ End(𝔥(X)) = CH_{d_{X}} (X × X) satisfies p = ^{t}p. Moreover, the morphism Ψ ∘ p : (X, p) → ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) is an isomorphism with inverse p ∘ Φ ∘ Ξ.

Proof. The second claim consists of the following identities: Ψ ∘ p ∘ p ∘ Φ ∘ Ξ = Ψ∘p∘Φ∘Ξ = Ψ∘Φ∘Ξ∘Ψ∘Φ∘Ξ = id∘id = id and p∘Φ∘Ξ∘Ψ∘p = p∘p∘p = p. As for the first claim, we have

p = \frac{1}{n} · Φ ∘ (1 + N + … + N^{d_{X}−d_{B}}) ∘ Ψ.

Recall that N = id − \frac{1}{n} · Ψ ∘ Φ, so that it is enough to see that ^{t}(Φ ∘ Ψ) = Φ ∘ Ψ. A straightforward computation gives

Φ ∘ Ψ = ∑_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘^{t}Γ_{f} ∘ Γ_{f} ∘ h^{i}.

We may then conclude by noting that the correspondence h ∈ CH_{d_{X}−1}(X ×X) satisfies h = ^{t}h. □

Finally, let us conclude with the following counterpart of Theorem 1.4 that deals with the Chow groups of quasi-projective varieties.

Proposition 1.6. Let f : X → B be a projective surjective morphism to a smooth quasi-projective variety B. Then the map

Φ_{∗} = ⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(B) ⟶ CH_{l}(X)

is split injective and its left-inverse is a polynomial function in f_{∗}, f^{∗} and h.

Proof. Thanks to Lemma 1.2 and to Lemma 1.3, there is a non-zero integer n such that

f_{∗} ∘ h^{i} ∘ f^{∗} : CH_{l}(B) → CH_{l+d_{X}−d_{B}−i}(B)

is multiplication by n if i = d_{X} − d_{B} and is zero if i < d_{X} − d_{B}. Let us write Ψ_{∗} for ⨁_{j=0}^{d_{X}−d_{B}} f_{∗} ∘ h^{j} : CH_{l}(X) → ⨁_{j=0}^{d_{X}−d_{B}} CH_{l−j}(B). In order to prove the injectivity of Φ_{∗}, it suffices to show that the composite

Ψ_{∗} ∘ Φ_{∗} : ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(B) ⟶ CH_{l}(X) ⟶ ⨁_{j=0}^{d_{X}−d_{B}} CH_{l−j}(B)

is an isomorphism. But then, as in the proof of Theorem 1.4, we see that Ψ_{∗}∘Φ_{∗} can be represented by a lower triangular matrix whose diagonal entries’ action on CH_{l−i}(B) is given by multiplication by n. □

Remark 1.7. Note that the conclusion of Proposition 1.6 also holds for a flat and projective surjective morphism f : X → B of quasi-projective varieties.

2. On the Chow groups of the fibres

In this section, we fix a universal domain Ω. The following statement was communicated to me by Burt Totaro.

Lemma 2.1. Let f : X → B be a morphism of varieties over Ω and let F be a geometric generic fibre of f. Then there is a subset U ⊆ B(Ω) which is a countable intersection of nonempty Zariski open subsets such that for each point b ∈ U, there is an isomorphism from the field Ω to the field \overline{Ω(B)} such that this isomorphism turns the scheme X_{b} over Ω into the scheme F over \overline{Ω(B)}. In other words, a very general fibre of f is isomorphic to F as an abstract scheme. Consequently, for each point p ∈ U, CH_{i}(X_{b}) is isomorphic to CH_{i}(F) for all integers i.

Proof. There exist a countable subfield K ⊂ Ω and varieties X_{0} and B_{0} defined over K together with a K-morphism f_{0} : X_{0} → B_{0} such that f = f_{0} ×_{Spec K} Spec Ω. Let us define U ⊆ B(Ω) to be ⋂_{Z_{0}}(B_{0}\Z_{0})_{Ω}(Ω), where the intersection runs through all proper K-subschemes Z_{0} of B_{0}. Note that there are only countably many such subschemes of B_{0} and that U is the set of Ω-points of B = B_{0} ×_{Spec K} Spec Ω that do not lie above a proper Zariski-closed subset of B_{0}. Let now b : Spec Ω → B be a Ω-point of B that lies in U, i.e. a point b such that the composite map β : Spec Ω ⟶^{b} B → B_{0} is dominant, or equivalently such that the composite map β factors as η_{B_{0}} ∘ α for some morphism α : Spec Ω → Spec K(B_{0}), where η_{B_{0}} : Spec K(B_{0}) → B_{0} is the generic point of B_{0}. Since X is pulled back from X_{0} along B → B_{0}, we see that X_{b} the fibre of f at b is the pull back of the generic fibre (X_{0})_{η_{B_{0}}} along α. Consider then \overline{η_{B}} : Spec \overline{Ω(B)} → B a geometric generic point of B such that X\overline{η_{B}} = F. Since the composite map Spec \overline{Ω(B)} → B → B_{0} factors through η_{B_{0}} : Spec K(B_{0}) → B_{0}, we see as before that F is the pull-back of the generic fibre (X_{0})_{η_{B_{0}}} along some morphism α′ : Spec \overline{Ω(B)} → Spec K(B_{0}). The fields \overline{Ω(B)} and Ω are algebraically closed fields of infinite transcendence degree over K(B_{0}) and there thus exists an isomorphism \overline{Ω(B)} ≅ Ω fixing K(B_{0}). Hence, the fibre X_{b} identifies with F after pullback by the isomorphism Spec Ω ≅ Spec \overline{Ω(B)} over Spec K(B_{0}). The last statement follows from the fact that the Chow groups of a variety X over a field only depend on X as a scheme. Precisely, if one denotes ψ_{b} : X_{b} → F an isomorphism of schemes, then the proper pushforward map (ψ_{b})_{∗} : CH_{i}(X_{b}) → CH_{i}(F) is an isomorphism with inverse (ψ_{b}^{−1})_{∗} : CH_{i}(F) → CH_{i}(X_{b}). □

The following lemma will be useful to refer to.

Lemma 2.2. Let f : X → B be a projective surjective morphism defined over Ω onto a quasi-projective variety B. Assume that CH_{l}(X_{b}) = Q (resp. CH_{l}(X_{b}) is finitely generated) for all b ∈ B(Ω). Then CH_{l}(X_{η_{D}}) = Q (resp. CH_{l}(X_{η_{D}}) is finitely generated) for all irreducible subvarieties D of X.

Proof. Let D be an irreducible subvariety of B and let \overline{η}_{D} → D be a geometric generic point of D. By Lemma 2.1 applied to X_{D} := X ×_{B} D, there is a closed point d ∈ D such that CH_{l}(X_{\overline{η}_{D}}) is isomorphic to CH_{l}(X_{d}). By assumption CH_{l}(X_{d}) = Q (resp. CH_{l}(X_{d}) is finitely generated). Therefore CH_{l}(X_{\overline{η}_{D}}) = Q (resp. CH_{l}(X_{\overline{η}_{D}}) is finitely generated), too. By a norm argument for Chow groups, the pullback map CH_{l}(X_{η_{D}}) → CH_{l}(X_{\overline{η}_{D}}) is injective. Hence CH_{l}(X_{η_{D}}) = Q (resp. CH_{l}(X_{η_{D}}) is finitely generated). □

The following definition is taken from Laterveer [15].

Definition 2.3. Let X be a variety over k. The Chow group CH_{i}(X) is said to have niveau ≤ r if there exists a closed subscheme Y ⊂ X of dimension i + r such that the proper pushforward map CH_{i}(Y_{Ω}) → CH_{i}(X_{Ω}) is surjective.

Proposition 2.4. Let f : X → B be a generically smooth, projective and dominant morphism onto a smooth quasi-projective variety B defined over Ω. Let n be a non-negative integer. The following statements are equivalent.

(1) If F is a general fibre, then CH_{0}(F) has niveau ≤ n;

(2) If F is a very general fibre, then CH_{0}(F) has niveau ≤ n;

(3) If F is a geometric generic fibre, then CH_{0}(F) has niveau ≤ n.

