On analogues of the Kato conjectures and proper base change for $1$-cycles on rationally connected varieties

TL;DR

This paper develops a new conjectural framework for 1-cycles on rationally connected varieties over algebraically closed fields, inspired by Kato's conjectures for 0-cycles, and proves some special cases using recent results.

Contribution

It introduces a novel set of conjectures for 1-cycles on rationally connected varieties, extending Kato's framework, and proves some cases leveraging recent advances.

Findings

Established special cases of the conjectures.

Extended Kato's framework to 1-cycles on rationally connected varieties.

Connected results to recent work of Kollár-Tian.

Abstract

In 1986, Kato set up a framework of conjectures relating (higher) $0$-cycles and \'etale cohomology for smooth projective schemes over finite fields or rings of integers in local fields through the homology of so-called Kato complexes. In analogy, we develop a framework of conjectures for $1$-cycles on smooth projective rationally connected varieties over algebraically closed fields and for families of such varieties over henselian discrete valuation rings with algebraically closed fields. This is partly motivated by results of Colliot-Th\'el\`ene-Voisin [3] in dimension $3$. We prove some special cases building on recent results of Koll\'ar-Tian [16].