# Varieties with representable CH_0-group and a question of Colliot-Thélène

Claire Voisin Pour

► To cite this version:

Claire Voisin Pour. Varieties with representable CH_0-group and a question of Colliot-Thélène. Algebraic Geometry, 2025, pp.888 - 901. 10.14231/ag-2025-027. hal-05343307

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# Varieties with representable $CH_0$-group and a question of Colliot-Thélène

Claire Voisin

Pour Gerard van der Geer, avec amitié

## ABSTRACT

We continue our investigation of the geometry of the Albanese morphism on 0-cycles. We provide an example of a smooth projective variety with representable $CH_0$-group but with no universal 0-cycle, which answers a question asked by Colliot-Thélène. Our construction relies on a counterexample to the integral Hodge conjecture provided by Benoist and Ottem.

## 1. Introduction

Let $X$ be a smooth projective complex variety of dimension $n$. The Albanese morphism

$$ \mathrm{alb}_X: X \to \mathrm{Alb}(X) \qquad (1.1) $$

is a morphism of algebraic varieties defined using a base-point in $X$. The induced morphism

$$ a_X: \mathrm{CH}_0(X)_{\mathrm{hom}} \to \mathrm{CH}_0(\mathrm{Alb}(X))_{\mathrm{hom}} \to \mathrm{Alb}(X), \qquad (1.2) $$

where the last map is the sum map, is the Griffiths–Abel–Jacobi map for 0-cycles on $X$, and can also be characterized as the universal functorial regular abelian quotient of $\mathrm{CH}_0(X)_{\mathrm{hom}}$ (see [Mur85, Ser60]). Indeed, the group morphism $a_X$ is surjective and regular, which means that for any smooth projective variety $W$ with base-point $w_0$ and codimension $n$ cycle $\Gamma$ in $\mathrm{CH}^n(W \times X)$, the map

$$ W \to \mathrm{Alb}(X), \quad w \mapsto a_X \circ \Gamma_*(w - w_0) $$

is a morphism of algebraic varieties. In [Voi24, Voi25], we introduced and studied the notion of “universal 0-cycle” (parameterized by the Albanese variety), as follows.

**DEFINITION 1.1.** A universal 0-cycle for $X$ is a codimension $n$ cycle $\Gamma \in \mathrm{CH}^n(\mathrm{Alb}(X) \times X)$, where $n := \dim(X)$, such that for any $t \in \mathrm{Alb}(X)$ we have

$$ a_X \circ \Gamma_*(\{t\} - \{0_{\mathrm{Alb}(X)}\}) = t \quad \text{in } \mathrm{Alb}(X). $$

**Example 1.2.** A smooth projective curve admits a universal 0-cycle given by the Poincaré divisor on $J(C) \times C$.

<footnote>Received 16 June 2025, accepted in final form 10 October 2025.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification 14C15, 14C25.</footnote>

<footnote>Keywords: Albanese morphism, universal zero-cycle.</footnote>

<footnote>This journal is © Foundation Compositio Mathematica 2025. This article is distributed with Open Access under the terms of the Creative Commons Attribution Non-Commercial License, which permits non-commercial reuse, distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial re-use, please contact the Foundation Compositio Mathematica.</footnote>

<footnote>The author is supported by the ERC Synergy Grant HyperK (Grant agreement No. 854361).</footnote>
UNIVERSAL 0-CYCLE ON VARIETIES WITH REPRESENTABLE CH₀-GROUP

We will review in Section 2 some basic facts concerning this definition. Let us just say that there always exists a universal 0-cycle with $\mathbb{Q}$-coefficients, and that an $X$ not having a universal 0-cycle provides a counterexample to the integral Hodge conjecture on $\text{Alb}(X) \times X$. Furthermore, we proved in [Voi25] that there exist smooth projective varieties not admitting a universal 0-cycle.

Recall that, following Mumford [Mum68] and Roitman [Roi80a], the variety $X$ has a representable $\text{CH}_0$-group if the morphism $a_X$ of (1.2) is an isomorphism. The examples provided in [Voi25] are surfaces whose Albanese variety is isomorphic to the intermediate Jacobian of a rationally connected threefold. These surfaces do not in general have a representable $\text{CH}_0$-group (like the surface of lines of a smooth cubic threefold, they rather tend to be of general type). In the paper [Col25], Colliot-Thélène asked the following.

*Question 1.3.* Does there exist a smooth complex projective variety $X$ with representable $\text{CH}_0$-group and not having a universal 0-cycle?

The purpose of this paper is to present such an example.

**THEOREM 1.4.** There exists a smooth projective threefold $X$ such that $\text{CH}_0(X)$ is representable and $X$ does not admit a universal 0-cycle.

Our examples have some torsion in their integral Betti cohomology, so one might ask whether such examples exist without torsion in their integral Betti cohomology. From the viewpoint of the integral Hodge conjecture, they also provide an example of a curve class (that is, a Hodge homology class of degree 2), on a fourfold with torsion canonical bundle, that is not algebraic modulo torsion. Benoist and Ottem construct in [BO20] examples of non-algebraic torsion curve classes on a threefold with torsion canonical bundle. Our construction in fact relies on the results of [BO20]. In Section 3.1 we will provide a mild generalization and a short proof of one of their statements (see Proposition 3.5).

The assumptions of Theorem 1.4 are very restrictive geometrically. Indeed, we have the following.

**LEMMA 1.5.** Let $X$ be a smooth projective complex variety with representable $\text{CH}_0$-group and nontrivial Albanese variety. Then there exist a smooth projective curve $C$ with $J(C) \cong \text{Alb}(X)$ and a dominant morphism $\phi_C: X \to C$ such that the Albanese morphism of $X$ is the composition of $\phi_C$ and the Albanese morphism of $C$.

*Proof.* When $\text{CH}_0(X)$ is representable, Mumford's theorem [Mum68] (see also [BS83] for an alternative proof) tells that $H^0(X, \Omega_X^i) = 0$ for $i \ge 2$. It follows that the Albanese morphism $\text{alb}_X: X \to \text{Alb}(X)$ of (1.1) has rank at most 1 everywhere, hence factors (using the Stein factorization) as the composition of a morphism $\phi_C: X \to C$ and a morphism $\psi: C \to \text{Alb}(X)$. Using the universal property of the Albanese morphism for $X$ and $C$, we conclude that $\psi$ induces an isomorphism $\text{Alb}(C) \cong \text{Alb}(X)$. $\square$

**Remark 1.6.** As the proof above shows, the conclusion of Lemma 1.5 holds under the assumption that $h^{i,0}(X) = 0$ for $i \ge 2$.