Proof. The implication (1) ⇒ (2) is obvious. Let us prove (2) ⇒ (3). Let X_{b} be a very general fibre and F a geometric generic fibre of f, and, by Lemma 2.1, let ψ_{b} : X_{b} → F be an isomorphism of schemes. Assume that there is a closed subscheme Z of dimension ≤ r in X_{b}, for some integer r, such that the proper pushforward CH_{0}(Z) → CH_{0}(X_{b}) is surjective. Then, denoting Z′ the image of Z in F under ψ_{b}, functoriality of proper pushforwards implies that CH_{0}(Z′) → CH_{0}(F) is surjective. We may then conclude by noting that the subscheme Z′ has dimension ≤ r in F. As for (3) ⇒ (1), let Y be a subvariety of F defined over \overline{η}_{B} such that CH_{0}(Y) → CH_{0}(F) is surjective. The technique of decomposition of the diagonal of Bloch–Srinivas [2] gives Δ_{F} = Γ_{1} + Γ_{2} ∈ CH_{dim F} (F × F), where Γ_{1} is supported on F × Y and Γ_{2} is supported on D × F for some divisor D in F. Consider a Galois extension K/Ω(B) over which the above decomposition and the morphism Y → F are defined, and consider an étale morphism U → B with Ω(U) = K such that f restricted to U is smooth. Let u be a Ω-point of U and let X_{u} be the fibre of f over u. Then the decomposition Δ_{F} = Γ_{1} + Γ_{2} specialises [7, §20.3] on X_{u} × X_{u} to a similar decomposition, where Γ_{1}|_{X_{u}×X_{u}} is supported on X_{u} × Y_{u} and Γ_{2}|_{X_{u}×X_{u}} is supported on D_{u} × X_{u}. Letting it act on zero-cycles, we see that CH_{0}(X_{u}) is supported on Y_{u}. □

Remark 2.5. When n = 0 or n = 1, the statements of Proposition 2.4 are further equivalent to CH_{0}(F) having niveau ≤ n for F the generic fibre of f. Indeed, if X is a smooth projective variety such that CH_{0}(X) has niveau ≤ 1, then CH_{0}(X) is supported on a one-dimensional linear section [11, Proposition 1.6]. In particular, CH_{0}(X) is supported on a one-dimensional subvariety of X which is defined over a field of definition of X. Note that, for general n, it is a consequence of the Lefschetz hyperplane theorem and of the Bloch–Beilinson conjectures that if CH_{0}(X) has niveau ≤ n, then CH_{0}(X) is supported on an n-dimensional linear section of X.

3. A generalisation of the projective bundle formula

We establish a formula that is analogous to the projective bundle formula for Chow groups. Our formula holds for flat morphisms, rather than Zariski locally trivial morphisms as is the case for the projective bundle formula. However, since a flat morphism does not have any local sections in general, it only holds with rational coefficients.

Proposition 3.1. Let f : X → B be a flat projective surjective morphism of quasi-projective varieties. Let l ≥ 0 be an integer. Assume that

CH_{l−i}(X_{η_{B_{i}}}) = Q

for all 0 ≤ i ≤ min(l, d_{B}) and for all closed irreducible subschemes B_{i} of B of dimension i, where η_{B_{i}} is the generic point of B_{i}. Then the map

Φ_{∗} = ⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(B) ⟶ CH_{l}(X)

is surjective.

Proof. The case when d_{B} = 0 is obvious. Let us proceed by induction on d_{B}. We have the localisation exact sequence

⨁_{D∈B^{1}} CH_{l}(X_{D}) ⟶ CH_{l}(X) ⟶ CH_{l−d_{B}}(X_{η_{B}}) ⟶ 0,

where the direct sum is taken over all irreducible divisors of B. If l ≥ d_{B}, let Y be a linear section of X of dimension l. By Lemma 1.1, f|_{Y} : Y → B is surjective. The restriction map CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}) is the direct limit of the flat pullback maps CH_{l}(X) → CH_{l}(X_{U}) taken over all open subsets U of B; see [3, Lemma 1A.1]. Therefore CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}) sends the class of Y to the class of Y_{η_{B}} inside CH_{l−d_{B}}(X_{η_{B}}). But then this class is non-zero because the restriction to η_{B} of a linear section of Y of dimension d_{B} has positive degree. Furthermore, if [B] denotes the class of B in CH_{d_{B}}(B), then the class of Y is equal to h^{d_{X}−l} ∘ f^{∗}[B] in CH_{l}(X). Thus, since by assumption CH_{l−d_{B}}(X_{η_{B}}) = Q, the composite map

CH_{d_{B}}(B) ⟶^{h^{d_{X} −l}∘f^{∗}} CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}})

is surjective. Consider now the fibre square

\xymatrix{X_{D} \ar[d]_{f_{D}} \ar[r]^{j′_{D}} & X \ar[d]^{f} \\ D \ar[r]^{j_{D}} & B.}

Then f_{D} : X_{D} → D is flat and its fibres above points of D satisfy the assumptions of the theorem. Therefore, by the inductive assumption, we have a surjective map

⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f_{D}^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(D) ⟶ CH_{l}(X_{D}).

Furthermore, since f is flat and j_{D} is proper, we have the formula [7, Prop. 1.7 & Th. 6.2]

j′_{D∗} ∘ h^{d_{X}−d_{B}−i} ∘ f_{D}^{∗} = h^{d_{X}−d_{B}−i} ∘ f^{∗} ∘ j_{D∗} : CH_{l−i}(D) → CH_{l}(X).

Hence, the image of Φ_{∗} contains the image of

⨁_{D∈B^{1}} ⨁_{i=0}^{d_{X}−d_{B}} j′_{D∗} ∘ h^{d_{X}−d_{B}−i} ∘ f_{D}^{∗} : ⨁_{D∈B^{1}} ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(D) ⟶ CH_{l}(X).

Altogether, this implies that the map Φ_{∗} is surjective. □

We can now gather the statements and proofs of Propositions 1.6 and 3.1 into the following.

Theorem 3.2. Let f : X → B be a flat and projective surjective morphism onto a quasi-projective variety B of dimension d_{B}. Let l ≥ 0 be an integer. Assume that

CH_{l−i}(X_{b}) = Q for all 0 ≤ i ≤ min(l, d_{B}) and for all points b in B(Ω). Then the map

Φ_{∗} = ⨁_{i=0}^{d_{X}−d_{B}} h^{d_{X}−d_{B}−i} ∘ f^{∗} : ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(B) ⟶ CH_{l}(X)

is an isomorphism. Moreover the map

Ψ_{∗} = ⨁_{i=0}^{d_{X}−d_{B}} f_{∗} ∘ h^{i} : CH_{l}(X) ⟶ ⨁_{i=0}^{d_{X}−d_{B}} CH_{l−i}(B)

is also an isomorphism. □

Proof. Let B_{i} be an irreducible closed subscheme of B of dimension i with 0 ≤ i ≤ min(l, d_{B}). Since CH_{l−i}(X_{b}) = Q for all points b ∈ B(Ω), Lemma 2.2 gives CH_{l−i}(X_{η_{B_{i}}}) = Q. Thus the theorem follows from a combination of Proposition 1.6 (and Remark 1.7) and Proposition 3.1. □

4. On the motive of quadric bundles

Let us first recall the following result.

Proposition 4.1 (Corollary 2.2 in [27]). Let m and n be positive integers. Let (Y, q) be a motive over k such that CH_{i}(Y_{Ω}, q_{Ω}) = 0 for all i < n and CH_{j}(Y_{Ω},^{t}q_{Ω}) = 0 for all j < m. Then there exist a smooth projective variety Z over k of dimension d_{X} − m − n and an idempotent r ∈ End(𝔥(Z)) such that (Y, q) is isomorphic to (Z, r, n). □

The main result of this section is the following theorem.

Theorem 4.2. Let f : X → B be a flat morphism of smooth projective varieties over k. Assume that there exists a positive integer n such that CH_{l}(X_{b}) = Q for all 0 ≤ l < n and for all points b ∈ B(Ω). Then there exists a smooth projective variety Z of dimension d_{X} − 2n and an idempotent r ∈ End(𝔥(Z)) such that the motive of X admits a direct sum decomposition

𝔥(X) ≅ ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) ⊕ (Z, r, n).

Proof. With the notations of Theorem 1.4 and its proof the endomorphism Ψ ∘ Φ ∈ End ( ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i)) admits an inverse denoted Ξ. Proposition 1.5 then states that p := Φ∘Ξ∘Ψ ∈ End(𝔥(X)) is a self-dual idempotent such that (X, p) ≅ ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i). By Theorem 3.2, (p_{Ω})_{∗} : CH_{l}(X_{Ω}) → CH_{l}(X_{Ω}) is an isomorphism for all l < n. It follows that CH_{l}(X_{Ω}, p_{Ω}) = CH_{l}(X_{Ω}) for all l < n and thus that CH_{l}(X_{Ω}, id_{Ω} − p_{Ω}) = 0 for all l < n. Because p = ^{t}p, we also have CH_{l}(X_{Ω}, id_{Ω} − ^{t}p_{Ω}) = 0 for all l < n. Proposition 4.1 then yields the existence of a smooth projective variety Z of dimension d_{X} − 2n such that (X, id − p) is isomorphic to a direct summand of 𝔥(Z)(n). □

Our original motivation was to establish Murre’s conjectures [18] for smooth projective varieties fibred by quadrics over a surface. The importance of Murre’s conjectures was demonstrated by Jannsen who proved [11] that these hold true for all smooth projective varieties if and only if Bloch and Beilinson’s conjecture holds true. In our covariant setting, Murre’s conjectures can be stated as follows.

(A) There exist mutually orthogonal idempotents π_{0}, …, π_{2d} ∈ CH_{d_{X}} (X × X) adding to the identity such that (π_{i})_{∗}H_{∗}(X) = H_{i}(X) for all i. We say that X has a Chow–Künneth decomposition. (B) π_{0}, …, π_{2l−1}, π_{d+l+1}, …, π_{2d} act trivially on CH_{l}(X) for all l. (C) F^{i} CH_{l}(X) := Ker (π_{2l}) ∩ … ∩ Ker (π_{2l+i−1}) doesn’t depend on the choice of the π_{j}’s. Here the π_{j}’s are acting on CH_{l}(X). (D) F^{1} CH_{l}(X) = CH_{l}(X)_{hom} := Ker ( CH_{l}(X) → H_{2l}(X)).

