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A surface with representable $\text{CH}_{0}$-group but no universal zero-cycle

Theodosis Alexandrou

TL;DR

The paper addresses the problem of universal $0$-cycles on smooth projective varieties by introducing a degeneration-based obstruction. It constructs a bielliptic-type $2$ surface $S$ with $ ext{Alb}(S)\cong E$ for some elliptic curve $E$ such that $ ext{CH}_{0}(S)$ is representable while $S$ lacks a universal $0$-cycle, providing a two-dimensional analogue of Voisin's counterexamples. This leads to a first example of a Kodaira dimension zero threefold with a non-torsion, non-algebraic Hodge class in degree $4$ on $E\times S$, highlighting a new degeneration-driven mechanism for obstructions. The results hinge on a carefully constructed degeneration of a bielliptic surface of type $2$ and a key obstruction from the Albanese-sum maps of the special fibre components, emphasizing the subtle interaction between representability of CH$_0$, universal cycles, and the integral Hodge conjecture in higher dimensions.

Abstract

We introduce a new obstruction to the existence of a universal $0$-cycle on a smooth projective complex variety. As an application, we construct a smooth projective complex surface whose Chow group of $0$-cycles is representable but which does not admit a universal $0$-cycle. This provides a two-dimensional analogue of Voisin's recent threefold counterexample to a question of Colliot-Thélène. As a further consequence, we exhibit the first example of a smooth projective threefold of Kodaira dimension zero carrying a non-torsion Hodge class of degree $4$ that is not algebraic. The construction relies on the geometry of bielliptic surfaces of type 2.

A surface with representable $\text{CH}_{0}$-group but no universal zero-cycle

TL;DR

The paper addresses the problem of universal -cycles on smooth projective varieties by introducing a degeneration-based obstruction. It constructs a bielliptic-type surface with for some elliptic curve such that is representable while lacks a universal -cycle, providing a two-dimensional analogue of Voisin's counterexamples. This leads to a first example of a Kodaira dimension zero threefold with a non-torsion, non-algebraic Hodge class in degree on , highlighting a new degeneration-driven mechanism for obstructions. The results hinge on a carefully constructed degeneration of a bielliptic surface of type and a key obstruction from the Albanese-sum maps of the special fibre components, emphasizing the subtle interaction between representability of CH, universal cycles, and the integral Hodge conjecture in higher dimensions.

Abstract

We introduce a new obstruction to the existence of a universal -cycle on a smooth projective complex variety. As an application, we construct a smooth projective complex surface whose Chow group of -cycles is representable but which does not admit a universal -cycle. This provides a two-dimensional analogue of Voisin's recent threefold counterexample to a question of Colliot-Thélène. As a further consequence, we exhibit the first example of a smooth projective threefold of Kodaira dimension zero carrying a non-torsion Hodge class of degree that is not algebraic. The construction relies on the geometry of bielliptic surfaces of type 2.
Paper Structure (11 sections, 10 theorems, 144 equations, 2 tables)

This paper contains 11 sections, 10 theorems, 144 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Cohomology
  5. Varieties with representable Chow group of $0$-cycles
  6. Bielliptic surfaces
  7. An obstruction to the existence of a universal $0$-cycle
  8. A degeneration of a Bielliptic surface of type 2
  9. Main results
  10. A bielliptic surface of type 2 admitting no universal $0$-cycle
  11. A counterexample to the integral Hodge conjecture

Key Result

Theorem 1.3

Let $E$ be a smooth complex elliptic curve with $\mathop{\mathrm{End}}\nolimits(E)=\mathbb{Z}$. There exists a smooth projective complex surface $S$ with $\textnormal{Alb}(S)\cong E$ for which $\operatorname{CH_{0}}(S)$ is representable, while $S$ admits no universal $0$-cycle.

Theorems & Definitions (22)

  • Definition 1.1: voisin_a, Definition 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • proof : Proof of Step \ref{['step:reduction']}
  • Corollary 3.2
  • ...and 12 more