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

This paper answers a question posed by Colliot-Théline by constructing a smooth projective variety with a representable CH₀-group that lacks a universal 0-cycle, utilizing a counterexample to the integral Hodge conjecture provided by Benoist and Ottem.

Original authors: Claire Voisin

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Claire Voisin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to organize a massive, complex party where every guest represents a specific point on a geometric shape (a "variety"). Mathematicians have long been trying to figure out if there is a perfect, universal "guest list" or "seating chart" that can describe how these points relate to each other in a specific, organized way.

This paper, written by mathematician Claire Voisin, solves a puzzle that another mathematician, Jean-Louis Colliot-Thélène, had been wondering about for a while.

Here is the breakdown of the story, the problem, and the solution, using simple analogies.

The Main Characters and the Goal

1. The "Albanese Map" (The Party Organizer)
Imagine your geometric shape is a huge, complex building. The "Albanese map" is like a special elevator system that takes any point in the building and sends it to a central "control room" (called the Albanese variety). This control room is a very organized, smooth space (like a torus or a donut shape) that captures the essential "loops" and connections of the building.

2. The "Universal 0-Cycle" (The Perfect Guest List)
Mathematicians want to know: Is there a single, perfect "guest list" (a mathematical object called a universal 0-cycle) that lives in the control room? If this list exists, it means you can take any location in the control room and use this list to perfectly reconstruct a corresponding set of points back in the original building. It's like having a master key that can unlock any specific configuration of guests.

3. The "Representable CH0-group" (The Organized Crowd)
Sometimes, the crowd of points in the building is so well-behaved that the elevator system (the Albanese map) is a perfect one-to-one match. Every point in the control room corresponds to exactly one unique group of points in the building, and vice versa. When this happens, we say the group is "representable."

The Big Question

For a long time, mathematicians knew two things:

Some buildings have a "perfect crowd" (representable group).
Some buildings do not have a "master guest list" (no universal 0-cycle).

But they didn't know if these two things could happen at the same time.
The Question: Can you have a building where the crowd is perfectly organized (representable), but you still cannot find a single master guest list (no universal 0-cycle)?

Most people suspected the answer was "No." They thought that if the crowd was organized enough to be representable, a master guest list must exist.

The Solution: A Counter-Example

Claire Voisin says: "Yes, it is possible."

She constructs a specific, 3-dimensional geometric building (a "threefold") that breaks the rule.

The Crowd: The points in this building are perfectly organized. The elevator system works flawlessly (the group is representable).
The Missing List: Despite this perfect organization, there is no master guest list that can describe all the points at once.

How She Built It (The Recipe)

To build this strange building, Voisin used a clever recipe involving two other shapes:

A K3 Surface: Think of this as a very complex, flat, 2D sheet with a special symmetry (like a pattern that repeats but flips).
An Elliptic Curve: Think of this as a simple loop or a donut shape.

She took these two shapes, twisted them together, and then applied a "folding" operation (a mathematical symmetry) that glued certain parts together. The result is a new 3D shape.

Why does this shape have no master list?
The proof relies on a deep connection to a famous unsolved problem in math called the Integral Hodge Conjecture.

Imagine trying to build a wall out of specific bricks. The Hodge Conjecture asks: "If a wall looks like it's made of bricks (mathematically speaking), is it actually made of bricks?"
In Voisin's example, the "wall" (the geometry of the shape) looks like it should be buildable with bricks, but it turns out it's impossible to build it using the specific "bricks" (algebraic cycles) available.
Because the "bricks" don't fit together to form the wall, the "master guest list" (the universal 0-cycle) cannot exist, even though the crowd is otherwise perfectly organized.

The "Index" Clue

The paper also discusses a concept called the "index."

Imagine you are trying to cover a floor with tiles. If the floor is 10 feet wide and your tiles are 1 foot wide, you need 10 tiles. If the floor is 10 feet wide and your tiles are 3 feet wide, you can't cover it perfectly without cutting them.
In Voisin's example, the "tiles" (the geometric pieces she uses to build the list) don't fit the "floor" perfectly. The "index" is 2, meaning there is a slight mismatch that prevents the creation of the universal list.

The Takeaway

This paper proves that perfect organization does not guarantee a master key.

Voisin showed that you can have a geometric world where the points are perfectly mapped to a control center, yet the fundamental "blueprint" (the universal 0-cycle) required to reconstruct those points from the center simply doesn't exist. This answers Colliot-Thélène's question with a definitive "Yes, such a thing exists," and it does so by using a counter-example to a deep theory about how shapes are built from smaller pieces.

Technical Summary: Varieties with Representable CH0-group and a Question of Colliot-Thélène

Problem Statement
The paper addresses a specific question posed by Colliot-Thélène regarding the relationship between the representability of the Chow group of zero-cycles ( $CH_0$ ) and the existence of a "universal 0-cycle" on smooth projective complex varieties.

Let $X$ be a smooth projective variety. The group $CH_0(X)$ is said to be representable if the Albanese morphism $a_X: CH_0(X)_{hom} \to Alb(X)$ is an isomorphism. A universal 0-cycle for $X$ is defined as a codimension $n$ cycle $\Gamma \in CH_n(Alb(X) \times X)$ (where $n = \dim X$ ) such that for any $t \in Alb(X)$ , the composition $a_X \circ \Gamma^* (\{t\} - \{0\})$ equals $t$ in $Alb(X)$.