The fact that $\text{Alb}(X)$ is isomorphic to the Jacobian of a curve makes it a priori easier for $X$ to have a universal 0-cycle. Indeed, for a Jacobian $J(C)$, or for any abelian variety which is a direct summand in a Jacobian, we can use the universal Poincaré divisor on $J(C) \times C$. This reduces the problem of constructing a codimension $n$ cycle in $\text{Alb}(X) \times X$ to the problem of constructing a codimension $n$ cycle in $C \times X$.
We will discuss in Section 2 various properties of a smooth projective variety $X$ related to the existence of a universal 0-cycle (see Proposition 2.2). We will relate in particular the existence of a universal 0-cycle for $X$ with the structure of the Künneth projector $\delta_1$ on $X$. Proposition 2.2 says that the property of having a universal 0-cycle has a good formulation in terms of the motive of $X$ when one knows furthermore that the Albanese variety of $X$ is a direct summand in the Jacobian of a curve, which is the case when $X$ has representable $CH_0$, by Lemma 1.5. Note that, thanks to [BdG23] and the recent work [EdGS25], it is now known that an abelian variety is not always a direct summand in the Jacobian of a curve (equivalently, by [BdG23] and [Voi24], the integral Hodge conjecture can be wrong for curve classes on abelian varieties). Motivated by Lemma 1.5, we will consider more generally in Section 2.2 smooth projective varieties $X$ whose Albanese map factors as the composition of a dominant morphism

$$ \phi_X : X \to C $$

and the Abel map of $C$, as it is the case when $CH_0(X)$ is representable, by Lemma 1.5. Recall that the index of a dominant morphism $X \to B$, where $B$ is irreducible, is the greatest common divisor of the degrees $\deg(Z/B)$ for all $Z \subset X$ closed irreducible dominant over $B$ and of dimension equal to $\dim B$. In the situation above, we will discuss the relation between the index of $\phi_X$ being 1 and the existence of a universal 0-cycle for $X$ (see Proposition 2.7). In one direction, we will prove that, if the Albanese map of $X$ factors through a morphism $\phi_X : X \to C$ as in Lemma 1.5 and $\phi_X$ has index 1, the variety $X$ has a universal 0-cycle. This statement has no converse, as we will also exhibit an example where the index of $\phi_X$ is not equal to 1, but $X$ has a universal 0-cycle.

We will prove Theorem 1.4 in Section 3. In Section 4 we will discuss the case of surfaces with representable $CH_0$-group, where more can be said thanks to the structure of surfaces with $q \neq 0$ and $p_g = 0$. Such surfaces $X$ are quotients of products of curves by a finite group, and we prove that the index of $\phi_X$ is 1 in the case of a cyclic group.

## 2. A discussion of universal 0-cycles

Let $X$ be a smooth projective variety of dimension $n$ over the complex numbers. The Albanese variety of $X$ is an abelian variety, which as a complex torus is constructed as the quotient

$$ \mathrm{Alb}(X) = H^n(X, \Omega_X^{n-1}) / H^{2n-1}(X, \mathbb{Z})_{\mathrm{tf}} = H^0(X, \Omega_X)^* / H_1(X, \mathbb{Z})_{\mathrm{tf}}, $$

where in both expressions, “tf” means “modulo torsion”. It follows that the Albanese map induces by construction an isomorphism of lattices

$$ H_1(X, \mathbb{Z})_{\mathrm{tf}} \cong H_1(\mathrm{Alb}(X), \mathbb{Z}). \qquad (2.1) $$

Here all the cohomology groups are integral Betti cohomology groups of the corresponding complex manifolds. The isomorphism (2.1) has an inverse in $\mathrm{Hom}(H_1(\mathrm{Alb}(X), \mathbb{Z}), H_1(X, \mathbb{Z})_{\mathrm{tf}})$ which provides an element $\gamma_X \in H^1(\mathrm{Alb}(X), \mathbb{Z}) \otimes H^{2n-1}(X, \mathbb{Z})_{\mathrm{tf}}$. As the isomorphism (2.1) is an isomorphism of Hodge structures, by Künneth decomposition, the class $\gamma_X$ provides a Hodge class of degree $2n$ on $\mathrm{Alb}(X) \times X$, also denoted by $\gamma_X$. The existence of a universal 0-cycle on $X$ is equivalent to the fact that $\gamma_X$ is algebraic, or rather that there exists an algebraic cycle $\Gamma$ of codimension $n$ in $\mathrm{Alb}(X) \times X$, such that the Kümneth component of type $(1, 2n-1)$ of $[\Gamma]$ equals $\gamma_X$. In particular, if a smooth projective variety $X$ has no universal 0-cycle, then the class $\gamma_X$ on $\mathrm{Alb}(X) \times X$ provides a counterexample to the integral Hodge conjecture. More precisely, it provides a counterexample to the integral Hodge conjecture modulo torsion, since
the class $\gamma_X$ is well defined in $H^{2n}(\mathrm{Alb}(X) \times X, \mathbb{Z})_{\mathrm{tf}}$ and is not algebraic there when $X$ has no universal 0-cycle.

If $X$, $Y$ are smooth projective complex varieties, with $\dim X = n$, and $Z \in \mathrm{CH}^n(Y \times X)$, the morphism of Hodge structures $[Z]_* : H_1(Y, \mathbb{Z})_{\mathrm{tf}} \to H_1(X, \mathbb{Z})_{\mathrm{tf}}$ induces a morphism of complex tori (which are in fact abelian varieties) that we will also denote by

$$[Z]_* : \mathrm{Alb}(Y) \to \mathrm{Alb}(X).$$

We will use the following construction.

**LEMMA 2.1.** Let $X$ be a smooth projective variety and $C$ a smooth curve with a morphism $j: C \to X$ inducing $j_*: J(C) \to \mathrm{Alb}(X)$. Assume that there exists a morphism of abelian varieties $s: \mathrm{Alb}(X) \to J(C)$ such that

$$s \circ j_* = k \mathrm{Id}_{\mathrm{Alb}(X)} \qquad (2.2)$$

for some integer $k$. Then there exists a codimension $n$ cycle $\Gamma \in \mathrm{CH}^n(\mathrm{Alb}(X) \times X)$ such that

$$[\Gamma]_* = k \mathrm{Id}_{\mathrm{Alb}(X)} : \mathrm{Alb}(\mathrm{Alb}(X)) = \mathrm{Alb}(X) \to \mathrm{Alb}(X).$$

*Proof.* The curve $C$ admits a universal 0-cycle $\Gamma_C \in \mathrm{CH}^1(J(C) \times C)$ (see Example 1.2). Then by (2.2) the codimension $n$ cycle

$$\Gamma_X := (\bar{s}, \mathrm{Id}_X)^* ((\mathrm{Id}_{J(C)}, j)_*(\Gamma_C)) \in \mathrm{CH}^n(\mathrm{Alb}(X) \times X)$$

has the property that

$$[\Gamma_X]_* = k \mathrm{Id}_{\mathrm{Alb}(X)} : \mathrm{Alb}(\mathrm{Alb}(X)) = \mathrm{Alb}(X) \to \mathrm{Alb}(X). \qquad \square$$

As a first application, let us prove that any smooth projective variety $X$ has a universal 0-cycle with rational coefficients. Indeed, choose a smooth curve $C$ which is a complete intersection of ample hypersurfaces in $X$, and denote by $j: C \to X$ the inclusion map; then

$$j_* : H_1(C, \mathbb{Z}) \to H_1(X, \mathbb{Z})_{\mathrm{tf}}$$

is a surjective morphism of Hodge structures. By the semisimplicity of polarized weight 1 Hodge structures, there exists a morphism of Hodge structures

$$s: H_1(\mathrm{Alb}(X), \mathbb{Z}) \cong H_1(X, \mathbb{Z})_{\mathrm{tf}} \to H_1(C, \mathbb{Z})$$

such that

$$j_* \circ s = k \mathrm{Id}_{H_1(X, \mathbb{Z})_{\mathrm{tf}}} \qquad (2.3)$$

for some nonzero integer $k$. The morphism $s$ induces a morphism $\bar{s}: \mathrm{Alb}(X) \to J(C)$ of abelian varieties, and the morphisms $j_*$ and $\bar{s}$ satisfy the assumptions of Lemma 2.1. Hence Lemma 2.1 provides a universal 0-cycle with rational coefficients.

