GENERIC CYCLES

M.ROVINSKY

It follows immediately from a theorem of Mumford [Mum], that if the sheaf of holomorphic 2-forms on a smooth projective surface S is very ample then for any general point p ∈ S, there exists no point q ≠ p, such that q is rationally equivalent to p.

This has been generalized by A.A.Roitman to the case of arbitrary projective varieties in [R1], where he got some upper bounds for dimensions of subvarieties of symmetric powers consisting of effective zero-cycles rationally equivalent to a sufficiently general fixed one.

The aim of this note is to present a refinement (see Proposition 2.2 below) of some results of [R1] using an approach of V.Srinivas [S], p.242.

In the first section we construct two versions of a class map on the Chow groups of varieties over a field K with a fixed model over a subfield k ⊂ K killing all cycles on divisors defined over k, i.e., class maps on the quotient group of “extremely transcendental” cycles. Assuming some of motivic conjectures, we show that the latter class maps are injective modulo torsion.

In the second section one of these classes is applied to the study of zero-cycles. In particular, we get a version of a theorem of Roitman [R1] on dimensions of subvarieties of symmetric powers of a variety consisting of rationally equivalent effective 0-cycles.

In the last section we give a few examples where the constructed classes are injective and deduce some corollaries of this injectivity.

I am grateful to J.P.Murre for his stimulating interest to this work, to A.Skorobogatov for calling my attention to the divisibility questions, to V.Srinivas for sending me a copy of the paper [S], to V.Voevodsky for helpful discussions on the motivic part, and to the Max-Planck-Institut in Bonn for its financial support and hospitality.

0.1. Notations and conventions. If x is a point of a scheme X then \overline{\{x\}} denotes the closure of the point x, which is a closed reduced irreducible subscheme of the scheme X.

For a scheme X and a non-negative integer q, the set of points of X whose closure has codimension (resp., dimension) q is denoted by X^{q} (resp., X_{q}).

For a vector space V over a field k the notation V^{∨} is used for the dual vector space Hom_{k}(k, V).

The Chow group CH^{q}(X) of a scheme X is defined as the cokernel of the divisor map

⨁_{x∈X^{q−1}} κ(x)^{×} ⟶ ⨁_{x∈X^{q}} ℤ,

where κ(x)^{×} is the multiplicative group of non-zero functions on the subscheme \overline{\{x\}} of X.

F^{•} and N^{•} will denote the Hodge and the coniveau filtrations on the de Rham cohomology, respectively.

1. Cycle classes c_{•}^{F} and c_{•}^{N}

Let X be a smooth d-dimensional projective variety over an algebraically closed field k of characteristic 0. Then, using duality H^{q}(X, Ω_{X}^{q}) = H^{d−q}(X, Ω_{X}^{d−q})^{∨}, one can define the class map cl : CH^{q}(X) ⟶ H^{q}(X, Ω_{X}^{q}) as [Z] ⟼ (ω ↦ ω|Z), where H^{d−q}(Z, Ω_{Z}^{d−q}) is canonically identified with k.

Fix an auxiliary smooth projective variety Y over k with the field of rational functions K. Decomposition

Ω_{X×_{k}Y/k}^{q} = ∑_{r=0}^{q} Ω_{X×_{k}Y/Y}^{q−r} ⊗_{O_{X×_{k}Y}} Ω_{X×_{k}Y/X}^{r}

induces via class map a homomorphism (at this point we may forget that K was finitely generated over k)

c̃_{q}^{F} : CH^{q}(X ×_{k} K) ⟶ ⨁_{r=0}^{q} H^{q}(X, Ω_{X}^{r}) ⊗_{k} Ω_{K/k}^{q−r}.  (1)

Suppose W is a subvariety of X of codimension p. Then the composition of the natural map CH^{q−p}(W ×_{k} K) ⟶ CH^{q}(X ×_{k} K) with the class map (1) factors through the subspace ⨁_{r=p}^{q} H^{q}(X, Ω_{X}^{r}) ⊗_{k} Ω_{K/k}^{q−r}, because for any cycle Z ∈ CH^{q−p}(W ×_{k} Y) the linear map [Z] : H^{d+dim Y −q}(X ×_{k} Y, Ω_{X×_{k}Y/k}^{d+dim Y −q}) ⟶ k factors through the restriction map H^{d+dim Y −q}(X ×_{k} Y, Ω_{X×_{k}Y/k}^{d+dim Y −q}) ⟶ H^{d+dim Y −q}(W ×_{k} Y, Ω_{W ×_{k}Y/k}^{d+dim Y −q}), and therefore, the kernel of this restriction contains both subspaces

H^{d−q}(X, Ω_{X/k}^{d−r}) ⊗ H^{dim Y} (Y, Ω_{Y/k}^{d+dim Y −q}) ⊆ H^{d+dim Y −q}(X ×_{k} Y, Ω_{X×_{k}Y/k}^{r+dim Y −q})

for any r < p. This shows that for any 0 ≤ q ≤ dim X, the sequence

⨁_{x∈X^{1}} CH^{q−1}(\overline{\{x\}} ×_{k} K) ⟶ CH^{q}(X ×_{k} K) \underrightarrow{c_{q}^{F}} H^{q}(X, O_{X}) ⊗_{k} H_{DR}^{q}(K/k),  (2)

where ĉ_{q}^{F} is induced by c̃_{q}^{F}, is a complex (this has been observed also by V.Srinivas [S]).

The following obvious lemma will allow us to define a class map c_{q}^{F} : CH^{q}(k(X)⊗_{k} K) ⟶ H^{q}(X, O_{X}) ⊗_{k} H_{DR}^{q}(K/k).

Lemma 1.1. 1. Let X be a reduced absolutely irreducible scheme over a field k. Then for any extension K of k the sequence

⨁_{x∈X^{1}} CH^{q−1}(\overline{\{x\}} ×_{k} K) ⟶ CH^{q}(X ×_{k} K) ⟶ CH^{q}(Spec(k(X) ⊗_{k} K)) ⟶ 0

is exact. This means that CH^{∗}(Spec(k(X) ⊗_{k} K)) is the Chow ring of “extremely transcendental cycles” on X.

2. If q > degtr(K/k) then CH^{q}(Spec(k(X) ⊗_{k} K)) = 0, and thus the sequence (2) is obviously exact.

Proof.

1. This is a limit case of the exact localization sequence for the Chow groups.

2. If the scheme X is finite-dimensional then Krull dimension of Spec(k(X) ⊗_{k} K) is the minimum of degtr(K/k) and dim X. Otherwise, it is degtr(K/k). □

Version:

Using duality H^{2q}(X) = H^{2d−2q}(X)^{∨}, one can define the class map cl_{H} : CH^{q}(X) ⟶ H^{2q}(X) as [Z] ⟼ (ω ↦ ω|Z), where H^{2d}(Z) is canonically identified with H^{0}(Z).