Definition 4.3. A variety X that satisfies conjectures (A), (B) and (D) is said to have a Murre decomposition.

In the particular case when f is a flat morphism whose geometric fibres are quadrics^{1}, Theorem 4.2 implies the following corollary. We write ⌊a⌋ for the greatest integer which is smaller than or equal to the rational number a.

Corollary 4.4. Let f : X → B be a flat morphism of smooth projective varieties over k. Assume that CH_{l}(X_{b}) = Q for all 0 ≤ l < \frac{d_{X}−d_{B}}{2} and for all points b ∈ B(Ω). For instance, the geometric fibres of f could either be quadrics or complete intersection of dimension 4 and bidegree (2, 2). Then

• If d_{B} = 1, then X is Kimura finite-dimensional [12].

• If d_{B} ≤ 2, then X has a Murre decomposition.

• If d_{B} = 3, d_{X} − d_{B} is odd and B has a Murre decomposition, then X has a Murre decomposition.

Proof. By Theorem 4.2, there is a variety Z and an idempotent r ∈ End(𝔥(Z)) such that the motive of X admits a direct sum decomposition

𝔥(X) ≅ ⨁_{i=0}^{d_{X}−d_{B}} 𝔥(B)(i) ⊕ (Z, r, ⌊\frac{d_{X} − d_{B} + 1}{2}⌋),

where

d_{Z} = \{ \begin{matrix}d_{B} − 1 & if d_{X} − d_{B} is odd; \\ d_{B} & if d_{X} − d_{B} is even. \end{matrix}

Thus, we only need to note that any direct summand of the motive of a curve is finite-dimensional [12] and that any direct summand of the motive of a surface has a Murre decomposition [27, Theorem 3.5]. Finally, let us mention that, when d_{B} = 1, it is not necessary to assume f to be flat to conclude that X is Kimura finite-dimensional; see Propositions 7.3 and 7.5 below. □

Remark 4.5. Examples of 3-folds having a Murre decomposition include products of a curve with a surface [19], 3-folds rationally dominated by a product of curves [28] and uniruled 3-folds [5].

1Actually if f is flat and if its closed geometric fibres are quadrics, then all of its geometric fibres are quadrics. Conversely if the geometric fibres of f are quadrics of dimension d_{X} −d_{B}, then f is flat.

Remark 4.6 (The case of smooth families). Suppose f : X → B is a smooth morphism between smooth projective varieties with geometric fibres being quadric hypersurfaces. Iyer [10] showed that f is étale locally trivial and deduced that f has a relative Chow–Künneth decomposition. By using the technique of Gordon–Hanamura–Murre [8], it is then possible to prove that

𝔥(X) ≅ \{ \begin{matrix}⨁_{l=0}^{d_{X}−d_{B}} 𝔥(B)(l) & if d_{X} − d_{B} is odd; \\ ⨁_{l=0}^{d_{X}−d_{B}} 𝔥(B)(l) ⊕ 𝔥(B)(\frac{d_{X}−d_{B}}{2} ) & if d_{X} − d_{B} is even. \end{matrix}

Remark 4.7. Suppose f : X → S is a complex morphism from a smooth projective 3-fold X to a smooth projective surface S whose fibres are conics. In that case, Nagel and Saito [20] identify (up to direct summands isomorphic to 𝟙 or 𝟙(1)) the motive (Z, r) in the proof of Corollary 4.4 with the 𝔥_{1} of the Prym variety P attached to a double-covering of the discriminant curve C of f. If now f : X → S is a flat complex morphism from a smooth projective variety X to a smooth projective surface S whose fibres are odd-dimensional quadrics, then, because the motive of a curve is Kimura finite-dimensional and by the Lefschetz (1, 1)-theorem, one would deduce an identification of the 𝔥_{1} of (Z, r) with 𝔥_{1}(P) from an isomorphism of Hodge structures H^{1}(Z, r) ≅ H^{1}(P). Here, P again is the Prym variety attached to a double-covering of the discriminant curve C of f. Such an identification is currently being investigated by J. Bouali [4] by generalising the methods of [20].

Corollary 4.8. Let f : X → B be a flat dominant morphism between smooth projective varieties defined over a finite field F whose geometric fibres are quadrics. If d_{B} ≤ 2, then numerical and rational equivalence agree on X.

Proof. As in the proof of Corollary 4.4, there is a direct sum decomposition

(3)  𝔥(X) ≅ ⨁_{i=0}^{d_{X}−2} 𝔥(B)(i) ⊕ (Z, r, ⌊\frac{d_{X} − d_{B} + 1}{2}⌋)

for some smooth projective variety Z, which is a curve if d_{X} − d_{B} is odd and a surface if d_{X} − d_{B} is even. Now the action of correspondences preserves numerical equivalence so that if α denotes the isomorphism from 𝔥(X) to the right-hand side of (3) and if β denotes its inverse, then we have CH_{l}(X)_{num} = β_{∗}α_{∗} CH_{l}(X)_{num} for all l. In particular, CH_{l}(X)_{num} = β_{∗}( ⨁_{i=0}^{d_{X}−2} CH_{l−i}(B)_{num} ⊕ r_{∗} CH_{l−m}(Z)_{num}), where m = ⌊\frac{d_{X}−d_{B}+1}{2}⌋. The corollary then follows from the fact that for any smooth projective variety Y defined over a finite field the groups CH_{0}(Y)_{num}, CH^{1}(Y)_{num} and CH^{0}(Y)_{num} are zero. □

5. On the motive of a smooth blow-up

Let X be a smooth projective variety over a field k and let j : Y ↪ X be a smooth closed subvariety of codimension r. We write τ : X̃_{Y} → X for the blow-up of X along Y. Manin [16] showed by an existence principle that the natural map, which is denoted Φ below, 𝔥(X) ⊕ ⨁_{i=1}^{r−1} 𝔥(Y)(i) ⟶ 𝔥(X̃_{Y}) is an isomorphism of Chow motives. Here, we make explicit the inverse to this isomorphism. An application is given by Proposition 5.4. The results of this section will not be used in the rest of the paper.

We have the following fibre square

\xymatrix{D \ar[r]^{j̃} \ar[d]_{τ_{D}} & X̃_{Y} \ar[d]^{τ} \\ Y \ar[r]^{j} & X}

where j̃ : D → X̃_{Y} is the exceptional divisor and where τ_{D} : D → Y is a P^{r−1}-bundle over Y. Precisely D = P(N_{Y/X}) is the projective bundle over Y associated to the normal bundle N_{Y/X} of Y inside X. The tautological line bundle on D = P(N_{Y/X}) is O_{P(N_{Y /X})}(−1) = OX̃_{Y} (D)|_{D}. Let

H^{r−1} ⊂ … ⊂ H^{i} ⊂ … ⊂ H ⊂ D

be linear sections of D corresponding to the relatively ample line bundle O_{P(N_{Y /X})}(1), where H^{i} has codimension i. Thus, if D ↪ P^{M} × Y is an embedding over Y corresponding to O_{P(N_{Y /X})}(1), then H^{i} denotes the smooth intersection of D with L^{i} × Y for some linear subspace L^{i} of codimension i inside P^{M}. Let us write ι^{i} : H^{i} ↪ D for the inclusion maps. As we will be using repeatedly Manin’s identity principle, let us mention that if Z is a smooth projective variety over k, then the blow-up of X × Z along Y × Z canonically identifies with τ × id_{Z} : X̃_{Y} × Z → X × Z. We write H_{Z}^{i} for H^{i} × Z.

Let us define the morphism of motives

Φ := ^{t}Γ_{τ} ⊕ ⨁_{i=1}^{r−1} Γj̃ ∘ h^{r−1−i} ∘^{t}Γ_{τ_{D}} : 𝔥(X) ⊕ ⨁_{i=1}^{r−1} 𝔥(Y)(i) ⟶ 𝔥(X̃_{Y}).

Here, h^{l} is the correspondence Γ_{ι^{l}} ∘^{t}Γ_{ι^{l}}; it coincides with the l-fold composite of h := Γ_{ι^{1}} ∘^{t}Γ_{ι^{1}} with itself. On the one hand, we have the following blow-up formula for Chow groups; see [16].

Proposition 5.1. The induced map

Φ_{∗} = τ^{∗} ⊕ ⨁_{i=1}^{r−1} j̃_{∗}h^{r−1−i}τ_{D}^{∗} : CH_{l}(X) ⊕ ⨁_{i=1}^{r−1} CH_{l−i}(Y) ⟶ CH_{l}(X̃_{Y})

is an isomorphism. □

On the other hand, we define

Ψ := Γ_{τ} ⊕ ⨁_{i=1}^{r−1}(−1) · Γ_{τ_{D}} ∘ h^{i−1} ∘^{t}Γj̃ : 𝔥(X̃_{Y}) ⟶ 𝔥(X) ⊕ ⨁_{i=1}^{r−1} 𝔥(Y)(i).

