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

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].