Composition of the class map cl_{H} with the Künneth decomposition induces a homomorphism

CH^{q}(X ×_{k} Y) ⟶ ⨁_{r=−q}^{q} H^{q+r}(X) ⊗ H^{q−r}(Y).  (3)

Suppose W is a subvariety of X of codimension p. Then the composition of the natural map CH^{q−p}(W ×_{k} Y) ⟶ CH^{q}(X ×_{k} Y) with the class map (3) commutes with composition of the corresponding class with X replaced by W with the Gysin map and factors through the subspace ⨁_{r=2p−q}^{q} N^{p}H^{q+r}(X) ⊗ H^{q−r}(Y), i.e., the following diagram commutes

[figure: CH^{q−p}(W ×_{k} Y); CH^{q}(X ×_{k} Y); H^{2(q−p)}(W ×_{k} Y); ⨁_{r=2p−q}^{q} N^{p}H^{q+r}(X) ⊗ H^{q−r}(Y); H^{2q}(X ×_{k} Y); Gysin⊗id; ⊆]

Therefore, interchanging the roles of X and Y, we can define the following class map (at this point, in the case H^{∗} = H_{DR}^{∗}, we may forget that K was finitely generated over k)

c̃_{q}^{N} : CH^{q}(X ×_{k} K) ⟶ ⨁_{r=−q}^{q} H^{q+r}(X) ⊗ H^{q−r}(K/k).

This shows that for any 0 ≤ q ≤ dim X, the sequence

⨁_{x∈X^{1}} CH^{q−1}(\overline{\{x\}} ×_{k} K) ⟶ CH^{q}(X ×_{k} K) \underrightarrow{c_{q}^{N}} H^{q}(k(X)/k) ⊗ H^{q}(K/k),  (4)

where ĉ_{q}^{N} is induced by c̃_{q}^{N}, is a complex.

1.1. Some consequences of motivic conjectures. Under some assumptions, it is sometimes clear, that the last map in the sequence (2) is non-trivial, whenever Γ(X, Ω_{X/k}^{q}) ≠ 0 and the field K is big enough. In particular, one has the following

Lemma 1.2. Let X be a smooth projective variety over a field k. Suppose that either

• for q as above there is a dominant rational map f : X ⟶ S to a smooth projective q-dimensional variety S over k with Γ(S, Ω_{S}^{q}) ≠ 0, or

• the inverse Lefschetz isomorphism H^{2d−q}(X)(d) ⟶^{∼} H^{q}(X)(q) is induced by a correspondence on X ×_{k} X.

Let K be an extension of the field of rational functions k(X), and Γ(X, Ω_{X}^{q}) ≠ 0. Then the composition of the cycle-class map (1) with the projection to H^{q}(X, O_{X}) ⊗_{k} Ω_{K/k}^{q}-summand is non-trivial.

Proof.

• Since the problem is birational, we may suppose that f : X ⟶ S is a morphism. Then we get a cycle Z = X ×_{S} X on X ×_{k} X of codimension q. Note that [Z] = (f ×f)^{∗}[Δ_{S}] ∈ CH^{q}(X ×_{k}X) and therefore, the image of [Z] under the map (2) coincides with the pullback of the image of [Δ_{S}] under the map (2) for S instead of X. Note further, that the homomorphism (f ×f)^{∗} : H^{q}(S, O_{S})⊗_{k} H_{DR}^{q}(k(S)/k) ⟶ H^{q}(X, O_{X})⊗_{k} H_{DR}^{q}(k(X)/k) is injective and the image of [Δ_{S}] under the map (2) is “the identity element in the algebra of endomorphisms of Γ(X, Ω_{X}^{d})” (see Lemma 2.1 formula (7) for details), so the last map in the complex (2) is not trivial.

• Let Z ⊂ X ×_{k} X be a cycle on X ×_{k} X of codimension q such that in the commutative

diagram

[figure: H^{2d−q}(X)(d); H^{q}(X)(q); H^{2d−q}(X ×_{k} X)(d); H^{2d+q}(X ×_{k} X)(d + q); ⊆; pr_{1}^{∗}; L^{−1}; ⋂ Z; pr_{2∗}]

the upper horizontal map L^{−1} is the inverse Lefschetz operator, that is, an isomorphism. Then its restriction induces an isomorphism

F^{0}H^{2d−q}(X)(d) ≅ H^{d−q}(X, Ω_{X}^{d}) ⟶^{∼} F^{0}H^{q}(X)(q) ≅ Γ(X, Ω_{X}^{q}).

Note, however, that the class of this isomorphism in H^{q}(X, O_{X}) ⊗_{k} Γ(X, Ω_{X}^{q}) coincides with the class of the cycle Z. □

Remark. If the space H^{q,q}(X × X) ⋂ H^{2q}(X × X; ℚ(q)) is generated by algebraic cycles then the inverse Lefschetz isomorphism H^{2d−q}(X)(d) ⟶^{∼} H^{q}(X)(q) is induced by a correspondence on X ×_{k} X.

(Proof. Since the multiplication by (d − q)-th power of a hyperplane section gives an isomorphism H^{q}(X)(q) ⟶^{∼} H^{2d−q}(X)(d), we get an isomorphism

id ⊗ L^{d−q} : H^{q}(X) ⊗ H^{q}(X)(q) ⟶^{∼} H^{q}(X) ⊗ H^{2d−q}(X)(d) ≅ EndH^{q}(X).

The identity of the algebra EndH^{q}(X) is a rational element of type (d, d). By pulling this class back via the isomorphism id ⊗ L^{d−q}, we get an element in H^{q,q}(X × X) ⋂ H^{2q}(X × X; ℚ(q)) inducing the inverse Lefschetz isomorphism. □)

It follows from the motivic conjectures, that the complex (2) is exact. Namely, in [Mur] Murre states the following conjectures. Conjecture A. For any smooth projective d-dimensional variety X there is a decompo-

sition Δ_{X} = ∑_{i=0}^{2d} π_{i}^{X} such that elements π_{i}^{X} ∈ CH^{d}(X ×_{k} X)_{ℚ} are orthogonal projectors: π_{i}^{X} ∘ π_{j}^{X} = δ_{i,j}π_{i}^{X} ∈ CH^{d}(X ×_{k} X) with classes [π_{i}^{X}] ∈ H^{2d−i}(X) ⊗ H^{i}(X) ⊆ H^{2d}(X × X).

Conjecture B(q). Elements π_{0}^{X}, …, π_{q−1}^{X} and π_{2q+1}^{X}, …, π_{2d}^{X} act trivially on CH^{q}(X)_{ℚ}. Conjecture D(q). CH_{hom}^{q}(X)_{ℚ} = Ker[π_{2q}^{X} : CH^{q}(X)_{ℚ} ⟶ CH^{q}(X)_{ℚ}]. In addition, we will need the following. Hodge Conjecture H(q). The subspace of H^{2q}(X; ℚ) generated by classes of algebraic

cycles coincides with the space H^{2q}(X; ℚ) ∩ F^{q}H^{2q}(X; ℂ).

Conjecture H(1, q)(Generalized Hodge Conjecture for coniveau 1). N^{1}H^{q}(X; ℚ) is the maximal Hodge substructure of H^{q}(X; ℚ) in F^{1}H^{q}(X; ℂ).

Proposition 1.1. 1. Conjectures A, B(q) and D(q) imply triviality of the canonical map

CH_{hom}^{q}(X ×_{k} Y) ⟶ CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ}  (5)

for any pair of k-varieties X and Y, and the existence of an embedding of CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ} into a canonical quotient of the group H^{q}(X; ℚ) ⊗ H^{q}(Y; ℚ(q)).