Let (Ψ ∘ Φ)_{i,j} be the (i, j)^{th} component of Ψ ∘ Φ, where 𝔥(X) is by definition the 0^{th} coordinate of 𝔥(X) ⊕ ⨁_{i=1}^{r−1} 𝔥(Y)(i). Thus, if i, j ≠ 0, then (Ψ ∘ Φ)_{i,j} is a morphism 𝔥(Y)(j) → 𝔥(Y)(i); if i ≠ 0, then (Ψ ∘ Φ)_{i,0} is a morphism 𝔥(X) → 𝔥(Y)(i); if j ≠ 0, then (Ψ ∘ Φ)_{0,j} is a morphism 𝔥(Y)(j) → 𝔥(X); and (Ψ ∘ Φ)_{0,0} is a morphism 𝔥(X) → 𝔥(X). The following lemma shows that Ψ ∘ Φ is a lower triangular matrix with invertible diagonal elements.

Lemma 5.2. We have

(Ψ ∘ Φ)_{i,j} = \{ \begin{matrix}0 & if i < j \\ 0 & if ij = 0 unless i = j = 0 \\ Δ_{X} & if i = j = 0 \\ Δ_{Y} & if i = j > 0 \end{matrix}

Proof. The proposition consists of the following relations:

(1) Γ_{τ} ∘^{t}Γ_{τ} = Δ_{X}.

(2) Γ_{τ_{D}} ∘ h^{i−1} ∘^{t}Γj̃ ∘ Γj̃ ∘ h^{r−1−j} ∘^{t}Γ_{τ_{D}} = 0 for all 1 ≤ i < j ≤ r − 1.
(3) Γ_{τ_{D}} ∘ h^{i−1} ∘^{t}Γj̃ ∘ Γj̃ ∘ h^{r−1−i} ∘^{t}Γ_{τ_{D}} = −Δ_{Y} for all 1 ≤ i ≤ r − 1.
(4) Γ_{τ} ∘ Γj̃ ∘ h^{r−1−i} ∘^{t}Γ_{τ_{D}} = 0 for all 1 ≤ i ≤ r − 1.

Let us establish them. The morphism τ is a birational morphism so that the identity (1) follows from the projection formula as in the proof of Lemma 1.2. The proof of (4) is a combination of the fact that τ ∘ j̃ = j ∘ τ_{D} and Lemma 1.3. As for (2) and (3), we claim that

tΓj̃ ∘ Γj̃ = −Γ_{ι} ∘^{t}Γ_{ι} = −h ∈ CH_{d_{X}−1}(D × D).

Indeed, the action of h on CH_{∗}(D) is given by intersecting with the class of H. Also, by [7, Prop. 2.6], the map j̃^{∗}j̃_{∗} : CH_{∗}(D) → CH_{∗−1}(D) is given by intersecting with the class of D|_{D} which is precisely −h. The same arguments for the smooth blow-up of X × Z along Y × Z (whose exceptional divisor is D × Z ↪ X̃_{Y} × Z), together with Manin’s identity principle, yield the claim. In view of the above claim, (2) follows from Lemma 1.3 and (3) follows from Lemma 1.2. □

Thus the endomorphism

N := ( \begin{matrix}Δ_{X} & 0 & 0 & ⋯ & 0 \\ 0 & Δ_{Y} & 0 & ⋯ & 0 \\ 0 & 0 & ... & ... & ⋮ \\ ⋮ & ⋮ & ... & ... & 0 \\ 0 & 0 & ⋯ & 0 & Δ_{Y} \end{matrix} ) − Ψ ∘ Φ

is a nilpotent endomorphism of 𝔥(X) ⊕ ⨁_{i=1}^{r−1} 𝔥(Y)(i) of index ≤ r − 1, i.e. N^{r−1} = 0. The morphism

Θ := (id + N + … + N^{r−2}) ∘ Ψ

thus gives a left-inverse to Φ.

The main result of this section is then the following theorem.

Theorem 5.3. The morphism Θ is the inverse of Φ.

Proof. Let p := Φ ∘ Θ ∈ End(𝔥(X̃_{Y})). Because Θ is a left-inverse to Φ, we see that p is an idempotent. The motive of X̃_{Y} thus splits as

𝔥(X̃_{Y}) = (X̃_{Y} , p) ⊕ (X̃_{Y} , id − p).

As a consequence of Proposition 5.1 and Lemma 5.2, we obtain that

CH_{∗}(X̃_{Y} , p) = CH_{∗}(X̃_{Y}).

Actually, since { \atop (X × Z)}_{Y ×Z} canonically identifies with X̃_{Y} ×Z, we get, thanks to [7, Prop. 16.1.1], that Φ × Δ_{Z} induces an isomorphism of Chow groups as in Proposition 5.1 and that Θ × Δ_{Z} is a left-inverse to Φ × Δ_{Z}. Thus, we also have that

CH_{∗}(X̃_{Y} × Z, p × Δ_{Z}) = CH_{∗}(X̃_{Y} × Z).

Therefore

CH_{∗}(X̃_{Y} × Z, ΔX̃_{Y} × Δ_{Z} − p × Δ_{Z}) = 0.

By Manin’s identity principle it follows that

p = ΔX̃_{Y} .

In other words, Θ is not only a left-inverse to Φ, it is also the inverse of Φ. □

Let us now use Theorem 5.3 to study the birational invariance of some groups of algebraic cycles attached to smooth projective varieties. For a smooth projective variety X over k, we write Griff_{l}(X) for its Griffiths group CH_{l}(X)_{hom}/ CH_{l}(X)_{alg}. We also write, when k ⊆ C,

T^{l}(X) := Ker (AJ^{l} : CH^{l}(X)_{hom} → J^{l}(X))

for the kernel of Griffiths’ Abel–Jacobi map to the intermediate Jacobian J^{l}(X) which is a quotient of H^{2l−1}(X, ℂ). If π : X̃ → X is a birational map, the projection formula implies that Γ_{π}∘^{t}Γ_{π} = Δ_{X}; see Lemma 1.2. Thus π_{∗}π^{∗} acts as the identity on CH_{l}(X), Griff_{l}(X) and on T^{l}(X). The following proposition shows that in some cases π_{∗} and π^{∗} are actually inverse to each other.

Proposition 5.4. Let π : X̃ → X be a birational map between smooth projective varieties. Then π^{∗}π_{∗} acts as the identity on CH_{0}(X̃), Griff_{1}(X̃), Griff^{2}(X̃), T^{2}(X̃), CH^{1}(X̃)_{hom} and CH^{0}(X̃).

Proof. By resolution of singularities, there are morphisms f : Y → X̃ and g : Y → X, which are composite of smooth blow-ups, such that g = π ∘ f. The groups considered in the proposition behave functorially with respect to the action of correspondences. Therefore it is enough to prove the proposition when π is a smooth blow-up X̃_{Y} → X as above. First note that Ψ_{i} := 𝔥(X̃_{Y}) → 𝔥(Y)(i) acts as zero on the groups considered in the proposition when 1 ≤ i ≤ r − 1, for dimension reasons. Having in mind that the first column and the first row of N are zero, and expanding Φ ∘ Θ, we see that Φ ∘ Θ acts like π^{∗}π_{∗} on the groups of the proposition. By Theorem 5.3, the correspondence Φ ∘ Θ acts as the identity on CH_{l}(X̃). Thus π^{∗}π_{∗} acts as the identity on the groups of the proposition. □

Remark 5.5. As a consequence of Theorem 5.3, we obtain an explicit Chow– Künneth decomposition for a smooth blow-up X̃_{Y} in terms of Chow–Künneth decompositions of X and Y. Precisely, assume that X and Y are endowed with Chow–Künneth decompositions {π_{X}^{i}, 0 ≤ i ≤ 2 dim X} and {π_{Y}^{i} , 0 ≤ i ≤ 2 dim Y }, respectively. Let us define

πi ; X̃_{Y} := Φ ∘ (π_{X}^{i} ⊕ ⨁_{j=1}^{r−1} π_{Y}^{i−2j}) ∘ Θ ∈ End(𝔥(X̃_{Y})).

Then {πi ; X̃_{Y} , 0 ≤ i ≤ 2 dim X̃_{Y} } is a Chow–Künneth decomposition of X̃_{Y}.

6. Chow groups of varieties fibred by varieties with small Chow groups

In this section, we consider a projective surjective morphism f : X → B onto a quasi-projective variety B. We are interested in obtaining information on the niveau of the Chow groups of X, under the assumption that the Chow groups of the fibres of f over generic points of closed subvarieties of B are either trivial (= Q) or finitely generated. Contrary to Section 3, we do not assume that f is flat. Let l be a non-negative integer and let d_{X} and d_{B} be the dimensions of X and B, respectively. In §6.1, we assume that CH_{l−i}(X_{η_{D_{i}}}) = Q for all 0 < i ≤ d_{B} and all irreducible subvarieties D_{i} ⊂ B of dimension i, and we deduce in Lemma 6.1 by a localisation sequence argument that CH_{l}(X) is spanned by CH_{l}(X_{b}) for all closed points b in B and by CH_{l}(H), where H ↪ X is a linear section of X of dimension d_{X} + l. We then move on to study the subspace of CH_{l}(X) spanned by CH_{l}(X_{b}) for all closed points b in B in the case when CH_{l}(X_{b}) = Q in §6.2, and in the case when CH_{l}(X_{b}) finitely generated in §6.3. The results of §§6.1 & 6.2 are combined into Proposition 6.5, while the results of §§6.1 & 6.3 are combined into Propositions 6.8 and 6.9 in §6.4. Finally, in §6.5, we use Lemma 2.1 to give statements that only involve the Chow groups of closed fibres when f is defined over the complex numbers; see Theorems 6.10, 6.12 and 6.13.

