Space of one cycles and coniveau filtrations

TL;DR

This paper investigates the structure of one cycles on certain algebraic varieties, establishing key equivalences in filtrations and contributing to the understanding of the integral Tate conjecture for these cycles.

Contribution

It provides a structural analysis of the space of one cycles on separably rationally connected varieties and proves the equivalence of coniveau filtrations in degree 3 homology.

Findings

Strong coniveau filtration equals coniveau filtration on degree 3 homology

Proves the integral Tate conjecture for homologically trivial one cycles

Characterizes the space of one cycles as a topological group or h-sheaf

Abstract

We prove a structural result about the space of one cycles of a separably rationally connected variety or a separably rationally connected fibration over a curve, either as a topological group or as an h-sheaf. This has the following consequences: a proof that the strong coniveau filtration agrees with the coniveau filtration on degree 3 homology, and a result on the integral Tate conjecture for homologically trivial one cycles.