2. Conjectures A, B(q), D(q) and H(q − 1) imply injectivity of the map

c_{q}^{N} : CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ} ⟶ (H^{q}(X; ℚ)/N^{1}H^{q}(X; ℚ)) ⊗ (H^{q}(Y; ℚ(q))/N^{1}H^{q}(Y; ℚ)).

3. Suppose c_{q}^{N} is injective and the generalized Hodge conjecture H(1, q) holds. Then the map

c_{q}^{F} : CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ} ⟶ H^{q}(X, O_{X}) ⊗ Γ(Y, Ω_{Y}^{q})

is injective. In particular, assuming also H(q), any embedding σ : k ↪ ℂ gives rise to the following equivalence of the category Ξ_{d}(k) of finitely generated fields of transcendence degree d over k (with morphisms given by Hom_{Ξ_{d}(k)}(K, L) = CH_{0}(K ⊗_{k} L)_{ℚ}) with a full subcategory of the category H_{ℚ}^{d} of pure polarizable ℚ-Hodge structures of weight d and length of the Hodge filtration 2d with no Hodge substructures of shorter Hodge filtration: for any smooth projective variety X over k

Φ_{σ} : Ξ_{d}(k) ⟶ H_{ℚ}^{d}, k(X) ⟼ H^{d}(X ×_{k,σ} ℂ; ℚ)/N^{1}H^{d}(X ×_{k,σ} ℂ; ℚ).

Proof.

1. By the Conjecture D(q), CH_{hom}^{q}(X ×_{k} Y) = Ker(∑_{s=0}^{2q} π_{s}^{X} × π_{2q−s}^{Y}). Clearly, the

sequence

CH^{∗}(X)_{ℚ} ⟶^{Δ−π_{s}} CH^{∗}(X)_{ℚ} ⟶^{π_{s}} CH^{∗}(X)_{ℚ}

is exact, so CH_{hom}^{q}(X ×_{k} Y)_{ℚ} = Im(Δ_{X×Y} − π_{2q}^{X×Y}) is a subgroup of the group ∑_{i=q}^{2q−1} Imπ_{i}^{X×Y} = ∑_{i=q}^{2q−1} ∑_{k=0}^{i} Im(π_{k}^{X} × π_{i−k}^{Y}). It is not hard to see (modulo A and B(q)) that elements of types π_{0}^{X} ×γ, …, π_{q−1}^{X}×γ and π_{2q+1}^{X} ×γ, …, π_{2d}^{X} ×γ induce trivial maps CH^{q}(X ×Y)_{ℚ} ⟶ CH^{q}(X ×_{k} k(Y))_{ℚ} for any cycle γ ∈ CH^{d_{Y}} (Y ×Y). Indeed, let α ∈ CH^{q}(X ×Y) and β is the image of (Δ_{X} ×γ)(α) in CH^{q}(X ×_{k} k(Y)). Then \overline{(π_{s}^{X} × γ)(α)} = π_{s}^{X}(β) = 0 for 0 ≤ s ≤ q −1 and 2q + 1 ≤ s ≤ 2d. Therefore, both k and i − k should be in the set {q, …, 2q} in order to induce non-trivial maps CH_{hom}^{q}(X × Y)_{ℚ} ⟶ CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ}. But then i ≥ 2q, so CH_{hom}^{q}(X × Y)_{ℚ} ⟶ CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ} is always trivial. In the commutative diagram

[figure: CH^{q}(X ×_{k} Y); CH^{q}(X ×_{k} Y); CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ}; H^{s}(X) ⊗ H^{t}(Y)(q); ⊕_{r=−q}^{q}H^{q+r}(X) ⊗ H^{q−r}(Y)(q); π_{s}^{X}×π_{t}^{Y}]

q ≤ s, t ≤ 2q if the composition of the maps in the first row is non-trivial and q ≤ s + t ≤ 2q if π_{s}^{X} × π_{t}^{Y} is non-trivial, so s = t = q. We see then, that the action of π_{s}^{X} × π_{t}^{Y} is non-trivial on CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ} only for s = t = q, and π_{s}^{X} × π_{t}^{Y} projects H^{∗}(X × Y) onto H^{s}(X) ⊗ H^{t}(Y).

2. The group CH^{q}(X × Y)/ ∼_{hom} of cycles modulo homological equivalence is embedded into H^{2q}(X × Y; ℚ(q)). The subgroup N^{1}H^{q}(X; ℚ) ⊗ H^{q}(Y; ℚ(q)) in the range of the latter embedding is generated by images under Gysin maps of H^{q−2}(D; ℚ) ⊗ H^{q}(Y; ℚ(q − 1)) for all (desingularizations of) divisors D on X. If the image \overline{α} of an element α ∈ CH^{q}(X × Y) in H^{2q}(X × Y; ℚ(q)) belongs in fact to this subgroup then, due to semi-simplicity of the category of pure polarizable Hodge structures and assuming the Hodge conjecture, \overline{α} is also the class of a cycle supported on a divisor of type D ×_{k} Y, so α should vanish in CH^{q}(k(X) ⊗_{k} k(Y))_{ℚ}.

3. Let γ be an algebraic cohomology class in the subspace F^{1}H^{q}(X; ℂ)⊗_{ℂ}H^{q}(Y; ℂ) of F^{1}H^{2q}(X×Y; ℂ). We may suppose that in the decomposition γ = ∑_{i} ϵ_{i}⊗_{ℚ}δ_{i}, where ϵ_{i} ∈ H^{q}(X; ℚ) and δ_{i} ∈ H^{q}(Y; ℚ), δ_{i} are ℚ-linearly independent, and therefore, they are linearly independent over ℂ. After the complexification, the component of γ in H^{0,q}(X) ⊗ H^{q}(Y) should be trivial, so ∑_{i} ϵ_{i}^{0,q} ⊗_{ℚ} δ_{i} = 0 and, due to ℂ-linear independence of δ_{i}, all elements ϵ_{i} belong to F^{1}H^{∗}(X; ℂ). In the same way, looking at the H^{q}(X) ⊗ H^{q,0}(Y)-component, we conclude that elements δ_{i} are in the subspace F^{1}H^{q}(Y; ℂ), i.e., γ ∈ F^{1}H^{q}(X; ℂ) ⊗_{ℂ} F^{1}H^{q}(Y; ℂ). γ defines a morphism of Hodge structures H^{2 dim Y −q}(Y)(dim Y) ⟶^{α} H^{q}(X)(q). Since the image of α is a Hodge substructure in F^{1}H^{q}(X; ℚ), the generalized Hodge conjecture for coniveau 1 asserts that the image of α is a subspace in N^{1}H^{q}(X; ℚ), and finally we get that γ is in the subspace N^{1}H^{q}(X; ℚ) ⊗_{ℚ} F^{1}H^{q}(Y; ℚ). □

2. 0-cycles

Lemma 2.1. Let X be a smooth projective variety over an algebraically closed field k and K is an extension of k(X). Then the set of 0-cycles on X ×_{k} K carries a natural decreasing filtration of length d:

F^{s}(X ×_{k} K)_{0} = ∐_{s≤t≤q} ∐_{x∈X^{t}} Hom_{fields/k}^{[0]}(k(x), K),  (6)

where Hom_{fields/k}^{[0]}(k(x), K) = \{σ : k(x) ↪ \overline{K}\}/Gal(\overline{K}/K); F^{0}(X ×_{k} K)_{0} = (X ×_{k} K)_{0}, F^{d}(X ×_{k} K)_{0} = X^{d}, F^{d+1}(X ×_{k} K)_{0} = ∅.