6.1. Some general statements.

Lemma 6.1. Let f : X → B be a projective surjective morphism onto a quasiprojective variety B and let H ↪ X be a linear section of dimension ≥ l + d_{B}. Assume that CH_{l−i}(X_{η_{D_{i}}}) = Q for all 0 < i ≤ d_{B} and all irreducible subvarieties D_{i} ⊂ B of dimension i. Then the natural map ⨁_{b∈B} CH_{l}(X_{b}) ⊕ CH_{l}(H) → CH_{l}(X) is surjective.

Proof. We prove the proposition by induction on d_{B}. If d_{B} = 0 then the statement is obvious. Let us thus consider a morphism f : X → B and a linear section ι : H ↪ X as in the statement of the proposition with d_{B} > 0. By Lemma 1.1, f restricted to H is surjective. We have the localisation exact sequence

⨁_{D∈B^{1}} CH_{l}(X_{D}) → CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}) → 0.

Here, B^{1} denotes the set of codimension-one closed irreducible subschemes of B. For any irreducible codimension-one subvariety D ⊂ B, the restriction of ι to D → B defines a linear section ι_{D} : H_{D} ↪ X_{D} of dimension ≥ l + d_{B} − 1 of X_{D}. The restriction of f : X → B to D → B defines a surjective morphism X_{D} → D which together with the linear section ι_{D} satisfies the assumptions of the proposition. Therefore, by the induction hypothesis applied to X_{D} → D, the map

⨁_{d∈D} CH_{l}(X_{d}) ⊕ CH_{l}(H_{D}) → CH_{l}(X_{D})

is surjective. This yields an exact sequence

⨁_{b∈B} CH_{l}(X_{b}) ⊕ ⨁_{D∈B^{1}} CH_{l}(H_{D}) → CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}) → 0.

Since each of the proper inclusion maps H_{D} → X factors through ι : H → X,

we see that the map ⨁_{D∈B^{1}} CH_{l}(H_{D}) ⟶^{⊕(ι_{D})_{∗}} CH_{l}(X) factors through ι_{∗} : CH_{l}(H) → CH_{l}(X). In order to conclude, it is enough to prove that the composite map

CH_{l}(H) → CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}})

is surjective. If l < d_{B}, then this is obvious. Let us then assume that l ≥ d_{B}. Let Y be an irreducible subvariety of H of dimension l such that the composite Y ↪ X → B is dominant. Because CH_{l−d_{B}}(X_{η_{B}}) = Q it is enough to see that the class of Y in CH_{l}(X) maps to a non-zero element in CH_{l−d_{B}}(X_{η_{B}}). But, as in the proof of Proposition 3.1, [Y] maps to [Y_{η_{B}}] ≠ 0 ∈ CH_{l−d_{B}}(X_{η_{B}}). □

Here is an improvement of Lemma 6.1:

Lemma 6.2. Let f : X → B be a projective surjective morphism onto a quasiprojective variety B and let H ↪ X be a linear section of dimension ≥ l + d_{B}. Assume that:

• CH_{l−i}(X_{η_{D_{i}}}) = Q for all i such that 0 < i < d_{B} and all irreducible subvarieties D_{i} ⊂ B of dimension i.

• CH_{l−d_{B}}(X_{η_{B}}) is finitely generated.

Then there exist finitely many closed subschemes Z_{j} of X of dimension l such that the natural map ⨁_{j} CH_{l}(Z_{j}) ⊕ ⨁_{b∈B} CH_{l}(X_{b}) ⊕ CH_{l}(H) → CH_{l}(X) is surjective.

Proof. If d_{B} = 0 then the statement is obvious. Let us thus consider a morphism f : X → B and a linear section ι : H ↪ X as in the statement of the proposition with d_{B} > 0. The morphism f restricted to H is surjective; see Lemma 1.1. As in the proof of Lemma 6.1 we have the localisation exact sequence

⨁_{D∈B^{1}} CH_{l}(X_{D}) → CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}) → 0.

Each of the morphisms X_{D} → D satisfies the assumptions of Lemma 6.1 and by the same arguments as in the proof of Lemma 6.1 we get that the image of the map ⨁_{b∈B} CH_{l}(X_{b}) ⊕ CH_{l}(H) → CH_{l}(X) contains the image of the map ⨁_{D∈B^{1}} CH_{l}(X_{D}) → CH_{l}(X). Let now Z_{j} be finitely many closed subschemes of X_{η_{B}} whose classes [Z_{j}] ∈ CH_{l−d_{B}}(X_{η_{B}}) generate CH_{l−d_{B}}(X_{η_{B}}). By surjectivity of the map CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}) there are cycles α_{j} ∈ CH_{l}(X) that map to [Z_{j}]. If Z_{j} is the support in X of any representative of α_{j}, we then have a surjective map ⨁_{j} CH_{l}(Z_{j}) → CH_{l}(X) → CH_{l−d_{B}}(X_{η_{B}}). It is then clear that the map ⨁_{j} CH_{l}(Z_{j}) ⊕ ⨁_{b∈B} CH_{l}(X_{b}) ⊕ CH_{l}(H) → CH_{l}(X) is surjective. □

6.2. Varieties fibred by varieties with Chow groups generated by a linear section.

Lemma 6.3. Let f : X → B be a projective surjective morphism onto a quasiprojective variety B. Assume that CH_{l}(X_{b}) = Q for all closed points b ∈ B. Then, if H ↪ X is a linear section of dimension ≥ l + d_{B}, we have

Im ( ⨁_{b∈B} CH_{l}(X_{b}) → CH_{l}(X)) ⊆ Im ( CH_{l}(H) → CH_{l}(X)).

Proof. Let b be a closed point of B and fix H ↪ X a linear section of dimension ≥ l + d_{B}. The morphism f restricted to H is surjective; see Lemma 1.1. Let Z_{l} be an irreducible closed subscheme of X of dimension l which is supported on X_{b}. Since f|_{H} : H → B is a dominant projective morphism, its fibre H_{b} over b is non-empty and has dimension ≥ l. By assumption CH_{l}(X_{b}) = Q, so that a rational multiple of [Z_{l}] is rationally equivalent to an irreducible closed subscheme of H_{b} of dimension l. Therefore [Z_{l}] ∈ CH_{l}(X_{b}) belongs to the image of the natural map CH_{l}(H_{b}) → CH_{l}(X_{b}). Thus the image of CH_{l}(X_{b}) → CH_{l}(X) is contained in the image of CH_{l}(H) → CH_{l}(X). □

Remark 6.4. It is interesting to decide whether or not it is possible to parametrise such l-cycles by a variety of dimension d_{B}; see Proposition 6.7.

Proposition 6.5. Let f : X → B be a projective surjective morphism onto a quasi-projective variety B. Assume that CH_{l−i}(X_{η_{D_{i}}}) = Q for all 0 ≤ i ≤ d_{B} and all irreducible subvarieties D_{i} ⊂ B of dimension i. Then, if H ↪ X is a linear section of dimension ≥ l + d_{B}, the pushforward map CH_{l}(H) → CH_{l}(X) is surjective. In particular, CH_{l}(X) has niveau ≤ d_{B}.

Proof. This is a combination of Lemma 6.1 and Lemma 6.3. □

6.3. An argument involving relative Hilbert schemes. Let f : X → B be a generically smooth, projective morphism defined over the field of complex numbers C onto a smooth quasi-projective variety B. Let B^{∘} ⊆ B be the smooth locus of f and let f^{∘} : X^{∘} → B^{∘} be the pullback of f : X → B along the open inclusion B^{∘} ↪ B so that we have a cartesian square

\xymatrix{X^{∘} \ar[d]_{f^{∘}} \ar@{^{(}->}[r] & X \ar[d]^{f} \\ B^{∘} \ar@{^{(}->}[r] & B.}

We assume that there is a non-negative integer l such that for all closed points b ∈ B^{∘}(C) the cycle class map CH_{l}(X_{b}) → H_{2l}(X_{b}) is an isomorphism.

Let π_{d} : Hilb_{l}^{d}(X/B) → B be the relative Hilbert scheme whose fibres over the points b in B parametrise the closed subschemes of X_{b} of dimension l and degree d, and let p_{d} : C_{l}^{d} → Hilb_{l}^{d}(X/B) be the universal family over Hilb_{l}^{d}(X/B); see [14, Theorem 1.4]. We have the following commutative diagram, where all the morphisms involved are proper:

\xymatrix{C_{l}^{d} \ar[d]_{p_{d}} \ar[r]^{q_{d}} & X. \ar[ddl] \\ Hilb_{l}^{d}(X/B) \ar[d]_{π_{d}} \\ B}

We then consider the disjoint unions Hilb_{l}(X/B) := ∐_{d≥0} Hilb_{l}^{d}(X/B) and C_{l} := ∐_{d≥0} C_{l}^{d}, and denote π : Hilb_{l}(X/B) → B, p : C_{l} → Hilb_{l}(X/B) and q : C_{l} → X the corresponding maps. Let us then denote Irr_{l}(X/B) := {H : H is an irreducible component of Hilb_{l}^{d}(X/B) for some d}. For a subset E ⊂ Irr_{l}(X/B), we define the following closed subscheme of B^{∘}:

Z_{E} := B^{∘} ∩ ⋂_{H∈E} π(H).

