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

Claire Voisin

TL;DR

The paper constructs a smooth projective $3$-fold with representable ${\rm CH}_0$ that nevertheless lacks a universal $0$-cycle, answering Colliot‑Thélène. It builds on the Benoist–Ottem obstruction to the integral Hodge conjecture, using quotients of products like $E\times S$ to produce the counterexample, and analyzes when a universal $0$-cycle exists via the Albanese map and index considerations. It further examines the relationship between universality and the curve-fibration structure, including the surface case where cyclic symmetry yields index one and a universal $0$-cycle. The results elucidate when the Albanese and motivic structures force universality and when torsion obstructions can obstruct it, with implications for integral Hodge theory on product varieties.

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.

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

TL;DR

The paper constructs a smooth projective -fold with representable that nevertheless lacks a universal -cycle, answering Colliot‑Thélène. It builds on the Benoist–Ottem obstruction to the integral Hodge conjecture, using quotients of products like to produce the counterexample, and analyzes when a universal -cycle exists via the Albanese map and index considerations. It further examines the relationship between universality and the curve-fibration structure, including the surface case where cyclic symmetry yields index one and a universal -cycle. The results elucidate when the Albanese and motivic structures force universality and when torsion obstructions can obstruct it, with implications for integral Hodge theory on product varieties.

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.

Paper Structure

This paper contains 7 sections, 15 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. A discussion of universal $0$-cycles
  3. Previous examples of smooth projective varieties with no universal $0$-cycle
  4. Varieties fibered over a curve
  5. A $3$-dimensional example for Question \ref{['questionCT']}
  6. On a result of Benoist and Ottem
  7. The case of surfaces

Key Result

Theorem 1.4

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

Theorems & Definitions (32)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Remark 2.3
  • ...and 22 more