Torsion higher Chow cycles modulo $\ell$

TL;DR

The paper investigates when higher Chow groups $\operatorname{CH}^{c}(X,p)$ have large torsion modulo a prime $n$, focusing on the action of higher Chow groups on refined unramified cohomology through morphic cohomology. The authors reduce the problem to injectivity statements for exterior product pairings and construct explicit degenerations of surfaces with controlled torsion in the Néron–Severi group to force injectivity. They prove that for every $p\ge1$, $d\ge p+4$, and $n\ge2$ there exists a smooth complex projective $d$-fold $X$ such that $\operatorname{CH}^{p+3}(X,p)[n]/n$ contains infinitely many order-$n$ elements in the range $p+3\le c\le d-1$, with bounds shown to be optimal. The work further provides morphic cohomology–theoretic extensions: for each $p\ge0$ there are examples with infinite $L^{p+2}H^{p+4}(X)/n$, and, via a degeneration argument, infinite $\ell^{\infty}$-torsion in higher Chow groups modulo $\ell$. The results illuminate deep connections between higher Chow groups, refined unramified cohomology, and morphic cohomology, and they yield new infinite families of cycles undetectable by classical cycle class maps, advancing understanding of torsion phenomena in algebraic cycles over complex varieties.

Abstract

We study the injectivity property of certain actions of higher Chow groups on refined unramified cohomology. As an application for every $p\geq1$ and for each $d\geq p+4$ and $n\geq2,$ we establish the first examples of smooth complex projective $d$-folds $X$ such that for all $p+3\leq c\leq d-1,$ the higher Chow group $\text{CH}^{c}(X,p)$ contains infinitely many torsion cycles of order $n$ that remain linearly independent modulo $n$. Our bounds for $c$ and $d$ are also optimal. A crucial tool for the proof is morphic cohomology.

Theorems & Definitions (74)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.3