We say that a finite subset E of Irr_{l}(X/B) is spanning at a point t ∈ Z_{E}(C) if H_{2l}(X_{t}) is spanned by the set { cl(q_{∗}[p^{−1}(u)]) : u ∈ H, H ∈ E, π(u) = t }. Note that, given H ∈ Irr_{l}(X/B) and u, u′ ∈ H such that π(u) = π(u′) = t, cl(q_{∗}[p^{−1}(u)]) = cl(q_{∗}[p^{−1}(u′)]) ∈ H_{2l}(X_{t}) if u and u′ belong to the same connected component in π^{−1}(t).

Claim. Let E be a finite subset of Irr_{l}(X/B) that is spanning at a closed point t ∈ Z_{E}(C). Then, for all points s ∈ Z_{E}(C) belonging to an irreducible component of Z_{E} that contains t, H_{2l}(X_{s}) is spanned by the set { cl(q_{∗}[p^{−1}(v)]) : v ∈ H, H ∈ E, π(v) = s }.

Indeed, consider any finite subset E of Irr_{l}(X/B). The local system of Q-vector spaces R_{2l}(f_{0})_{∗}Q on B^{∘} restricts to a local system (R_{2l}(f_{0})_{∗}Q)|_{Z_{E}} on Z_{E}. If t is a complex point on Z_{E}, let r_{t} be the rank of the subspace of H_{2l}(X_{t}) spanned by { cl(q_{∗}[p^{−1}(u)]) : u ∈ H, H ∈ E, π(u) = t }. If we see this latter set as a set of sections at t of the local system R_{2l}(f_{0})_{∗}Q, then these sections extend locally to constant sections of the local system (R_{2l}(f_{0})_{∗}Q)|_{Z_{E}} on Z_{E}. This shows that the rank r_{t} is locally constant. If E is spanning at the point t ∈ Z_{E}(C), then r_{t} is maximal, equal to dim_{Q} H_{2l}(X_{t}). The subset of Z_{E}(C) consisting of points s in Z_{E}(C) for which r_{s} = dim_{Q} H_{2l}(X_{s}) is therefore both open and closed in Z_{E}. It contains then the irreducible components of Z_{E} that contain t.

Lemma 6.6. There exists a finite subset E of Irr_{l}(X/B) such that B^{∘} = Z_{E} and such that E is spanning at every point t ∈ B^{∘}(C).

Proof. By working component-wise, we may assume that B is irreducible. By assumption on f : X → B, H_{2l}(X_{t}) is spanned by algebraic cycles on X_{t} for all points t ∈ B^{∘}(C). Thus, for all points t ∈ B^{∘}(C), there is a finite subset E_{t} of Irr_{l}(X/B) that is spanning at t. For each point t, choose an irreducible component Y_{E_{t}} of Z_{E_{t}} that contains t. According to the claim above, E_{t} is spanning at every point s ∈ Y_{E_{t}}(C). Now, we have B^{∘}(C) = ∐_{t∈B^{∘}(C)} Y_{E_{t}}(C). Since there are only countably many finite subsets of Irr_{l}(X/B) and since Z_{E} has only finitely many irreducible components, we see that the latter union is in fact a countable union. This yields that B^{∘} = Y_{E} for some finite subset E of Irr_{l}(X/B) that is spanning at every point in Y_{E}(C). We then conclude that B^{∘} = Z_{E} and that E is spanning at every point t ∈ Y_{E}(C) = B^{∘}(C). □

Proposition 6.7. Let f : X → B be a generically smooth and projective morphism defined over C onto a smooth quasi-projective variety B. Let B^{∘} ⊆ B be the smooth locus of f. Assume that there is an integer l ≤ d_{X} − d_{B} such that for all closed points b ∈ B^{∘}(C) the cycle class map CH_{l}(X_{b}) → H_{2l}(X_{b}) is an isomorphism. Then Im ( ⨁_{b∈B^{∘}} CH_{l}(X_{b}) → CH_{l}(X)) is supported on a closed subvariety of X of dimension d_{B} + l. If moreover X is smooth, then there exist a smooth quasi-projective variety B̃ of dimension d_{B} and a correspondence Γ ∈ CH_{d_{B}+l}(B̃ × X) such that Γ_{∗} : CH_{0}(B̃) → CH_{l}(X) is well-defined and

Im ( ⨁_{b∈B^{∘}} CH_{l}(X_{b}) → CH_{l}(X)) ⊆ Im (Γ_{∗} : CH_{0}(B̃) → CH_{l}(X)).

Proof. By Lemma 6.6, there exists a finite set E of irreducible components of Hilb_{l}(X/B) such that B^{∘} = Z_{E} and such that for all points t ∈ B^{∘}(C) the set { cl(q_{∗}[p^{−1}(u)]) : u ∈ H, H ∈ E, π(u) = t } spans H_{2l}(X_{t}). Denote H_{i} the irreducible components of Hilb_{l}(X/B) that belong to E and let H̃_{i} → H_{i} be resolutions thereof. For all i, pick a smooth linear section B̃_{i} → H̃_{i} of dimension d_{B}. Lemma 1.1 shows that r_{i} : B̃_{i} → H̃_{i} → H_{i} → B is surjective and a refinement of its proof shows that, for all points b ∈ B(C), r_{i}^{−1}(b) contains a point in every connected component of H̃_{i,b}. Consider then p_{i} : (C_{l})|B̃_{i} → B̃_{i} the pullback of the universal family p : C_{l} → Hilb_{l}(X/B) along B̃_{i} ↪ H̃_{i} → H_{i} ↪ Hilb_{l}(X/B). For each i, we have the following picture

\xymatrix{(C_{l})|B̃_{i} \ar[r]^{q_{i}} \ar[d]_{p_{i}} & X \\ B̃_{i}}

and we have

Im ( ⨁_{b∈B^{∘}} CH_{l}(X_{b}) → CH_{l}(X)) ⊆ ∑_{i} Im ((q_{i})_{∗} : CH_{l}((C_{l})|B̃_{i}) → CH_{l}(X))

so that the group on the left-hand side is supported on the union of the schemetheoretic images of the morphisms q_{i}. If X is smooth, we define Γ_{i} ∈ CH_{d_{B}+l}(B̃_{i} × X) to be the class of the image of (C_{l})|B̃_{i} inside B̃_{i} × X. Because q : C_{l}^{d} → X is proper for all d ≥ 0, Γ_{i} has a representative which is proper over X. It is therefore possible [7, Remark 16.1] to define maps (Γ_{i})_{∗} : CH_{0}(B̃_{i}) → CH_{l}(X) for all i. In fact, we have (Γ_{i})_{∗} = (q_{i})_{∗}p_{i}^{∗}. Finally, we define B̃ to be the disjoint union of the B̃_{i}’s and Γ ∈ CH_{d_{B}+l}(B̃ ×X) to be the class of the disjoint union of the correspondences Γ_{i}. □

6.4. Complex varieties fibred by varieties with small Chow groups. From now on, the base field k is assumed to be the field of complex numbers C.

Proposition 6.8. Let f : X → C be a generically smooth, projective morphism defined over C to a smooth curve. Assume that

• CH_{l}(X_{c}) is finitely generated for all closed points c ∈ C,

• CH_{l}(X_{c}) → H_{2l}(X_{c}) is an isomorphism for a general closed point c ∈ C,

• CH_{l−1}(X_{η}) is finitely generated, where η is the generic point of C.

Then CH_{l}(X) has niveau ≤ 1.

Proof. We have the localisation exact sequence

⨁_{c∈C} CH_{l}(X_{c}) ⟶ CH_{l}(X) ⟶ CH_{l−1}(X_{η}) ⟶ 0.

Let Z_{1}, …, Z_{n} be irreducible closed subschemes of X_{η} of dimension l − 1 that span CH_{l−1}(X_{η}) and let Z_{1}, …, Z_{n} be closed subschemes of X of dimension l that restrict to Z_{1}, …, Z_{n} in X_{η}. Then by flat pullback the class of Z_{j} in CH_{l}(X) maps to the class of Z_{j} in CH_{l−1}(X_{η}) so that the composite map ⨁_{j=1}^{n} CH_{l}(Z_{j}) → CH_{l}(X) → CH_{l−1}(X_{η}) is surjective. Let U ⊆ C be a Zariski-open subset of C such that for all closed points c ∈ U the cycle class map CH_{l}(X_{c}) → H_{2l}(X_{c}) is an isomorphism. Up to shrinking U, we may assume that f|_{U} : X|_{U} → U is smooth. We may then apply Proposition 6.7 to get a closed subscheme ι : D ↪ X of dimension l + 1 such that ι_{∗} CH_{l}(D) ⊇ Im ( ⨁_{c∈U} CH_{l}(X_{c}) → CH_{l}(X)). As such, we have a surjective map

⨁_{j=1}^{n} CH_{l}(Z_{j}) ⊕ ⨁_{c∈C\U} CH_{l}(X_{c}) ⊕ CH_{l}(D) ⟶ CH_{l}(X)

and it is straightforward to conclude. □

The next proposition is a generalisation of Proposition 6.8 to the case when the base variety B has dimension greater than 1.