Let us now discuss the link between universal 0-cycles and the Künneth decomposition of the Albanese motive. Assume that the smooth projective complex variety $X$ of dimension $n$ has the property that $H^*(X, \mathbb{Z})$ has no torsion. The Künneth components $\delta_i \in H^i(X, \mathbb{Z}) \otimes H^{2n-i}(X, \mathbb{Z})$ of the diagonal of $X$ are then well defined, and it is known that $\delta_1$ is algebraic (see [Kle72]), at least with $\mathbb{Q}$-coefficients.

**PROPOSITION 2.2.** The following statements are equivalent:

(i) The Künneth projector $\delta_1$ is algebraic on $X \times X$ and is the class of a cycle supported on $X \times C$ for some curve $C \subset X$.
(ii) The variety $X$ has a universal 0-cycle, and the Albanese variety of $X$ is a direct summand in the Jacobian of a curve.

(iii) The motive of $X$ contains a direct summand $M_1$ with the property that $M_1$ is a direct summand in the motive of a curve and $\mathrm{Alb}(X) = \mathrm{Alb}(M_1)$.

In the statement above, we allow nonconnected curves, so “Jacobian of a curve” stands for “product of Jacobians of curves”.

*Remark 2.3.* If $X$ has representable $\mathrm{CH}_0$-group, then it is automatic that the Albanese variety of $X$ is a direct summand in the Jacobian of a curve by Lemma 1.5, so in this case we see that $X$ has a universal 0-cycle if and only if the motive of $X$ is the direct sum of a motive with $\mathrm{CH}_0 = 0$ and a motive which is a direct summand in the motive of a curve.

*Proof of Proposition 2.2.* Assume that there exist a curve $C \subset X$ and a codimension 1 cycle $Z \subset X \times C$ such that the class of $Z$ in $X$ is $\delta_1$. We can replace by desingularization $C$ with a smooth curve $\tilde{C}$ with a morphism $\tilde{j}: \tilde{C} \to X$, such that $Z$ lifts to a codimension 1 cycle $\tilde{Z} \subset X \times C$. Then, as the class of $Z$ in $X$ is $\delta_1$, for any $\alpha \in H_1(X, \mathbb{Z})$, we have

$$ \alpha = \delta_{1*}(\alpha) = \tilde{j}_*([\tilde{Z}]_*(\alpha)) \quad \text{in } H_1(X, \mathbb{Z}). \qquad (2.4) $$

It follows from (2.4) that, via $\tilde{j}_*: J(\tilde{C}) \to \mathrm{Alb}(X)$, the Albanese variety $\mathrm{Alb}(X)$ is a direct summand of $J(\tilde{C})$, with right inverse given by $\sigma := [\tilde{Z}]_* : \mathrm{Alb}(X) \to J(\tilde{C})$, which proves the second statement in item (ii). Next, we apply Lemma 2.1 to $\tilde{j}$ and $\sigma$ and consider the cycle

$$ \Gamma_1 := (\sigma, \mathrm{Id}_C)^* \circ (\mathrm{Id}_{J(C)}, \tilde{j})_* (P_{\tilde{C}}) \in \mathrm{CH}^n(\mathrm{Alb}(X) \times X). $$

It has the property that $[\Gamma_1]_* : \mathrm{Alb}(\mathrm{Alb}(X)) = \mathrm{Alb}(X) \to \mathrm{Alb}(X)$ is the identity, since $\tilde{j}_* \circ \sigma = \mathrm{Id}_{\mathrm{Alb}(X)}$. Hence $X$ has a universal 0-cycle. Thus item (i) implies item (ii).

In the other direction, let $\Gamma \in \mathrm{CH}^n(\mathrm{Alb}(X) \times X)$ be a universal 0-cycle for $X$. This means that $[\Gamma]_* : H_1(\mathrm{Alb}(X), \mathbb{Z}) \to H_1(X, \mathbb{Z})$ is the natural isomorphism. Let $D$ be a (not necessarily connected) smooth projective curve such that $\mathrm{Alb}(X)$ is a direct summand of $J(D)$, and let

$$ \sigma: \mathrm{Alb}(X) \to J(D), \quad \pi: J(D) \to \mathrm{Alb}(X) $$

be such that $\pi \circ \sigma = \mathrm{Id}_{\mathrm{Alb}(X)}$. Finally, let $i_D: D \to J(D)$ be the natural embedding, and consider the cycle $\Gamma_1 := \Gamma \circ \pi \circ i_D \in \mathrm{CH}^n(D \times X)$. We have

$$ [\Gamma_1]_* = [\Gamma]_* \circ \pi_* \circ i_{D*} = \pi: J(D) \to \mathrm{Alb}(X). $$

Next, let $P_D \in \mathrm{CH}^1(J(D) \times D)$ be a universal 0-cycle (or Poincaré divisor) for $D$, and consider the cycle $\Gamma_2 = P_D \circ \sigma \circ \mathrm{alb}_X \in \mathrm{CH}_n(X \times D)$. We have $[\Gamma_2]_* = \sigma: \mathrm{Alb}(X) \to J(D)$. It follows that $\Gamma_X := \Gamma_1 \circ \Gamma_2 \in \mathrm{CH}^n(X \times X)$ has the property that

$$ [\Gamma_X]_* = [\Gamma_1]_* \circ [\Gamma_2]_* = \pi \circ \sigma = \mathrm{Id}_{\mathrm{Alb}(X)}. \qquad (2.5) $$

Finally, $\Gamma_X$ is supported on $X \times D'$, where $D' \subset X$ is the curve in $X$ defined as $\mathrm{pr}_2(\mathrm{Supp}(\Gamma_1))$. Furthermore, one easily checks that $[\Gamma_X]^* = 0$ on $H^i(X, \mathbb{Z})$ for $i \neq 1$. It follows that $[\Gamma_X] = \delta_1$. This proves that item (ii) implies item (i).

The equivalence of properties (ii) and (iii) is now clear. If we have a direct summand $M_1$ as in item (iii), then $M_1$ has a universal 0-cycle, being a direct summand in the motive of a curve $C$, and its Albanese variety is a direct summand of $J(C)$. Hence property (ii) holds. Conversely, if property (ii) holds, then we construct $\Gamma_1$ and $\Gamma_2$ as above, and (2.5) says exactly that $[\Gamma_2]$ realizes motivically $H_1(X, \mathbb{Z})$ as a direct summand in $H_1(D, \mathbb{Z})$. $\square$
## 2.1 Previous examples of smooth projective varieties with no universal 0-cycle

The first example of a smooth projective surface not admitting a universal 0-cycle is constructed in [Voi25]. It builds on the fact that there exist rationally connected 3-folds $Y$ with no universal codimension 2 cycle parameterized by the intermediate Jacobian $J(Y)$ (such examples are constructed in [Voi15]). Here a universal codimension 2 cycle $\mathcal{Z}_{\text{univ}} \in \text{CH}^2(J(Y) \times Y)$ is characterized by the condition that the composite map

$$J(Y) \to \text{CH}^2(Y)_{\text{alg}} \to J(Y), \quad t \mapsto \psi_Y(\mathcal{Z}_{\text{univ},t} - \mathcal{Z}_{\text{univ},0(Y)})$$

is the identity, where $\psi_Y$ is the Abel-Jacobi morphism (an isomorphism in this case) of $Y$. The second ingredient is the following result proved in [Voi25].