This filtration is multiplicative:

F^{s}CH^{p}(X ×_{k} K) × F^{t}CH^{q}(Y ×_{k} K) ⟶ F^{s+t}CH^{p+q}(X ×_{k} Y ×_{k} K).

The group G of automorphisms of the field \overline{K} over the field k acts transitively on each subset Hom_{fields/k}^{[0]}(k(x), \overline{K}) of the set (X ×_{k} \overline{K})_{0}.

Proof. Any reduced irreducible effective zero-cycle on X ×_{k} K is determined by the kernel of a homomorphism of k-algebras O(U) ⊗_{k} K ⟶ \overline{K} identical on k ⊗_{k} K for an appropriate open subset U in X.

Such homomorphism is given by some O(U) ⟶^{id⊗1} O(U) ⊗_{k} K ⟶ \overline{K}. Let k(x) be the fraction field of the quotient of O(U) by the kernel of this homomorphism. Then we are reduced to homomorphisms k(x) ⊗_{k} K ⟶ \overline{K} identical on k ⊗_{k} K and such that \overline{K} is an algebraic extension of the field of fractions of their images.

Let σ and σ′ be some embeddings k(x) ↪ \overline{K} with the same kernel of the corresponding homomorphisms k(x) ⊗_{k} K ⟶ \overline{K}.

First note that Gal(\overline{K}/K) obviously respects the kernels.

Secondly, we need to show that there exists an automorphism τ ∈ Gal(\overline{K}/K) such that σ′ = τ ∘ σ. For any element g of k(x) let ∑_{i} h_{i}y^{i} be a minimal polynomial of σ′(g) with coefficients in the field K. Then ∑_{i} g^{i} ⊗ h_{i} ∈ Ker(σ′ · id) = Ker(σ · id), and therefore ∑_{i} h_{i}y^{i} is a minimal polynomial also of σ(g) with coefficients in K, and thus, there is an element τ ∈ Gal(\overline{K}/K) such that σ′ = τ ∘ σ, so we get the identity (6). □

Lemma 2.2. When q = dim X, the class map (1) is compatible with the natural composition law

ℤ[Hom_{fields/k}^{[0]}(L, K)] × ℤ[Hom_{fields/k}^{[0]}(K, F)] ⟶ ℤ[Hom_{fields/k}^{[0]}(L, F)],

and it is induced by

(x, σ) ⟼ [ω ⟼ ∑_{σ′∈Gal(\overline{K}/K)-orbit of σ} σ′ (ω|\overline{\{x\}})] ∈ H^{d}(X, Ω_{X}^{r}) ⊗_{k} Ω_{K/k}^{d−r},  (7)

where x ∈ X^{t}, σ : k(x) ⟶ \overline{K}, ω ∈ Γ(X, Ω_{X/k}^{d−r}), and the space H^{d}(X, Ω_{X}^{r}) is identified with the space Γ(X, Ω_{X/k}^{d−r})^{∨}.

Proof. Let (x, σ) correspond to a closed point of X ×_{k} K and K is the function field of a smooth projective variety Y. Then the closure of (x, σ) is the graph of a rational map Y ⟶ \overline{\{x\}}. The rest is obvious. □

Now let X be a smooth projective variety over an algebraically closed field k of characteristic 0, and k(X) be the field of rational functions on X.

The following proposition will not be used until some speculations in the next section.

Proposition 2.1. (compare with [R1] Theorem 3, [BS] Prop.1, [J] Theorem 3.5...) If for an extension K of k(X) all rational equivalence classes of 0-cycles come from r-subvarieties defined over k, i.e.,

⨁_{x∈X_{r}} CH_{0}(\overline{\{x\}} ×_{k} K) ⟶ CH_{0}(X ×_{k} K),

then H^{p}(X) = N^{1}H^{p}(X) (and therefore, Γ(X, Ω_{X}^{p}) = 0) for all p ≥ r + 1 and there is a finite set of cycles S ⊂ X_{r} such that

⨁_{x∈S} CH_{0}(\overline{\{x\}} ×_{k} K) ⟶ CH_{0}(X ×_{k} K).

The latter assertion holds if the groups CH_{0}( · ) are replaced with the groups CH_{0}( · )/n for some positive integer n.

Proof. The diagonal Δ ∈ CH^{dim X}(X ×_{k} X) acts on the cohomology as the identity. The kernel of the composition

CH^{dim X}(X ×_{k} X) ⟶ CH^{dim X}(X ×_{k} k(X)) ⟶ CH^{dim X}(X ×_{k} K)

(at least, modulo torsion) consists of cycles supported on X × D for all divisors D. Denote by \overline{Δ} the image of Δ under this composition.

Under the assumptions as above, ⨁_{x∈X_{r}} CH_{0}(\overline{\{x\}} ×_{k} K) ⟶ CH_{0}(X ×_{k} K). So \overline{Δ} is a linear combination of cycles in CH_{0}(X ×_{k} K) sitting on an r-cycle C defined over k. Then the restriction of \overline{Δ} to CH_{0}((X − C) ×_{k} K) is a torsion element and the restriction of \overline{Δ} to CH_{0}((X − C) ×_{k} k(X)) is also a torsion element.

Thus (a multiple of) Δ is rationally equivalent to a sum Z_{1} + Z_{2}, where Z_{1} is supported on C × X and Z_{2} is supported on X × D for an r-dimensional subscheme C and a divisor D. Clearly, the map H^{∗}(X) ⟶ H^{∗}(X) induced by the correspondence Z_{1} factors through the restriction H^{∗}(X) ⟶ H^{∗}(Supp(C)), and therefore, its image is in the subspace N^{1}H^{∗}(X) for p > r. The image of Z_{2} is also in the same subspace N^{1}H^{∗}(X), so the same applies to the image of Δ.

Consider the (identity) map from CH_{0}(second X) to CH_{0}(first X) induced by Z_{1} + Z_{2}. If the support of a cycle does not intersect D then it is sent to a cycle on C. Since any 0-cycle is rationally equivalent to another one with the support disjoint with the support of D this implies that CH_{0}(C) generates CH_{0}(X). □

Corollary 2.1. ([R1]) If Γ(X, Ω_{X}^{r}) ≠ 0 for some r > 0 and K contains an algebraic closure of k(X), then the group CH_{0}((X ×_{k} K)\(∐_{x∈X_{r−1}} \overline{\{x\}} ×_{k} K)) is of infinite rank (and nontrivial for any extension K of k(X)).

Proof. This group is non-trivial, just because any embedding σ : k(X) ⟶ K induces an embedding Γ(X, Ω_{X}^{r}) ⟶ H_{DR}^{r}(K/k).