Proposition 6.9. Let f : X → B be a generically smooth, projective morphism defined over C to a smooth quasi-projective variety B. Assume that the singular locus of f in B is finite and let U be the maximal Zariski-open subset of B over which f is smooth. Assume also that

• CH_{l}(X_{b}) is finitely generated for all closed points b ∈ B,

• CH_{l}(X_{b}) → H_{2l}(X_{b}) is an isomorphism for all closed points b ∈ U,

• CH_{l−i}(X_{η_{D_{i}}}) = Q for all i such that 0 < i < d_{B} and all irreducible subvarieties D_{i} ⊂ B of dimension i.

• CH_{l−d_{B}}(X_{η_{B}}) is finitely generated.

Then CH_{l}(X) has niveau ≤ d_{B}.

Proof. Let H ↪ X be a linear section of dimension ≥ l + d_{B}. The restriction of f to H is surjective. Thanks to Lemma 6.2, there are finitely many closed subschemes Z_{j} of X of dimension l such that the natural map ⨁_{j} CH_{l}(Z_{j}) ⊕ ⨁_{b∈B} CH_{l}(X_{b}) ⊕ CH_{l}(H) → CH_{l}(X) is surjective. By Proposition 6.7, there exists a closed subscheme ι : B̃ ↪ X of dimension d_{B} + l such that the image of the map ⨁_{b∈U} CH_{l}(X_{b}) → CH_{l}(X) is contained in the image of the map ι_{∗} : CH_{l}(B̃) → CH_{l}(X). Therefore the map

⨁_{j} CH_{l}(Z_{j}) ⊕ CH_{l}(B̃) ⊕ ⨁_{b∈B\U} CH_{l}(X_{b}) ⊕ CH_{l}(H) ⟶ CH_{l}(X)

is surjective. It is then straightforward to conclude. □

6.5. The main results. The field of complex numbers is a universal domain and in view of Section 2 we restate some of the results above in a more comprehensive way.

First, we deduce from Proposition 6.5 the following.

Theorem 6.10. Let f : X → B be a complex projective surjective morphism onto a quasi-projective variety B. Assume that CH_{i}(X_{b}) = Q for all i ≤ l and all closed point b ∈ B . Then CH_{i}(X) has niveau ≤ d_{B} for all i ≤ l.

Proof. By Lemma 2.2, CH_{i}(X_{η_{D}}) = Q for all i ≤ l and all irreducible subvarieties D of X. Proposition 6.5 implies that CH_{i}(X) has niveau ≤ d_{B}. □

The following lemma gives a criterion for the second point in the assumptions of Propositions 6.8 and 6.9 to be satisfied.

Lemma 6.11. Let X be a smooth projective complex variety. Assume that CH_{i}(X) is finitely generated for all i ≤ l. Then the cycle class map CH_{l}(X) → H_{2l}(X) is an isomorphism.

Proof. The proof follows the same pattern as the proof of [24, Theorem 3.4] once it is noted that if CH_{i}(X) is finitely generated then the group of algebraically trivial cycles CH_{i}(X)_{alg} is representable in the sense of [24, Definition 2.1]. Concretely, we get that the Chow motive of X is isomorphic to 𝟙 ⊕ 𝟙(1)^{b_{2}} ⊕ … ⊕ 𝟙(l)^{b_{2l}} ⊕ N(l + 1) where b_{i} is the i-th Betti number of X and N is an effective motive. This yields that the cycle class map CH_{i}(X) → H_{2i}(X) is an isomorphism for all i ≤ l. □

As a consequence of Proposition 6.8, we obtain

Theorem 6.12. Let f : X → C be a complex generically smooth projective surjective morphism onto a smooth complex curve. Assume that CH_{i}(X_{c}) is finitely generated for all closed points c ∈ C and all i ≤ l, then CH_{i}(X) has niveau ≤ 1 for all i ≤ l.

Proof. Let D be an irreducible component of C. Lemma 2.2 shows that CH_{i}(X_{η_{D}}) is finitely generated for all i ≤ l. Let U ⊆ C be the smooth locus of f. It is an open dense subset of C. Then, for c ∈ U, the closed fibre X_{c} is smooth and the groups CH_{i}(X_{c}) are finitely generated for all i ≤ l. By Lemma 6.11 the cycle class maps CH_{i}(X_{c}) → H_{2i}(X_{c}) are isomorphisms for all c ∈ U and all i ≤ l. Proposition 6.8 implies that CH_{i}(X) has niveau ≤ 1. □

And, as a consequence of Proposition 6.9, we obtain

Theorem 6.13. Let f : X → B be a complex projective surjective morphism onto a smooth quasi-projective variety B. Assume that the singular locus of f in B is finite, and assume also that

• CH_{i}(X_{b}) is finitely generated for all closed points b ∈ B and all i ≤ l,

• CH_{i}(X_{b}) = Q for all but finitely many closed points b ∈ B and all i < l.

Then CH_{i}(X) has niveau ≤ d_{B} for all i ≤ l.

Proof. Let D be an irreducible subvariety of X of positive dimension. By Lemma 2.2, we have CH_{i}(X_{η_{D}}) = Q for all i < l. If we denote by U the smooth locus of f, then, by Lemma 6.11, the cycle class maps CH_{i}(X_{b}) → H_{2i}(X_{b}) are isomorphisms for all b ∈ U and all i ≤ l. Proposition 6.9 implies that CH_{i}(X) has niveau ≤ d_{B} for all i ≤ l. □

7. Applications

7.1. Varieties with small Chow groups. In this section, we review the known results about varieties with Chow groups having small niveau; see Definition 2.3. Varieties are defined over an algebraically closed field k of characteristic zero and Ω denotes a universal domain over k. In that case, Grothendieck’s standard conjectures for a smooth projective variety X over k reduce to the Lefschetz standard conjecture for X; see Kleiman [13].

Theorem 7.1. Let X be a smooth projective variety of dimension d. Assume that the Chow groups CH_{0}(X_{Ω}), …, CH_{l}(X_{Ω}) have niveau ≤ n.

• If n = 3 and l = ⌊\frac{d−4}{2}⌋, then X satisfies the Hodge conjecture.

• If n = 2 and l = ⌊\frac{d−3}{2}⌋, then X satisfies the Lefschetz standard conjecture.

• If n = 1 and l = ⌊\frac{d−3}{2}⌋, then X has a Murre decomposition.

• If n = 1 and l = ⌊\frac{d−2}{2}⌋, then X is Kimura finite-dimensional.

Proof. The fourth item is proved in [26]. It is also proved there that if X is as in the third item, then X has a Chow–Künneth decomposition. That such a decomposition satisfies Murre’s conjectures (B), (C) and (D) is proved in [25, §4.4.2]. The first item is proved in [15]. We couldn’t find a reference for the proof of the second item so we include a proof here. Since it is enough to prove the conclusion of the theorem for X_{Ω}, we may assume that X is defined over Ω. Laterveer used the assumptions on the niveau of the Chow groups to show [15, 1.7] that the diagonal Δ_{X} admits a decomposition as follows : there exist closed and reduced subschemes V_{j}, W^{j} ⊂ X with dim V_{j} ≤ j + 2 and dim W^{j} ≤ n − j, there exist correspondences Γ_{j} ∈ CH_{n}(X × X) for 0 ≤ j ≤ ⌊\frac{n−3}{2}⌋ and Γ′ ∈ CH_{n}(X × X) such that each Γ_{j} is in the image of the pushforward map CH_{n}(V_{j} × W^{j}), Γ′ is in the image of the pushforward map CH_{n}(X × W^{⌊ \frac{n−1}{2} ⌋}), and

Δ_{X} = Γ_{0} + … + Γ_{⌊ \frac{n−3}{2} ⌋} + Γ′.