**THEOREM 2.4.** Given a smooth projective rationally connected threefold $Y$ over $\mathbb{C}$, there exist a smooth projective surface $S$ and a codimension 2 cycle $\mathcal{Z} \in \text{CH}^2(S \times Y)$ such that the induced Abel-Jacobi map $[\mathcal{Z}]_* : \text{Alb}(S) \to J(Y)$ is an isomorphism.

The varieties $Y$ and $S$ being as above, if $Y$ does not have a universal codimension 2 cycle, $S$ does not have a universal 0-cycle, since a universal 0-cycle

$$\mathcal{Z}_S \in \text{CH}^2(\text{Alb}(S) \times S) = \text{CH}^2(J(Y) \times S)$$

would produce a universal codimension 2 cycle $\mathcal{Z}_{\text{univ}} = \mathcal{Z} \circ \mathcal{Z}_S \in \text{CH}^2(J(Y) \times Y)$ for $Y$.

A particular case of this construction also solves in the negative the following question asked by Colliot-Thélène in the original version of [Col25].

**Question 2.5.** Given any smooth projective variety $X$ over the complex numbers, and any function field $K = \mathbb{C}(C)$, where $C$ is a curve, is the natural morphism $\text{CH}_0(X_K)_0 \to \text{Alb}(X)(K)$ surjective?

Indeed, we first observe the following.

**PROPOSITION 2.6.** Assume that the smooth projective variety $X$ has the property that $\text{Alb}(X)$ is a direct summand, as an abelian variety, in the Jacobian $J(C)$ of a smooth projective curve $C$. Then the following properties are equivalent:

(i) The variety $X$ has a universal 0-cycle.

(ii) For any curve $D$, the natural morphism $a_X : \text{CH}_0(X_K)_0 \to \text{Alb}(X)(K)$ is surjective, where $K = \mathbb{C}(D)$.

We give the proof for completeness, although similar statements and arguments can be found in [Voi24, Col25].

**Proof.** Property (i) implies property (ii) in an obvious way, since given a morphism $\phi: D \to \text{Alb}(X)$ and a universal 0-cycle $\Gamma \in \text{CH}^n(\text{Alb}(X) \times X)$ of relative degree 0 over $\text{Alb}(X)$, where $n := \dim X$, we get by restriction to $D \times X$ a cycle $\Gamma_D \in \text{CH}^n(D \times X)$, of relative degree 0 over $D$. By the definition of a universal 0-cycle, $\Gamma_D$ has the property that the composed morphism $\text{alb}_X \circ \Gamma_{D_*}: J(D) \to \text{Alb}(X)$ equals $\phi_*: J(D) \to \text{Alb}(X)$. It follows that, for a chosen inclusion $j_D: D \to J(D)$, the two morphisms $\text{alb}_X \circ \Gamma_{D_*} \circ j_D: D \to \text{Alb}(X)$ and $\phi$ differ by a translation. Correcting $\Gamma_D$ by the adequate cycle $D \times z_0$, where $z_0 \in \text{CH}_0(X)_{\text{hom}}$, gives a cycle $\Gamma'_D \in \text{CH}^n(D \times X)$ such that

$$\text{alb}_X \circ \Gamma_{D_*} \circ j_D = \phi: D \to \text{Alb}(X). \qquad (2.6)$$
Equation (2.6) says exactly that the image of $\Gamma'_D$ in $CH_0(X_K)_0$ maps to $\phi_K \in \mathrm{Alb}(X)(K)$.

In the other direction, assume that $\mathrm{Alb}(X)$ is a direct summand in $J(C)$, that is, for some abelian variety $B$, we have

$$ \mathrm{Alb}(X) \times B \cong J(C), $$

with inclusion $i: \mathrm{Alb}(X) \to J(C)$ and projection $p: J(C) \to \mathrm{Alb}(X)$, such that $p \circ i = \mathrm{Id}_{\mathrm{Alb}(X)}$. We observe that the curve $C$ has a universal 0-cycle $\Gamma_C$ (see Example 1.2). Let $i_C$ be the natural inclusion of $C$ in $J(C)$ (it is in fact defined up to translation). The composite morphism $p_C = p \circ i_C: C \to \mathrm{Alb}(X)$ gives a $K$-point in $\mathrm{Alb}(X)(K)$, where $K = \mathbb{C}(C)$. Assuming that property (ii) holds, there is a codimension $n$ cycle $Z_C \in \mathrm{CH}^n(C \times X)$, of relative degree 0 over $C$, such that

$$ \mathrm{alb}_X \circ Z_{C_*} = p_C: C \to A. \qquad (2.7) $$

We next consider the composition $Z_{C,\mathrm{Alb}(X)} := Z_C \circ \Gamma_C \circ i \in \mathrm{CH}^n(\mathrm{Alb}(X) \times X)$; it follows from (2.7) that $\mathrm{alb}_X \circ Z_{C,\mathrm{Alb}(X)_*} = \mathrm{Id}_{\mathrm{Alb}(X)}: \mathrm{Alb}(X) \to \mathrm{Alb}(X)$. $\square$

We now get a negative answer to Question 2.5 by considering a smooth projective rationally connected 3-fold $Y$ which does not admit a universal codimension 2 cycle while the intermediate Jacobian $J(Y)$ has dimension 3, so that it is the Jacobian of a curve. Such examples are constructed in [Voi15]: one can take for $Y$ the desingularization of a very general quartic double solid with seven nodes. We then introduce as before a surface $S$ as in Theorem 2.4, which has $\mathrm{Alb}(S) \cong J(Y)$, hence has its Albanese variety isomorphic to the Jacobian of a curve. Furthermore, $S$ has no universal 0-cycle because $Y$ does not have a universal codimension 2 cycle; hence $S$ provides the desired example using Proposition 2.6.

## 2.2 Varieties fibered over a curve

We now consider smooth projective varieties $X$ such that the Albanese map of $X$ factors through a curve as in Lemma 1.5. We first prove the following.

**PROPOSITION 2.7.** If the Albanese map of $X$ factors through a morphism $\phi_X: X \to C$, where $C$ is a smooth projective curve such that $\phi_{X_*}: \mathrm{Alb}(X) \to J(C)$ is an isomorphism, and $\phi_X$ has index 1, then $X$ has a universal 0-cycle.

*Proof.* By definition, if $X$ and $B$ are smooth projective and $\phi: X \to B$ is a dominant morphism of relative dimension $d$ and index 1, there exists a cycle $\Gamma \in \mathrm{CH}^d(X)$ such that $\phi_*\Gamma = B$ in $\mathrm{CH}^0(B)$. The cycle $\Gamma$ has cohomology class $[\Gamma]$ acting by cup-product on the integral cohomology of $X$, and by the projection formula we have

$$ \phi_*([\Gamma] \cup \phi^*\alpha) = \alpha \qquad (2.8) $$

for any $\alpha \in H^*(B, \mathbb{Z})$.