If the group was only finite-dimensional, then all possible embeddings of fields σ : k(X) ⟶ \overline{K} could span only a finite-dimensional k-vector subspace in the infinite-dimensional vector space H^{d}(X, Ω_{X}^{d−r}) ⊗_{k} H_{DR}^{r}(K/k). Therefore, after identifying H^{d}(X, Ω_{X}^{d−r}) with Γ(X, Ω_{X}^{r})^{∨}, we see, that images of Γ(X, Ω_{X}^{r}) under all possible σ’s span only a finite-dimensional ksubspace in H_{DR}^{r}(K/k). But this is obviously false, when K contains an algebraic closure of k(X). □

2.1. Dimensions of subvarieties consisting of rationally equivalent 0-cycles. Before we proceed to the main result of this section we need the following lemma in linear algebra.

Lemma 2.3. Let V_{1}, …, V_{N} be a collection of subspaces of a K-space W. Fix an integer r > 1 and for each 1 ≤ i ≤ N fix a non-trivial element ω_{i} ∈ ∧^{r}V_{i}. Let P be a subspace of the sum V_{1} + ⋯ + V_{N} of these spaces such that ∑_{i} ω_{i} ∈ ∧^{r}P. Suppose that t = dim Ker(V_{1} ⊕

⋯ ⊕ V_{N} \underrightarrow{∑} W) ≤ r − 2. Then dim P ≥ r · N − t.

Remarks. 1. In the case dim V_{1} = ⋯ = dim V_{N} = r and ∑ V_{i} = W this bound is obviously precise.

2.If N = 2 and t = r − 1 then value r (< r · 2 − (r − 1) = r + 1) of dim P does occur. Proof. First suppose that t = 0. For each 1 ≤ i ≤ N we may choose a surjection π_{i} : V_{i} ⟶ V′_{i} onto an r-dimensional

K-space V′_{i} such that the r-vectors π_{i}(ω_{i}) ∈ ∧^{r}V′_{i} are all non-trivial.

Let P′ = (⊕_{i=1}^{N}π_{i})(P) ⊆ ⊕_{i=1}^{N}V′_{i}. Then the r-vector ω = ∑_{i=1}^{N} π_{i}(ω_{i}) ∈ ∧^{r}P′ is non-trivial. In some dual bases {x_{1}^{(i)}, …, x_{r}^{(i)}} and {ξ_{1}^{(i)}, …, ξ_{r}^{(i)}} of the spaces V′_{i} and V′_{i}^{∨}, respectively, we have π_{i}(ω_{i}) = x_{1}^{(i)} ∧ ⋯ ∧ x_{r}^{(i)} and the map

⊕_{i=1}^{N}V′_{i}^{∨} ⟶^{⟨ · ,ω⟩} ⊕_{i=1}^{N} ∧^{r−1} V′_{i} ⊂ ∧^{r−1}(⊕_{i=1}^{N}V′_{i}), ξ_{j}^{(i)} ⟼ x_{1}^{(i)} ∧ ⋯ ∧ x̂_{j}^{(i)} ∧ ⋯ ∧ x_{r}^{(i)},

is obviously injective (in fact, an isomorphism). But if dim P′ < rN then there exists a non-zero element v ∈ ⊕_{i=1}^{N}V′_{i}^{∨} such that ⟨v, P′⟩ = 0, and thus if ω ∈ ∧^{r}P′ then ⟨v, ω⟩ = 0, contradicting injectivity.

Sublemma 2.1. Let V_{1}, …, V_{N} be a collection of subspaces of a K-space W. Suppose that r − 1 > t = dim Ker(V_{1} ⊕ ⋯ ⊕ V_{N} ⟶ W). Then the evident map ⊕_{i=1}^{N} ∧^{r−1} V_{i} ⟶ ∧^{r−1}W is injective.

Remark. If t = r − 1 then the latter map can be not injective for any N and any spaces V_{i} of dimension ≥ r − 1.

Proof. Without loss of generality, we may suppose that W = V_{1}+⋯+V_{N}. Let dim V_{i} = d_{i} and V_{i} = V′_{i} + V′′_{i}, where V_{1} = V′′_{1}, V′_{i} ⊂ ∑_{j=1}^{i−1} V_{j} and V′′_{i} ∩ (∑_{j=1}^{i−1} V_{j}) = 0. Then dim W = ∑_{i=1}^{N} d_{i} −t = dim V′′_{1} +dim V′′_{2} +dim V′′_{3} +⋯+dim V′′_{N}. If dim V′′_{i} ≤ d_{i} −r+1 for some i then dim W ≤ ∑_{i=1}^{N} d_{i} − (r − 1) < ∑_{i=1}^{N} d_{i} − t, so contradiction shows that dim V′′_{i} ≥ d_{i} − r + 2 for all i and therefore, dim V′_{i} ≤ r − 2. This implies that in the decomposition

∧^{r−1}V_{i} = ∧^{r−1}V′_{i} + ∧^{r−2}V′_{i} ⊗ V′′_{i} + ∧^{r−3}V′_{i} ⊗ ∧^{2}V′′_{i} + ∧^{r−4}V′_{i} ⊗ ∧^{3}V′′_{i} + …

∧^{r−1}V′_{i} = 0, and thus ∧^{r−1}(V_{1} + ⋯ + V_{i−1}) ∩ ∧^{r−1}V_{i} = 0. In other words,

∧^{r−1}(V_{1} + ⋯ + V_{i−1}) ⊕ ∧^{r−1}V_{i} ↪ ∧^{r−1}(V_{1} + ⋯ + V_{i}).

Iterating the process we get a sequence of injections

∧^{r−1}V_{1} ⊕ ∧^{r−1}V_{2} ⊕ ⋯ ⊕ ∧^{r−1}V_{N} ↪ ∧^{r−1}(V_{1} + V_{2}) ⊕ ∧^{r−1}V_{3} ⊕ ⋯ ⊕ ∧^{r−1}V_{N} ↪ …

… ↪ ∧^{r−1}(V_{1} + ⋯ + V_{N−1}) ⊕ ∧^{r−1}V_{N} ↪ ∧^{r−1}(V_{1} + ⋯ + V_{N}) = ∧^{r−1}W.
□

We use this Sublemma to reduce the case of general t to the case t = 0 as follows. Consider the commutative diagram (again, we suppose that W = V_{1} + ⋯ + V_{N})

[figure: ⊕_{i=1}^{N}V′_{i}^{∨}; ⊕_{i=1}^{N}V_{i}^{∨}; W^{∨}; ⊕_{i=1}^{N} ∧^{r−1} V′_{i}; ⊕_{i=1}^{N} ∧^{r−1} V_{i}; ∧^{r−1}W; ∧^{r−1}(W/ ∑_{i=1}^{N} Ker(V_{i} ⟶ V′_{i})); ⊆; ⊕⟨ · ,π_{i}(ω_{i})⟩; ≅; ⊕⟨ · ,ω_{i}⟩; ⟨ · ,∑ ω_{i}⟩; ⊇; codim=t; α; onto; β]

Let L be the subspace of W^{∨} of functionals vanishing on P. Then L ⊆ Ker⟨ · , ∑ ω_{i}⟩, and thus,

dim P = dim W − dim L ≥ dim W − dim Ker⟨ · , ∑ ω_{i}⟩ = dim Im⟨ · , ∑ ω_{i}⟩.