Given j such that 0 ≤ j ≤ ⌊\frac{n−3}{2}⌋, let Ṽ_{j} and W̃^{j} denote desingularisations of V_{j} and W^{j} respectively. The action of Γ_{j} on H^{k}(X) then factors through H^{k}(Ṽ_{j}) and through H_{2n−k}(W̃^{j}). On the one hand, we have H_{2n−k}(W̃^{j}) = H^{k−2j}(W̃^{j}) and hence if k ≤ 2j + 1 then the action of Γ_{j} on H^{k}(X) factors through the H^{0} or the H^{1} of a smooth projective variety. Since the Lefschetz standard conjecture is true in degrees ≤ 1, it follows that the action of Γ_{j} on H^{k}(X) factors through the H_{0} or the H_{1} of a smooth projective variety. On the other hand, we have H^{k}(Ṽ_{j}) = H_{4+2j−k}(Ṽ_{j}) and hence if k ≥ 2j + 2 then Γ_{j} factors through the H_{0}, the H_{1} or the H_{2} of a smooth projective variety. Concerning the action of Γ′ on H^{k}(X), it factors through H_{2n−k}(W̃^{⌊ \frac{n−1}{2} ⌋}) which vanishes for dimension reasons if k < n when n is odd and if k < n − 1 when n is even. When n is even and k = n − 1, the action of Γ′ on H^{k}(X) factors through the H_{1} of a curve. Indeed this follows from a combination of the fact that it factors through H_{n+1}(W̃^{⌊ \frac{n−1}{2} ⌋}) = H^{1}(W̃^{\frac{n−2}{2}} ) and of the validity of the Lefschetz standard conjecture in degree 1. By the Lefschetz hyperplane theorem, we get that for k < n the cohomology groups H^{k}(X) are generated algebraically (that is through the action of correspondences) by the H_{0} of points, the H_{1} of curves and the H_{2} of surfaces. We may then conclude with [28, Proposition 3.19]. □

7.2. Varieties fibred by low-degree complete intersections. As explained by Esnault–Levine–Viehweg in the introduction of [6], it is expected from general conjectures on algebraic cycles, that if Y ⊂ P_{k}^{n} is a complete intersection of multidegree d_{1} ≥ … ≥ d_{r} ≥ 2, then CH_{l}(Y) = Q for all l < ⌊\frac{n−∑_{i=2}^{r} d_{i}}{d_{1}} ⌋, see also Paranjape [22] and Schoen [23]. If there is no proof of the above for the moment, the following theorem however was proved.

Theorem 7.2 (Esnault–Levine–Viehweg [6]). Let Y ⊂ P_{k}^{n} be a complete intersection of multidegree d_{1} ≥ … ≥ d_{r} ≥ 2.

• If either d_{1} ≥ 3 or r ≥ l + 1, assume that ∑_{i=1}^{r} ( \begin{matrix}l + d_{i} \\ l + 1 \end{matrix} ) ≤ n.

• If d_{1} = … = d_{r} = 2 and r ≤ l, assume that ∑_{i=1}^{r} ( \begin{matrix}l + d_{i} \\ l + 1 \end{matrix} ) =

r(l + 2) ≤ n − l + r − 1. Then CH_{l′}(Y) = Q for all 0 ≤ l′ ≤ l. □

Let us consider f : X → B a dominant morphism between smooth projective complex varieties whose closed fibres are complete intersections. Theorem 6.10, together with Theorem 7.2, shows that the niveau of the first Chow groups of X have niveau ≤ dim B. When X is fibred by very low-degree complete intersections, we can thus expect X to satisfy the assumptions of Theorem 7.1. In the remainder of this paragraph, we inspect various such cases.

7.2.1. Varieties fibred by quadric hypersurfaces. Let Q ⊂ P^{n} be a quadric hypersurface. Then CH_{l}(Q) = Q for all l < \frac{dim Q}{2}.

Proposition 7.3. Let f : X → B be a dominant morphism between smooth projective complex varieties whose closed fibres are quadric hypersurfaces.

• If dim B ≤ 1, then X is Kimura finite-dimensional and satisfies Murre’s conjectures.

• If dim B ≤ 2, then X satisfies Grothendieck’s standard conjectures.

• If dim B ≤ 3, then X satisfies the Hodge conjecture.

Proof. The fibres of f have dimension ≥ dim X − dim B, so that X satisfies the assumptions of Theorem 6.10 with l = ⌊\frac{d_{X}−d_{B}−1}{2}⌋. Thus the Chow groups CH_{0}(X), CH_{1}(X), …, CH_{⌊ \frac{d_{X} −d_{B} −1}{2} ⌋}(X) have niveau ≤ d_{B}. We can therefore conclude by Theorem 7.1. □

7.2.2. Varieties fibred by cubic hypersurfaces. Let X ⊂ P^{n} be a cubic hypersurface. Then

• CH_{0}(X) = Q for dim X ≥ 2.

• CH_{1}(X) = Q for dim X ≥ 5.
• CH_{2}(X) = Q for dim X ≥ 8.

Note that Theorem 7.2 only gives CH_{2}(X) = Q for dim X ≥ 9. The bound on the dimension of X was improved to dim X = 8 by Otwinowska [21].

Proposition 7.4. Let f : X → B be a dominant morphism between smooth projective complex varieties whose closed fibres are cubic hypersurfaces.

• If dim X = 6 and dim B = 1, then X satisfies Grothendieck’s standard conjectures and has a Murre decomposition.

• If dim X = 7 and if dim B ≤ 2, then X satisfies the Hodge conjecture.

• If dim X = 9 and if dim B ≤ 1, then X satisfies the Hodge conjecture.

Proof. We use Theorem 6.10 as in the proof of Proposition 7.3. In the first case, we get that CH_{0}(X) and CH_{1}(X) have niveau ≤ 1. In the second case we get that CH_{0}(X) and CH_{1}(X) have niveau ≤ 2 and in the third case we get that CH_{0}(X), CH_{1}(X) and CH_{2}(X) have niveau ≤ 1. We can then conclude in all three cases by Theorem 7.1. □

7.2.3. Varieties fibred by complete intersections of bidegree (2, 2). Let X ⊂ P^{n} be the complete intersection of two quadrics. By Theorem 7.2, CH_{0}(X) = Q; and if dim X ≥ 4, then CH_{1}(X) = Q.

Proposition 7.5. Let f : X → B be a dominant morphism between smooth projective complex varieties whose closed fibres are complete intersections of bidegree (2, 2).

• If dim B ≤ 1 and dim X ≤ 5, then X is Kimura finite-dimensional.

• If dim B ≤ 1 and dim X ≤ 6, then X satisfies Murre’s conjectures.

• If dim B ≤ 2 and dim X ≤ 6, then X satisfies Grothendieck’s standard conjectures.

• If dim B ≤ 3 and dim X ≤ 7, then X satisfies the Hodge conjecture.

Proof. The variety X satisfies the assumptions of Theorem 6.10 with l = 1 for dim X − dim B ≥ 4 and with l = 0 in any case. Thus the Chow group CH_{0}(X) has niveau ≤ d_{B} and CH_{1}(X) has niveau ≤ d_{B} for dim X − dim B ≥ 4. We can therefore conclude by Theorem 7.1. □

7.2.4. Varieties fibred by complete intersections of bidegree (2, 3). Let X ⊂ P^{n} be the complete intersection of a quadric and of a cubic. If dim X ≥ 6, then Hirschowitz and Iyer [9] showed CH_{l}(X) = Q for l ≤ 1. (The result of Esnault– Levine–Viehweg only says that CH_{l}(X) = Q for l ≤ 1 when dim X ≥ 7).

Proposition 7.6. Let f : X → C be a dominant morphism from a smooth projective complex variety X to a smooth projective complex curve C whose closed fibres are complete intersections of bidegree (2, 3) of dimension 6. Then X satisfies the Hodge conjecture.

Proof. By Theorem 6.10, we see that the Chow groups CH_{0}(X) and CH_{1}(X) have niveau ≤ 1. We can thus conclude by Theorem 7.1. □

7.3. Varieties fibred by cellular varieties. Let f : X → B be a complex dominant morphism from a smooth projective variety X to a smooth projective variety B whose closed fibres are cellular varieties (not necessarily smooth). In other words, X is a smooth projective complex variety fibred by cellular varieties over B. For example, if Σ ⊂ B is the singular locus of f, then X could be such that X|_{B\Σ} is a rational homogeneous bundle over B\Σ (e.g. a Grassmann bundle) and the closed fibres of f over Σ (the degenerate fibres) are toric. That kind of situation is reminiscent of the setting of [8].

Proposition 7.7. Let f : X → B be a dominant morphism between smooth projective complex varieties whose closed fibres are cellular varieties.

• Assume B is a curve, then X is Kimura finite-dimensional and X satisfies Murre’s conjectures.

• Assume dim B ≤ 2 and dim X ≤ 6. If f is connected and smooth away from finitely many points in B, then X satisfies Grothendieck’s standard conjectures.

• Assume dim B ≤ 3 and dim X ≤ 7. If f is connected and smooth away from finitely many points in B, then X satisfies the Hodge conjecture.

Proof. The Chow groups of cellular varieties are finitely generated. The first statement thus follows from Theorems 6.12 and 7.1. Let us now focus on the cases when dim B is either 2 or 3. It is a consequence of Mumford’s theorem [17] that a connected smooth projective complex variety with finitely generated Chow group of zero-cycles actually has Chow group of zero-cycles generated by a point. Thus the second and third statements follow from Theorem 6.13 with l = 1, and from Theorem 7.1. □

Acknowledgements

Thanks to Reza Akhtar and Roy Joshua for asking a question related to the Chow–Künneth decomposition of a smooth blow-up, to Sergey Gorchinskiy for suggesting parts of Proposition 2.4, to Mingmin Shen for a discussion related to the proof of Lemma 6.6, and to Burt Totaro for suggesting the statement of Lemma 2.1. Finally, many thanks to the referee for helpful comments and suggestions. This work was supported by EPSRC Early Career Fellowship number EP/K005545/1.

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Charles Vial DPMMS University of Cambridge Wilberforce Road Cambridge CB3 0WB UK c.vial@dpmms.cam.ac.uk 1554

Documenta Mathematica 18 (2013)