We now consider the case where $B$ is a curve $C$ and the morphism $\phi = \phi_X$ induces an isomorphism $\mathrm{Alb}(X) \cong J(C)$. If the index of $\phi_X$ is 1, the construction above gives a cycle

$$ \Gamma' = (\phi_X, \mathrm{Id}_X)_* \Gamma \in \mathrm{CH}_1(C \times X) $$

satisfying, by (2.8),

$$ \phi_{X_*} \circ [\Gamma']_* = \mathrm{Id}: H^1(C, \mathbb{Z}) \to H^1(C, \mathbb{Z}). \qquad (2.9) $$

As before we use a Poincaré divisor $Z_C \in \mathrm{CH}^1(J(C) \times C)$ such that $Z_{C_*}: H_1(J(C), \mathbb{Z}) \to H_1(C, \mathbb{Z})$ is the natural isomorphism, inverse of $\mathrm{alb}_{C_*}: H_1(C, \mathbb{Z}) \to H_1(J(C), \mathbb{Z})$. It follows from (2.9) that
the cycle

$$ \Gamma'' := Z_C \circ \Gamma' \in \mathrm{CH}^n(J(C) \times X), \quad n = \dim X $$

has the property that

$$ [\Gamma'']_*: H_1(J(C), \mathbb{Z}) \to H_1(X, \mathbb{Z}) $$

satisfies $\phi_{X_*} \circ [\Gamma'']_* = Z_{C_*}$. By assumption, $\phi_{X_*}$ induces an isomorphism $H_1(X, \mathbb{Z})_{\mathrm{tf}} \cong H_1(C, \mathbb{Z})$; we conclude that $[\Gamma'']_*: H_1(J(C), \mathbb{Z}) = H_1(\mathrm{Alb}(X), \mathbb{Z}) \to H_1(X, \mathbb{Z})_{\mathrm{tf}}$ is the inverse of the isomorphism $\mathrm{alb}_{X_*}: H_1(X, \mathbb{Z})_{\mathrm{tf}} \cong H_1(\mathrm{Alb}(X), \mathbb{Z})$. Hence $\Gamma''$ is a universal 0-cycle for $X$. $\square$

A particular case of Proposition 2.7 is the case of rationally connected fibrations over a curve, which is considered in [Col25, Voi24]. They admit universal 0-cycles by [GHS03] and Proposition 2.7.

We now show that the implication of Proposition 2.7 is strict by constructing an example of a smooth projective surface $S$ with a morphism $\phi: S \to C$ to a curve, inducing an isomorphism $\phi_*: \mathrm{Alb}(S) \cong J(C)$, such that the index of $\phi$ is $d > 1$ and $S$ admits a universal 0-cycle. We start with any curve $C$ of genus $> 0$ and choose a smooth curve

$$ \Gamma \subset C \times C \qquad (2.10) $$

such that the second projection $p: \Gamma \to C$ has degree $d$ and $\Gamma^*: H^1(C, \mathbb{Z}) \to H^1(C, \mathbb{Z})$ is the identity. This exists for $d$ large enough, by taking (for example) $\Gamma$ to be a smooth member of the linear system $|\mathcal{O}_{C \times C}((d-1)(c \times C) + d'(C \times c') + \Delta_C)|$ for some points $c, c' \in C$ and some large enough integer $d'$, where $\Delta_C$ is the diagonal of $C$. We now choose an embedding $i$ of $C$ in $\mathbb{P}^3$, which gives an embedding $(\mathrm{Id}_C, i)$ of $C \times C$, hence of $\Gamma$, in $C \times \mathbb{P}^3$. For some very ample line bundle $L$ on $C$, we choose the surface $S \subset C \times \mathbb{P}^3$ to be a very general complete intersection of members of the linear system $|L \boxtimes \mathcal{O}_{\mathbb{P}^3}(d)|$ containing $(\mathrm{Id}_C, i)(\Gamma)$. By the Lefschetz theorem on hyperplane sections, the first projection $\phi: S \to C$ induces an isomorphism $\mathrm{Alb}(S) \cong J(C)$. By construction we have $C \times \Gamma \subset C \times S$, and $C \times \Gamma$ contains the transpose ${}^t\Gamma_{p'} \cong \Gamma$ of the graph of the morphism $p': \Gamma \to C$ given by the first projection in (2.10). Hence ${}^t\Gamma_{p'}$ is contained in $C \times S$, and it follows from the equalities

$$ \phi|_{\Gamma} = p, \quad \mathrm{pr}_{1}{}^{t}\Gamma_{p'} = p', $$

that its image in $C \times C$ under $(\mathrm{Id}_C, \phi)$ is the original curve $\Gamma \subset C \times C$. Hence it acts as the identity on $H^1(C, \mathbb{Z})$, so $S$ has a universal 0-cycle. Finally, we argue as in [Lop92], for example, to conclude that for very general $S$, the Néron–Severi group of $S$ is generated by a fiber of $\phi$, the line bundle $\mathcal{O}_{\mathbb{P}^3}(1)$ and the class of the curve $(\mathrm{Id}_C, i)(\Gamma)$. This immediately implies that the index of $\phi$ is $d$, since $\Gamma$ has degree $d$ over $C$.

### 3. A 3-dimensional example for Question 1.3

In this section we give an example of a smooth projective 3-fold $X$ with representable $\mathrm{CH}_0$ and no universal 0-cycle, thus proving Theorem 1.4. Let $S$ be a projective $K3$ surface over the complex numbers equipped with a fixed-point-free antisymplectic involution $g$. We denote by $\Sigma := S/g$ the corresponding Enriques surface. Let $E$ be an elliptic curve with a translation $t_\xi$ by a point $\xi$ of order 2. We set $E_\xi := E/t_\xi$.

The variety we will consider is

$$ X := (E \times S)/(t_\xi, g). \qquad (3.1) $$
It admits two natural morphisms

$$p_{E_{\xi}}: X \to E_{\xi}, \quad p_{\Sigma}: X \to \Sigma. \qquad (3.2)$$

**LEMMA 3.1.** (i) The morphism $p_{E_{\xi}}$ induces an isomorphism

$$p_{E_{\xi}*}: \mathrm{CH}_0(X) \to \mathrm{CH}_0(E_{\xi}).$$

In particular, $\mathrm{CH}_0(X)$ is representable.

(ii) One has $\mathrm{Alb}(X) \cong E_{\xi}$, and the morphism $p_{E_{\xi}}$ identifies (up to translation) with the Albanese map of X.

*Proof.* The morphism $p_{E_{\xi}}: X \to E_{\xi} = E/t_{\xi}$ has all its fibers isomorphic to the K3 surface S, which is connected and simply connected. It follows that $p_{E*}: H_1(X, \mathbb{Z}) \to H_1(E_{\xi}, \mathbb{Z})$ is an isomorphism, which proves item (ii). As item (ii) is proved, in order to prove item (i), it suffices to prove that the kernel $\mathrm{CH}_0(X)_{\mathrm{alb}}$ of the Albanese morphism $\mathrm{CH}_0(X)_{\mathrm{hom}} \to \mathrm{Alb}(X)$ is trivial. As this kernel has no torsion by [Roi80b], it injects by pull-back in the invariant part under $(t_{\xi}, g)$ of the group $\mathrm{CH}_0(E \times S)_{\mathrm{alb}}$. It thus suffices to show that the involution $(t_{\xi}, g)$ acts by $-\mathrm{Id}$ on the group $\mathrm{CH}_0(E \times S)_{\mathrm{alb}} \otimes \mathbb{Q}$. The involution $t_{\xi}$ acts as the identity on $\mathrm{CH}_0(E) \otimes \mathbb{Q}$. The involution $g$ acts by $-\mathrm{Id}$ on the group $\mathrm{CH}_0(S)_{\mathrm{hom}}$ by [BKL76]. With rational coefficients, choosing a 0-cycle $o_S$ of degree 1 which is $g$-invariant, we have a decomposition

$$\mathrm{CH}_0(E \times S) \otimes \mathbb{Q} = \mathrm{CH}_0(E)_{\mathbb{Q}} \otimes o_S \oplus \mathrm{Im}(\mathrm{CH}_0(E)_{\mathbb{Q}} \otimes \mathrm{CH}_0(S)_{\mathrm{hom}}) \to \mathrm{CH}_0(E \times S)_{\mathbb{Q}}, \quad (3.3)$$

and the group $\mathrm{CH}_0(E \times S)_{\mathrm{alb}} \otimes \mathbb{Q}$ is contained in the second summand of the decomposition (3.3), on which $(t_{\xi}, g)$ acts by $-\mathrm{Id}$, which concludes the proof. $\square$

We now prove the main result.