Note that Ker(⊕_{i=1}^{N}V′_{i} ⟶ W/ ∑_{i=1}^{N} Ker(V_{i} ⟶ V′_{i})) is the image of Ker(⊕_{i=1}^{N}V_{i} ⟶ W), so its dimension is < r − 1, and thus we apply the Sublemma to see that the slope arrow is injective and the composition of ⟨ · , ⊕ω_{i}⟩ with α and β is injective on the subspace ⊕_{i=1}^{N}V′_{i}^{∨}, and therefore,

dim Im⟨ · , ∑ ω_{i}⟩ ≥ dim(W^{∨} ∩ (⊕_{i=1}^{N}V′_{i}^{∨})) ≥ rN − t.
□

Proposition 2.2. (compare with [R1] Theorem 1)

1. If the canonical sheaf Ω_{X}^{d} defines a rational map φ of X, then for any “generic” (i.e., in the complement to the union of all divisors defined over k) point x ∈ X(K) and for any point y in its rational equivalence class φ is defined at y and φ(x) = φ(y).

2. Suppose that Γ(X, Ω_{X}^{r}) ≠ 0 for some r > 1. Fix such N subvarieties Y_{1}, …, Y_{N} of X defined over k that each restriction map Γ(X, Ω_{X}^{r}) ⟶ Γ(Y_{i}, Ω_{Y_{i}}^{r}) is not trivial. Let y_{1} +⋯+y_{N} be a sum of “generic” points on Y_{i}’s corresponding to a point in the complement to all subvarieties in Y_{1} ×_{k} ⋯ ×_{k} Y_{N} of codimension t + 1 < r defined over k. Then the set of effective 0-cycles on X in the rational equivalence class of y_{1} + ⋯ + y_{N} is either finite or countable union of subvarieties of codimension ≥ r · N − t in the N-th symmetric power Σ^{N}X of the variety X.

Proof.

1. Due to Lemma 2.1, \overline{K}-valued points of X outside divisors defined over k are described as embeddings of fields σ : k(X) ⟶ \overline{K}. If the canonical sheaf defines a rational map of X, then the field of rational functions on X is an extension of the field L generated by fractions of global sections of the canonical sheaf. Therefore, restriction of any embedding of fields σ : k(X) ⟶ \overline{K} to the field L is uniquely defined by the induced embedding Γ(X, Ω_{X/k}^{dim X}) ⟶ Ω_{\overline{K}/k}^{dim X}, and its prolongations back to the field k(X) are points of the fibre of the canonical map φ over the point σ|_{L}. By the second part of the same Lemma, this implies that for generic points (=embeddings of fields) in distinct fibres of φ their images under the last map in the sequence (2) are distinct and non-zero. On the other hand, the last map in the sequence (2) kills all non-generic (i.e., sitting on a divisor over k) points.

2. Fix N embeddings τ_{i} : k(Y_{i}) ⟶ K with images generating a field of transcendence degree ∑_{i} d_{i} − t over k, where d_{i} = dim Y_{i}. We may always suppose that the field k is countable. Now fix a holomorphic r-form ω on X with non-trivial restrictions to each of the subvarieties Y_{i}. If a cycle ∑_{i}[σ_{i} : k(x_{i}) ⟶ K] is rationally equivalent to the fixed one then ∑_{i} τ_{i}(ω|_{Y_{i}}) = ∑_{i} σ_{i}(ω|\overline{\{x_{i}\}}). We apply Lemma 2.3 as follows. Set V_{i} = K ⊗_{τ_{i}(k(Y_{i}))} Ω_{τ_{i}(k(Y_{i}))/k}^{1}, and

P = (∑_{i=1}^{N} K ⊗_{σ_{i}(k(x_{i}))} Ω_{σ_{i}(k(x_{i}))/k}^{1}) ⋂ (∑_{i=1}^{N} K ⊗_{τ_{i}(k(Y_{i}))} Ω_{τ_{i}(k(Y_{i}))/k}^{1}) .

Due to the Lemma, dim P ≥ rN − t, so

dim ((∑_{i=1}^{N} K ⊗_{σ_{i}(k(x_{i}))} Ω_{σ_{i}(k(x_{i}))/k}^{1}) + (∑_{i=1}^{N} K ⊗_{τ_{i}(k(Y_{i}))} Ω_{τ_{i}(k(Y_{i}))/k}^{1})) ≤ ∑_{i} d_{i} + N(d − r),

and therefore,

degtr(σ_{1}(k(x_{1})) … σ_{N}(k(x_{N}))τ_{1}(k(Y_{1})) … τ_{N}(k(Y_{N}))/τ_{1}(k(Y_{1})) … τ_{N}(k(Y_{N}))) ≤ N(d−r)+t,

in other words, there are ≤ N(d−r)+t degrees of freedom in the choice of the field σ_{1}(k(x_{1})) … σ_{N}(k(x_{N}))τ_{1}(k(Y_{1})) … τ_{N}(k(Y_{N})). Finally, since the number of the embeddings σ_{i}(k(x_{i})) ↪ σ_{1}(k(x_{1})) … σ_{N}(k(x_{N}))τ_{1}(k(Y_{1})) … τ_{N}(k(Y_{N})) is countable, we are done. □

Corollary 2.2. If the canonical sheaf Ω_{X}^{d} defines a birational map of X, then any “generic” in the above sense point x ∈ X(K) is the only one in its rational equivalence class. □

Remarks. 1. If Γ(X, Ω_{X}^{r}) ≠ 0 is the non-trivial space of holomorphic forms of the maximal degree, r > 1 is odd and M is big (M > dim Γ(X, Ω_{X}^{r})) then for the M-th symmetric power Y = Σ^{M}X of X we have Γ(Y, Ω_{Y}^{rM}) = 0, but Proposition 2.2 (2) still gives the same assertions for effective 0-cycles, when applied to Y. Namely, if p : Σ^{N}Y ⟶ CH_{0}(X) is the natural map, then for any generic point x ∈ Σ^{N}Y the set p^{−1}(p(x)) is a countable union of subvarieties of codimension ≥ (rM)N in Σ^{N}Y, since p factors through Σ^{MN}X ⟶ CH_{0}(X).

2. One can easily construct an example of a variety with the canonical map to the point, where arbitrary set of rationally equivalent effective zero-cycles of degree N is Zariski-dense in Σ^{N}X. So the first part of the Proposition 2.2 is precise, and in the second part one cannot replace countable unions of subvarieties by finite ones.

