Refined unramified cohomology of schemes

TL;DR

This paper introduces refined unramified cohomology for algebraic schemes, linking it to cycle groups and using the Bloch--Kato conjecture to analyze torsion cycles and their invariants on smooth projective varieties.

Contribution

It defines refined unramified cohomology, proves comparison theorems, and establishes new results on torsion cycles, Abel--Jacobi invariants, and their filtrations, extending previous work and conjectures.

Findings

Identification of certain refined unramified cohomology groups with cycle groups

Any homologically trivial torsion cycle with trivial Abel--Jacobi invariant has coniveau 1 on complex projective varieties

Finite filtration of homologically trivial torsion cycles modulo algebraic equivalence, with graded quotients determined by higher Abel--Jacobi invariants

Abstract

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This recovers for cycles of low codimensions on smooth projective varieties previous results of Bloch--Ogus, Colliot-Th\'el\`ene--Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch--Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel--Jacobi invariant has coniveau 1. This establishes a torsion version of a conjecture of Jannsen originally formulated with rational coefficients. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel--Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$-adic analogues of these results over any field which contains all $\ell$-power roots of unity.