**THEOREM 3.2.** For a fixed K3 surface S with involution g over an Enriques surface $\Sigma$, there exists an elliptic curve E with a translation $t_{\xi}$ of order 2, such that the variety X constructed above does not admit a universal 0-cycle.

*Proof.* Given S, E and $\xi$, let us assume that X has a universal 0-cycle. By Lemma 3.1, a universal 0-cycle for X is given by a 1-cycle $\Gamma \in \mathrm{CH}_1(E_{\xi} \times X)$ which has the property that $[\Gamma]_*: H_1(E_{\xi}, \mathbb{Z}) \to H_1(X, \mathbb{Z})_{\mathrm{tf}}$ is inverse to the isomorphism $p_{E_{\xi}*}: H_1(X, \mathbb{Z})_{\mathrm{tf}} \to H_1(E_{\xi}, \mathbb{Z})$, that is,

$$p_{E_{\xi}*} \circ [\Gamma]_* = \mathrm{Id}: H_1(E_{\xi}, \mathbb{Z}) \to H_1(E_{\xi}, \mathbb{Z}). \qquad (3.4)$$

Dualizing (3.4) and passing to $\mathbb{Z}/2$-coefficients, we get that

$$[\Gamma]^* \circ p_{E_{\xi}}^* = \mathrm{Id}: H^1(E_{\xi}, \mathbb{Z}/2) \to H^1(E_{\xi}, \mathbb{Z}/2). \qquad (3.5)$$

The étale double covers $S \to \Sigma$, $E \to E_{\xi}$, $q = (p_{E_{\xi}}, p_{\Sigma})$: $X \to E_{\xi} \times \Sigma$ are given by elements

$$\sigma_{\Sigma} \in H^1(\Sigma, \mathbb{Z}/2), \quad \sigma_{E_{\xi}} \in H^1(E_{\xi}, \mathbb{Z}/2), \quad \sigma_X \in H^1(E_{\xi} \times \Sigma, \mathbb{Z}/2)$$

that are characterized by the fact that they are nonzero and they are annihilated, respectively, by the pull-back maps

$$H^1(\Sigma, \mathbb{Z}/2) \to H^1(S, \mathbb{Z}/2), \quad H^1(E_{\xi}, \mathbb{Z}/2) \to H^1(E, \mathbb{Z}/2), \\ q^*: H^1(E_{\xi} \times \Sigma, \mathbb{Z}/2) \to H^1(X, \mathbb{Z}/2).$$

Recalling the notation of (3.2), we claim that

$$\sigma_X = \mathrm{pr}_{E_{\xi}}^*(\sigma_{E_{\xi}}) + \mathrm{pr}_{\Sigma}^*(\sigma_{\Sigma}) \quad \text{in } H^1(E_{\xi} \times \Sigma, \mathbb{Z}/2), \qquad (3.6)$$
where $\mathrm{pr}_{E_\xi}$ and $\mathrm{pr}_\Sigma$ are the projections from $E_\xi \times \Sigma$ to its factors, so that

$$p_{E_{\xi}} = \mathrm{pr}_{E_{\xi}} \circ q, \quad p_{\Sigma} = \mathrm{pr}_{\Sigma} \circ q. \qquad (3.7)$$

In order to prove (3.6), we write

$$\sigma_X = \mathrm{pr}_{E_\xi}^*(\alpha) + \mathrm{pr}_{\Sigma}^*(\beta) \quad \text{in } H^1(E_\xi \times \Sigma, \mathbb{Z}/2)$$

for some classes $\alpha \in H^1(E_\xi, \mathbb{Z}/2)$ and $\beta \in H^1(\Sigma, \mathbb{Z}/2)$. It follows that, for any $e \in E_\xi$ and $s \in \Sigma$, we have $\sigma_{X|\{e\} \times \Sigma} = \beta$ and $\sigma_{X|E_\xi \times \{s\}} = \alpha$. Finally, we observe that the fiber of $p_{E_\xi}$ over $e$ is the double cover of $\Sigma$ isomorphic to $S$ and the fiber of $p_\Sigma$ over $s$ is the double cover of $E_\xi$ isomorphic to $E$, so that $\beta = \sigma_\Sigma$ and $\alpha = \sigma_{E_\xi}$, which proves the claim.

The meaning of (3.6) is that

$$q^*(\mathrm{pr}_{E_\xi}^*(\sigma_{E_\xi}) + \mathrm{pr}_{\Sigma}^*(\sigma_\Sigma)) = 0 \quad \text{in } H^1(X, \mathbb{Z}/2), \qquad (3.8)$$

which thanks to (3.7) can be rewritten as

$$p_{E_{\xi}}^{*}(\sigma_{E_{\xi}}) + p_{\Sigma}^{*}(\sigma_{\Sigma}) = 0 \quad \text{in } H^{1}(X, \mathbb{Z}/2). \qquad (3.9)$$

Let

$$\Gamma_E := (\mathrm{Id}_{E_\xi}, p_{E_\xi})_*(\Gamma) \in \mathrm{CH}_1(E_\xi \times E_\xi), \quad \Gamma_\Sigma := (\mathrm{Id}_{E_\xi}, p_\Sigma)_*(\Gamma) \in \mathrm{CH}_1(E_\xi \times \Sigma).$$

It follows from (3.9) that

$$\Gamma^*(p_{E_\xi}^*(\sigma_{E_\xi})) + \Gamma^*(p_\Sigma^*(\sigma_\Sigma)) = 0 \quad \text{in } H^1(E_\xi, \mathbb{Z}/2),$$

which can be written as

$$\Gamma_E^*(\sigma_{E_\xi}) + \Gamma_\Sigma^*(\sigma_\Sigma) = 0 \quad \text{in } H^1(E_\xi, \mathbb{Z}/2). \qquad (3.10)$$

By equation (3.5), the morphism $\Gamma_E^*$ acts as the identity on $H^1(E_\xi, \mathbb{Z}/2)$, hence from (3.10) we finally get

$$\sigma_{E_{\xi}} = \Gamma_{\Sigma}^{*}(\sigma_{\Sigma}) \quad \text{in } H^{1}(E_{\xi}, \mathbb{Z}/2). \qquad (3.11)$$

The results of [BO20, Propositions 1.1 and 2.1] now tell us that there exist $E_\xi$ and $\sigma_{E_\xi}$ such that (3.11) does not hold for any 1-cycle $\Gamma_\Sigma \in \mathrm{CH}_1(E_\xi \times \Sigma)$ (see also Proposition 3.5 for an alternative proof and generalization of this statement). The analysis above thus shows that for this choice of $S$ and of pair $(E_\xi, \sigma_{E_\xi})$ (determining a double cover $E$ with translation $t_\xi$), the variety $X$ does not have a universal 0-cycle. $\square$

**COROLLARY 3.3.** For $S$ fixed and $E$ very general, the index of $p_{E_\xi}: X \to E_\xi$ is 2.

*Proof.* The index of $p_{E_\xi}$ is either 1 or 2, since the image of a curve $E \times S$ in $X$ for some $s \in S$ has degree 2 over $E_\xi$. However, the index cannot be 1, as otherwise Proposition 2.7 would provide a contradiction with Theorem 3.2. $\square$

We finish with the following statement, which combined with Corollary 3.3 shows that the index of $p_{E_\xi}$ is not dictated by topological or Hodge-theoretic reasons.