Namely, let X be a smooth model of the quotient of an even-dimensional abelian variety A by the involution x ⟼ −x. Clearly, AlbX = 0 and Γ(X, Ω_{X}^{d}) is a one-dimensional vector space. Due to a theorem of Roitman ([R2]), the group CH_{0}(X) is torsion-free. On the other hand, any integer n ≠ 0 induces an injection (in fact, an isomorphism) n_{∗} : CH_{0}(A)_{ℚ} ⟶ CH_{0}(A)_{ℚ}. (For any non-zero z in Kern_{∗} one has n^{∗}n_{∗}z = z∗∑_{nζ=0}[ζ] = n^{2d}·z+z∗∑_{nζ=0}([ζ]−

[0]) = 0 ∈ CH_{0}(A)_{ℚ}, where [a] ∗ [b] := [a + b]. According to Bloch [B1], z ∈ (CH_{0}(X)_{ℚ}^{0})^{∗^{i}}

but z ∉ (CH_{0}(X)_{ℚ}^{0})^{∗^{i+1}} for some i ≥ 0. So n^{2d} · z = z ∗ ∑_{nζ=0}([ζ] − [0]) = 0 ∈ CH_{0}(A)_{ℚ}.) Therefore, images of a point of A and its translates by torsion elements of A in X are rationally equivalent. Adding N generic points in general position on X gives the example.

To construct odd-dimensional examples, one can use a smooth model X of the quotient of the cube of the elliptic curve ℂ/ℤ[(−1 + \sqrt{−3})/2] by the automorphism of third order (z_{1}, z_{2}, z_{3}) ⟼ (−1 + \sqrt{−3})/2 · (z_{1}, z_{2}, z_{3}).

3. When X is a general surface of degree ≥ 6 then rational equivalence classes of distinct points are distinct (see [V]).

4. When X is an abelian variety and N ≤ 3 there are more precise bounds, see [AP].

3. Complex (2), Bloch’s conjecture and divisibility problems

Bloch’s conjecture asserts that for any smooth projective surface X over ℂ with vanishing H^{2}(X, O_{X}) the group CH_{0}(X)^{0} is generated by images of a finite number of Jacobians of curves on X.

As it was shown in Proposition 2.1, if for some embedding σ : k ⟶ ℂ the group CH_{0}(X ×_{k,σ} ℂ) was generated by CH_{0}-groups of all divisors defined over k then in fact it was generated by CH_{0}-groups of a finite number of divisors. Therefore, the exactness of the complex (2) is a generalization of Bloch’s conjecture.

Even more particularly, this implies that • for any pair of k-varieties X and Y, and any q the group CH^{q}(k(X) ⊗_{k} k(Y)) is finitely

generated and neglects base changes k ⟼ K, and

• if the numerical equivalence coincides with the homological one modulo torsion then the ℚ-algebras CH_{0}(k(X) ⊗_{k} k(X))_{ℚ} are semisimple, when non-trivial.

In case q = 1 the exactness of the sequence (2) is easily verified:

Lemma 3.1. Let 𝔛 and S be smooth projective varieties over \overline{ℚ} and λ : 𝔛 ⟶ S is a structure of S-scheme on 𝔛 such that generic fibre is smooth and absolutely irreducible. Suppose that K is an extension of the field k = \overline{ℚ}(S) and X = 𝔛 ×_{S} k. Then the sequence

Pic^{0}𝔛/Pic^{0}S ⟶ Pic^{0}(X ×_{k} K) ⟶ H_{DR}^{2}(X ×_{k} K/ℚ)  (8)

induced by the sheaf homomorphism O_{X×_{k}K}^{×} ⟶^{d log} Ω_{X×_{k}K/ℚ}^{1}, is exact. In particular,

Pic^{0}(X ×_{k} K)/Pic^{0}(X) ⟶ H^{1}(X, O_{X}) ⊗_{k} H_{DR}^{1}(K/k)  (9)

is an embedding.

Proof. Without loss of generality, suppose that K is finitely generated over k and let T be a model of K over \overline{ℚ} with a morphism f : T ⟶ S. Consider a Cartesian square

[figure: 𝔛 ×_{S} T = 𝔛_{T}; 𝔛; ◇; T; S; λ; f]

By the definition of the Neron–Severi group, one has a commutative diagram with exact rows

[figure: 0; Pic^{0}𝔛_{T}; Pic𝔛_{T}; NS(𝔛_{T}); 0; 0; Pic^{0}T; PicT; NS(T); 0.]

Therefore, the sequence

Pic^{0}𝔛_{T} /Pic^{0}T ⟶ Pic𝔛_{T} /PicT ⟶ NS(𝔛_{T})/NS(T) ⟶ 0

is exact.

The next step is an identifying of the group Pic^{0}𝔛_{T} /Pic^{0}T. Consider the Leray spectral sequence of the projection 𝔛_{T} ⟶ T: E_{2}^{p,q}(T) = H^{p}(T, f^{∗}R^{q}λ_{∗}ℚ) ⇒ H^{p+q}(𝔛_{T}; ℚ).

Since the fibres of the map λ are connected, E_{2}^{p,0}(T) is identified via pull-back f^{∗} with the space H^{p}(T; ℚ). Moreover, since any surjective morphism of smooth projective varieties induces injective pull-back on the cohomology, E_{2}^{p,0}(T) is a subspace of H^{p}(𝔛_{T}; ℚ) and hence E_{2}^{p.0}(T) = E_{∞}^{p,0}(T) and E_{2}^{0,1}(T) = E_{∞}^{0,1}(T).

Note also, that E_{2}^{0,p}(T) = f^{∗}E_{2}^{0,p}(S). Therefore we get

E_{∞}^{0,1}(T) = H^{1}(𝔛_{T}; ℚ)/H^{1}(T; ℚ) ≅ E_{∞}^{0,1}(S) = H^{1}(𝔛; ℚ)/H^{1}(S; ℚ),

and thus Pic^{0}𝔛_{T} /Pic^{0}T = Pic^{0}𝔛/Pic^{0}S.

One can identify the group Pic𝔛_{T} /PicT with the group PicX_{K} to get an exact sequence

Pic^{0}𝔛/Pic^{0}S ⟶ PicX_{K} ⟶ NS(𝔛_{T})/NS(T) ⟶ 0.

The rest is straightforward. □

Corollary 3.1. If X is an abelian variety over k, α ∈ CH^{q}(X ×_{k} K) and a is a k-rational point of AlbX, then t_{a}α − α ∈ F^{1}CH^{q}(X ×_{k} K), when q = 0, 1, dim X. In other words, in these cases translations by elements of the group AlbX(k) act trivially on CH^{q}(k(X) ⊗_{k} K).

Proof. Due to the Lemma, for any a ∈ X(k) and any divisor D on X ×_{k} K the difference t_{a}D − D is in the subgroup F^{1}CH^{1}(X ×_{k} K), and thus t_{a}D_{1} ⋯ t_{a}D_{q} − D_{1} ⋯ D_{q} ∈ F^{1}CH^{q}(X ×_{k} K) for any divisors D_{1}, …, D_{q}. But as it is shown in [B1] p.216,

Pic(X)^{⊗^{g−1}} ⊗ Pic(X)^{0} ⟶ CH_{0}(A)^{0},

so if ∑_{α} D_{1}^{α} ⋯ D_{g}^{α} = [x] − [b] ∈ CH_{0}(X ×_{k} K) for some b ∈ X(k) then

[x + a] − [b + a] − [x] + [a] = ∑_{α} t_{a}D_{1}^{α} ⋯ t_{a}D_{g}^{α} − ∑_{α} D_{1}^{α} ⋯ D_{g}^{α} ∈ F^{1}CH_{0}(X ×_{k} K)

due to the multiplicativity property of the filtration F (see the remark on p.6). □

Example (Bloch’s conjecture for certain quotients of abelian varieties). Let A be an odddimensional abelian variety. Then there exist some divisors D_{1} and D_{2} on varieties Σ^{2}A and A/⟨−1⟩, respectively, such that the inclusions induce surjections CH_{0}(D_{1}) ⟶ CH_{0}(Σ^{2}A) and CH_{0}(D_{2}) ⟶ CH_{0}(A/⟨−1⟩).