While it is known that varieties without universal 0-cycles exist (e.g., certain surfaces constructed in prior work by the author), and that varieties with representable $CH_0$ groups exist, it was an open question whether these two properties are mutually exclusive. Specifically, Colliot-Thélène asked: Does there exist a smooth complex projective variety $X$ with a representable $CH_0$ group that does not admit a universal 0-cycle?

Methodology and Theoretical Framework
The author approaches this problem by analyzing the geometric constraints imposed by the representability of $CH_0$ .

Geometric Reduction: Using Mumford's theorem, the paper establishes that if $CH_0(X)$ is representable and $Alb(X)$ is non-trivial, the Albanese map of $X$ factors through a smooth projective curve $C$ . Specifically, there exists a dominant morphism $\phi: X \to C$ such that $Alb(X) \cong J(C)$ .
Index of Morphisms: The existence of a universal 0-cycle is linked to the index of the morphism $\phi: X \to C$ . The index is defined as the greatest common divisor of the degrees of irreducible subvarieties of $X$ dominating $C$ . The paper proves that if this index is 1, a universal 0-cycle exists.
Connection to Hodge Theory: The non-existence of a universal 0-cycle is shown to be equivalent to the non-algebraicity of a specific Hodge class $\gamma_X \in H^{2n}(Alb(X) \times X, \mathbb{Z})_{tf}$ (modulo torsion). This class arises from the Künneth decomposition of the diagonal. Thus, constructing a counterexample requires finding a variety where this specific class is not algebraic, providing a counterexample to the integral Hodge conjecture modulo torsion.
Counterexample Construction: The construction relies on a counterexample to the integral Hodge conjecture provided by Benoist and Ottem. The author constructs a 3-fold $X$ as a quotient of a product $E \times S$ , where $E$ is an elliptic curve and $S$ is a K3 surface equipped with a fixed-point-free antisymplectic involution $g$ .

Key Contributions and Results

Theorem 1.4 (Main Result): The paper proves the existence of a smooth projective threefold $X$ such that $CH_0(X)$ is representable, but $X$ does not admit a universal 0-cycle. This provides an affirmative answer to Colliot-Thélène's question.
- Construction: $X = (E \times S) / (t_\xi, g)$ , where $E$ is an elliptic curve with a translation $t_\xi$ of order 2, and $S$ is a K3 surface with a fixed-point-free involution $g$ . The quotient is an Enriques surface $\Sigma = S/g$ and an elliptic curve quotient $E_\xi = E/t_\xi$ .
- Representability: The morphism $p_{E_\xi}: X \to E_\xi$ induces an isomorphism on $CH_0$ , proving representability.
- Non-existence of Universal 0-cycle: The proof demonstrates that if a universal 0-cycle existed, it would imply the existence of a correspondence $\Gamma_\Sigma \in CH_1(E_\xi \times \Sigma)$ satisfying a specific cohomological condition involving the étale double covers. This condition is shown to be impossible for a general choice of parameters, relying on results from Benoist and Ottem regarding non-algebraic torsion curve classes.
Proposition 2.2: The paper establishes an equivalence between the existence of a universal 0-cycle and the algebraicity of the Künneth projector $\delta_1$ on $X \times X$ , provided the Albanese variety is a direct summand in the Jacobian of a curve. This links the geometric property to the motivic structure of $X$ .
Proposition 2.7: It is proven that if the Albanese map factors through a curve $C$ with index 1, a universal 0-cycle exists. The paper notes that the converse is not true, providing an example of a surface with index $d > 1$ that still admits a universal 0-cycle.
Proposition 3.5: The author provides a short proof and mild generalization of a result by Benoist and Ottem, stating that for a smooth projective variety $W$ and a non-zero class $\beta \in H^1(W, \mathbb{Z}/k)$ , there are only countably many elliptic curves $E$ admitting a correspondence $\Gamma$ such that $[\Gamma]^*\beta \neq 0$ . This technical lemma is crucial for ensuring the non-existence of the required correspondence in the main construction.
Analysis of Surfaces (Section 4): The paper investigates surfaces with $p_g=0$ and $q \neq 0$ . It shows that if such a surface is a quotient of a product of curves by a cyclic group, the index of the Albanese fibration is 1, implying the existence of a universal 0-cycle. This contrasts with the 3-fold example, highlighting the restrictive nature of the counterexample.

Significance and Claims
The paper claims to resolve a specific open question in the geometry of zero-cycles by demonstrating that the property of having a representable $CH_0$ group does not guarantee the existence of a universal 0-cycle.

The construction relies on the fact that the variety $X$ possesses torsion in its integral Betti cohomology. Consequently, the non-existence of a universal 0-cycle for $X$ yields a counterexample to the integral Hodge conjecture modulo torsion for curve classes (Hodge homology classes of degree 2) on the fourfold $Alb(X) \times X$ .

The author notes that the assumptions of the main theorem are "very restrictive geometrically," as the representability of $CH_0$ forces the Albanese variety to be isomorphic to the Jacobian of a curve. The paper does not claim to construct examples without torsion in integral Betti cohomology, leaving that as an open inquiry. The work serves to clarify the subtle distinctions between representability, the structure of the Albanese map, and the integral Hodge conjecture.