**LEMMA 3.4.** There exists an integral Hodge class $\alpha \in H^4(X, \mathbb{Z})$ of degree 4 on $X$ such that $p_{E_\xi*\alpha} = 1_{E_\xi}$ in $H^0(E_\xi, \mathbb{Z})$.

*Proof.* We note that, because $H^2(X, \mathcal{O}_X) = 0$, any integral Betti cohomology class $\alpha \in H^4(X, \mathbb{Z})$ is Hodge. So it suffices to prove the existence of an integral Betti cohomology class $\alpha \in H^4(X, \mathbb{Z})$ such that $p_{E_\xi*\alpha} = 1_{E_\xi}$ in $H^0(E_\xi, \mathbb{Z})$. This statement is topological, so it suffices to prove the
result for a specific elliptic curve E with involution $t_ξ$. We choose a $K3$ surface with involution $g$ over an Enriques surface $S$ satisfying the property that $\mathrm{NS}(S)$ is $g$-invariant. We now observe that $S$ has elliptic pencils, which must be globally invariant under $g$. The action of $g$ on the base $\mathbb{P}^1$ of the pencil has two fixed points, and each one produces a possibly singular elliptic curve $E \subset S$, invariant under $g$. As $g$ acts without fixed point on $E$, the curve $E$ must be smooth, and $g$ acts as an order 2 translation $t_ξ$ on it. Indeed, the only possibility for a singular fixed fiber $E$ would be that $E$ is the union of two smooth rational curves meeting in two points, where $g$ exchanges the two components. This however would contradict the fact that $\mathrm{NS}(S)$ is $g$-invariant. We now consider the diagonal inclusion of $E$ in $E \times S$. It is invariant under $(t_ξ, g)$, and we get a curve $E/(t_ξ, g) \subset (E \times S)/(t_ξ, g) = X$, which is isomorphic to $E_ξ := E/t_ξ$ via $p_{E_ξ}$. The class $\alpha := [E/(t_ξ, g)] \in \mathbb{H}^4(X, \mathbb{Z})$ is the desired one. $\square$

## 3.1 On a result of Benoist and Ottem

Our proof of Theorem 1.4 relies on the results of Benoist and Ottem in [BO20]. In this section we give a short proof and a mild generalization of the statement in [BO20] that we have been using.

**PROPOSITION 3.5.** Let $W$ be a smooth projective complex variety of dimension $d$, and let $0 \neq \beta \in \mathbb{H}^1(W, \mathbb{Z}/k)$ for some prime integer $k > 1$. Then there exist only countably many elliptic curves $E$ such that there exists a correspondence $\Gamma \in \mathrm{CH}^d(E \times W)$ with

$$[\Gamma]^*\beta \neq 0 \text{ in } \mathbb{H}^1(E, \mathbb{Z}/k).$$

*Proof.* We argue by contradiction, and, using standard spreading arguments, we get that there exist a smooth projective curve $B$, a smooth nonisotrivial elliptically fibered surface $f: T \to B$ and a correspondence $\Gamma \in \mathrm{CH}^d(T \times W)$, such that the class $\beta' := [\Gamma]^*\beta \in \mathbb{H}^1(T, \mathbb{Z}/k)$ does not vanish on the smooth fibers $T_b$ of $f$. We can furthermore assume that $f$ has a section, by performing an extra base-change $B' \to B$ if necessary. The class $\beta'$ induces an étale cyclic cover $e: T' \to T$ of degree $k$, and the map $f' := f \circ e: T' \to B$ has connected elliptic general fiber $T'_b$, obtained as the étale degree $k$ cyclic cover of the fiber $T_b$ induced by the class $\beta'_{|T_b}$ (which is nontrivial by assumption). The two surfaces satisfy

$$\mathbb{H}^i(T, \mathcal{O}_T) \cong \mathbb{H}^i(T', \mathcal{O}_{T'}), \quad i = 0, 1, 2. \qquad (3.12)$$

This indeed follows from the fact that the order $k$ automorphism $i$ of $T'$ over $T$ acts by a finite-order translation of the smooth fibers $T'_b$. It thus acts as the identity on $\mathrm{CH}_0(T'_b)_{\mathbb{Q}}$ for a general $b$, hence also on $\mathrm{CH}_0(T')_{\mathbb{Q}}$, which implies by [Mum68] that it acts as the identity on $\mathbb{H}^i(T', \mathcal{O}_{T'})$. Formula (3.12) gives that

$$\chi(T', \mathcal{O}_{T'}) = \chi(T, \mathcal{O}_T),$$

while we also have

$$\chi(T', \mathcal{O}_{T'}) = k\chi(T, \mathcal{O}_T)$$

since $e: T' \to T$ is étale of degree $k$. We thus conclude that $\chi(T, \mathcal{O}_T) = 0$. However, the Kodaira canonical bundle formula [SS10, Uen73] shows that a surface $T$ admitting a nonisotrivial elliptic fibration with a section cannot have $\chi(T, \mathcal{O}_T) = 0$. $\square$

**Proposition 3.5** applies for example to quintic Godeaux surfaces, quotients of quintic surfaces $S$ in $\mathbb{P}^3$ by the action of a certain order 5 automorphism $g$ acting freely. It is proved in [Voi92] that such a surface $\Sigma$ has trivial $\mathrm{CH}_0$-group; hence the whole argument described previously would work as well, thanks to Proposition 3.5, for a general quotient $X = (E \times S)/(t_ξ, g)$,
where E is very general and $t_\xi$ is a translation by a point of order 5. Such an X has representable CH₀-group and has no universal 0-cycle.

## 4. The case of surfaces

A ruled surface, and more generally a rationally connected fibration over a curve, admits a universal 0-cycle, thanks to Lemma 2.7 (see also [Col25, Voi24]). We also have the following.

**PROPOSITION 4.1.** Let S be a smooth projective surface with $p_g(S) = 0$ and $q(S) \neq 0$. Assume that the morphism

$$ \phi_S: S \to C, \quad J(C) = \mathrm{Alb}(S) $$

given by Lemma 1.5 has the property that the class $[S_c] \in H^2(S, \mathbb{Z})_{\mathrm{tf}}$ of the fiber $S_c = \phi_S^{-1}(c)$ is primitive. Then the index of $\phi_S$ is 1, and S admits a universal 0-cycle.

*Proof.* By Proposition 2.7, the first statement implies the second one. Let us now prove that under our assumptions, the index of $\phi_S$ is 1. As the class $[S_c]$ is primitive, Poincaré duality on $H^2(S, \mathbb{Z})$ tells us that there exists a class $\gamma \in H^2(S, \mathbb{Z})$ such that $\langle \gamma, [S_c] \rangle = 1$. As $p_g(S) = 0$, the theorem on (1,1)-classes tells that $H^2(S, \mathbb{Z}) \cong \mathrm{NS}(S)$, hence $\gamma = [D]$ for some divisor class $D \in \mathrm{NS}(S)$. Thus D has degree 1 along the fibers of $\phi_S$. $\square$

We now consider the non-uniruled case and first recall the following result due to Beauville [Bea10].