Proof. We use an explicit decomposition of the diagonal in a g-dimensional abelian variety Δ ∈ CH^{g}(A ×_{k} A) into a sum of orthogonal projectors Δ = ∑_{s=0}^{2g} π_{s} from [K], where

π_{s} = \frac{(−1)^{g}}{(2g)!} \binom{2g}{s}l^{s} ∘ ^{t}l^{2g−s} = \frac{(−1)^{g}}{(2g)!} \binom{2g}{s}pr_{A×_{k}A∗}(pr_{A×_{k}Â}^{∗}(l^{s}) · pr_{Â×_{k}A}^{∗}(^{t}l^{2g−s})),

and l is the first Chern class of the Poincaré line bundle on A ×_{k} Â. Clearly, for s > g the support of any representative of the class l^{s} cannot map surjectively onto A, and thus the support of some representative of the class π_{s} sits in a product D_{s} ×_{k} A for some divisor D_{s}. In particular, π_{s} induces the trivial map CH_{0}(A) ⟶ CH_{0}(A). Using the identity π_{s} = ^{t}π_{2g−s}, we get that π_{s} maps CH_{0}(A) to the image of CH_{0}(D_{s}) for s < g. Let X be the quotient of an odd-dimensional abelian variety A by the involution a ⟼ −a. Since CH^{∗}(X ×_{k} X)_{ℚ} = CH^{∗}(A ×_{k} A)_{ℚ}^{(ℤ/2ℤ)^{2}} and the elements (1, 0) and (0, 1) of (ℤ/2ℤ)^{2} change the sign of π_{g}, we see that all zero-cycles on X come from the divisor ∑_{s=0}^{g−1} \overline{D_{s}} on X.

For the symmetric square of A, we note that all but a finite number of fibres of the summation map are isomorphic to X and the monodromy acts as a finite group. All 0-cycles on a fibre come from a divisor. Since almost all fibres are identified up to a finite action of the monodromy, all 0-cycles on the symmetric square of A come from a divisor. □

Remarks. 1. This implies in particular that for a sufficiently large field K the group CH_{0}(k(A^{N}) ⊗_{k} K) is dominated by ∧^{N}CH_{0}(k(A) ⊗_{k} K).

Even more particularly, if E is an elliptic curve over k then from a commutative diagram

[figure: ∧^{N}CH_{0}(k(E) ⊗_{k} K); CH_{0}(k(E^{N}) ⊗_{k} K); H^{1}(E, O_{E})^{⊗N} ⊗ ∧^{N}H_{DR}^{1}(K/k); H^{N}(E^{N}, O_{E^{N}}) ⊗ H_{DR}^{N}(K/k),; ⟶; ⊂]

if Y is a projective variety of dimension ≥ N such that ∧^{N}F^{1}H_{DR}^{1}(Y/k) ⟶ H_{DR}^{N}(Y/k) is an injection, we get that ∧^{N}CH^{1}(k(E) ⊗_{k} k(Y)) ⟶ CH_{0}(k(E^{N}) ⊗_{k} k(Y)) is an injection. The lower horizontal map is obviously not injective, even on the image of ∧^{N}CH_{0}(k(E)⊗_{k}K) (if Y is a hyperplane section of E^{N}, embeddings σ_{i} : k(E) ↪ K(Y)’s correspond to the projections pr_{i} : Y ⟶ E, and ω ∈ Γ(E, Ω_{E}^{1}), then σ_{1}(ω), …, σ_{N}(ω) generate an N-dimensional ksubspace in Ω_{k(Y)/k}^{1} but an (N − 1)-dimensional k(Y)-subspace in Ω_{k(Y)/k}^{1}).

2. Due to a theorem of Pirola, there are no hyperelliptic curves on a generic abelian variety of dimension ≥ 3 in characteristic zero. Therefore, there are rational curves neither on A/⟨−1⟩ nor on the symmetric square of a generic abelian variety A.

3.1. Divisibility problems.

Proposition 3.1. Suppose that the cohomological l-dimension of the field K is < q and the class c_{q}^{N} : CH^{q}(k(X) ⊗_{k} K) ⟶ H^{q}(X)/N^{1}H^{q}(X) ⊗ H^{q}(K/k) is injective modulo torsion. Then the group CH^{q}(k(X) ⊗_{k} K) is l-divisible modulo l-torsion.

Proof. Let Y_{i} be a smooth projective variety over k with the field of rational functions k(Y_{i}) ⊆ K. If the class c_{q}^{N} : CH^{q}(k(X) ⊗_{k} k(Y_{i})) ⟶ H^{q}(X)/N^{1}H^{q}(X) ⊗ H^{q}(Y_{i})/N^{1}H^{q}(Y_{i}) is injective modulo torsion then we have an embedding

lim_{⟵_{n}} CH^{q}(k(X) ⊗_{k} k(Y_{i}))/l^{n} ⟶ H^{q}(X)/N^{1}H^{q}(X) ⊗ lim_{⟵_{n}}(H^{q}(Y_{i})/l^{n})

modulo l-torsion.

Let (α_{i}) be an inductive system of non-trivial elements of CH^{q}(k(X) ⊗_{k} k(Y_{i}))/l^{n} for a projective system of varieties Y_{i} as above with function fields approximating the field K. Then lim_{i} α_{i} is a non-trivial element of the group lim_{⟶_{i}} CH^{q}(k(X) ⊗_{k} k(Y_{i}))/l^{n} ↪ H^{q}(X)/N^{1}H^{q}(X) ⊗ lim_{⟶_{i}} H^{q}(k(Y_{i}); ℤ/l^{n}). But the latter group coincides with the group H^{q}(X)/N^{1}H^{q}(X) ⊗ H^{q}(∪_{i}k(Y_{i}); ℤ/l^{n}) and should be zero by the assumption on the cohomologiical l-dimension of the field K. □

Remarks. 1. Under assumptions of the Proposition, if k is algebraically closed and K is finitely generated over k, then the group CH^{q}(k(X) ⊗_{k} K) is trivial (Lemma 1.1).

2. In the case of algebraically closed field K this is obvious, since any Chow variety is define over the field k, so the divisible group CH_{alg}^{q}(X ×_{k} K) surjects onto the groups CH^{q}(X ×_{k} K)/CH^{q}(X) ⟶ CH^{q}(k(X) ⊗_{k} K).

3. If K is a p-adic field and l ≠ p is another prime, then the group CH_{alg}^{q}(X ×_{k} K) is ldivisible modulo (finite) torsion, and, as above, we have the following sequence of surjections CH_{alg}^{q}(X ×_{k} K) ⟶ CH^{q}(X ×_{k} K)/CH^{q}(X) ⟶ CH^{q}(k(X) ⊗_{k} K).

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