**THEOREM 4.2** ([Bea10, Proposition 11]). Let S be a non-uniruled smooth projective surface such that $p_g(S) = 0$ and $q(S) \neq 0$. Then S is birational to a quotient $(C_1 \times C_2)/G$, where G is a finite group acting by automorphisms of both $C_1$ and $C_2$, without fixed points on $C_1 \times C_2$. Furthermore, at least one of the two curves is elliptic, and (up to exchanging $C_1$ and $C_2$) we have

$$ C_1/G \cong E, \quad C_2/G \cong \mathbb{P}^1, $$

where E is elliptic and $\mathrm{Alb}(S) = J(E) \cong E$.

**Remark 4.3.** We can assume that G acts faithfully on both $C_1$ and $C_2$, because, otherwise, denoting by $H$ the kernel of morphism $G \to \mathrm{Aut}(C_1)$, we can replace $C_2$ with $C_2/H$, and $\bar{G} := G/H$ acts on $C_1$ and $C_2$ with $(C_1 \times C_2)/G \cong (C_1 \times C_2/H)/\bar{G}$. Assuming the faithfulness condition, if $C_2$ is not elliptic, then the group G is commutative, because $C_1$ is elliptic and $C_1/G$ is also elliptic.

We now have the following.

**PROPOSITION 4.4.** Assume that S is as above and the group G is cyclic. Then the index of $\phi_S$ is equal to 1; hence S has a universal 0-cycle.

*Proof.* Again the second statement follows from the first. Let G be cyclic generated by $g$, and assume that the order of $g$ is $d$. Using the notation of Theorem 4.2, we know that $C_2/G \cong \mathbb{P}^1$. An irreducible degree $d$ cyclic cover of $\mathbb{P}^1$ has an affine version with equation

$$ u^d = (t - t_1)^{a_1} \cdots (t - t_k)^{a_k}, $$

where the gcd of $\{d, a_i, i = 1, \dots, k\}$ is 1 since otherwise the curve is not irreducible. Let $r_i := \gcd(d, a_i)$. After normalization, the ramification order of $C_2 \to \mathbb{P}^1$ over $t_i$ is $d/r_i$, and $d/r_i$ is the order of the subgroup $G_i \subset G$ fixing any point $c_i \in C_2$ over $t_i \in \mathbb{P}^1$. The surface S contains the quotient $(C_1 \times \{c_i\})/G_i$, which has degree $r_i$ over $C_1/G$. As the gcd of the set $\{r_i\}$ is 1, we conclude that the index of $\phi_S$ is 1. $\square$
ACKNOWLEDGEMENTS

I thank Jean-Louis Colliot-Thélène for his remarks and careful reading. I also thank the referee for their useful comments and corrections.

REFERENCES

Bea10 A. Beauville, *Surfaces algébriques complexes*, in *Algebraic surfaces*, C.I.M.E. Summer Sch., vol. 76 (Springer, Heidelberg 2010), 5-56; doi:10.1007/978-3-642-11087-0_1.

BdG23 T. Beckmann and O. de Gaay Fortman, *Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties*, Compos. Math. **159** (2023), no. 6, 1188-1213; doi:10.1112/s0010437x23007133.

BO20 O. Benoist and J. C. Ottem, *Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero*, Comment. Math. Helv. **95** (2020), no. 1, 27-35; doi:10.4171/cmh/479.

BKL76 S. Bloch, A. Kas, and D. Lieberman, *Zero cycles on surfaces with $p_g = 0$*, Compos. Math. **33** (1976), no. 2, 135-145.

BS83 S. Bloch and V. Srinivas, *Remarks on correspondences and algebraic cycles*, Amer. J. Math. **105** (1983), no. 5, 1235-1253; doi:10.2307/2374341.

Col25 J.-L. Colliot-Thélène, *Notes sur l'application d'Albanese pour les zéro-cycles*, 2025, arXiv:2504.21726.

EdGS25 P. Engel, O. de Gaay Fortman, and S. Schreieder, *Matroids and the integral Hodge conjecture for abelian varieties*, 2025, arXiv:2507.15704.

GHS03 T. Graber, J. Harris, and J. Starr, *Families of rationally connected varieties*, J. Amer. Math. Soc. **16** (2003), no. 1, 57-67; doi:10.1090/S0894-0347-02-00402-2.

Kle72 S. L. Kleiman, *Motives*, in *Algebraic geometry, Oslo 1970* (Proc. Fifth Nordic Summer School in Math.) (Wolters-Noordhoff Publishing, Groningen, 1972), 53-96.

Lop92 A. F. Lopez, *On the curves lying on a general surface containing a fixed space curve*, Ric. Mat. **41** (1992), no. 1, 21-40.

Mum68 D. Mumford, *Rational equivalence of 0-cycles on surfaces*, J. Math. Kyoto Univ. **9** (1968), 195-204; doi:10.1215/kjm/1250523940.

Mur85 J. P. Murre, *Applications of algebraic K-theory to the theory of algebraic cycles*, Algebraic Geometry (Sitges, 1983), Lecture Notes in Math., vol. 1124 (Springer, Berlin, Heidelberg, 1985), 216-261; doi:10.1007/BFb0075002.

Roi80a A. A. Roitman, *Rational equivalence of zero-dimensional cycles*, Mat. Zametki **28** (1980), no. 1, 85-90, 169 (Russian).

Roi80b __________, *The torsion of the group of 0-cycles modulo rational equivalence*, Ann. of Math. (2) **111** (1980), no. 3, 553-569; doi:10.2307/1971109.

SS10 M. Schütt and T. Shioda, *Elliptic surfaces*, in *Algebraic Geometry in East Asia—Seoul 2008*, Adv. Stud. Pure Math., vol. 60 (Math. Soc. Japan, Tokyo, 2010), 51-160; doi:10.2969/aspm/06010051.

Ser60 J.-P. Serre, *Morphismes universels et variétés d'Albanese*, in *Séminaire C. Chevalley, IIIème année: 1958/59. Variétés de Picard* (Éc. Norm. Supér., Paris, 1960), 1-22.

Uen73 K. Ueno, *Classification of algebraic varieties. I*, Compos. Math. **27** (1973), 277-342.

Voi92 C. Voisin, *Sur les zéro-cycles de certaines hypersurfaces munies d'un automorphisme*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) **19** (1992), no. 4, 473-492.

Voi15 __________, *Unirational threefolds with no universal codimension 2 cycle*, Invent. Math. **201** (2015), no. 1, 207-237; doi:10.1007/s00222-014-0551-y.

Voi24 __________, *Cycle classes on abelian varieties and the geometry of the Abel-Jacobi map*, Pure Appl. Math. Q. **20** (2024), no. 5, 2469-2496; doi:10.4310/pamq.241105235723.
UNIVERSAL 0-CYCLE ON VARIETIES WITH REPRESENTABLE CH₀-GROUP

Voi25 ________, *Geometric representability of 1-cycles on rationally connected threefolds*, in *Perspectives on four decades of Algebraic Geometry. Vol. 2. In Memory of Alberto Collino*, Progr. Math., vol. 352, (Birkhäuser/Springer, Cham, 2025), 467–492; doi:10.1007/978-3-031-66234-8_13.

Claire Voisin claire.voisin@imj-prg.fr

Institut de Mathématiques de Jussieu-Paris Rive Gauche, 4 place Jussieu, 75005 Paris, France