Zero cycles on rationally connected varieties over Laurent fields

Abstract

We study zero-cycles on rationally connected varieties defined over characteristic zero Laurent fields with algebraically closed residue fields. We show that the degree map induces an isomorphism for rationally connected threefolds defined over such fields. In general, the degree map is an isomorphism if rationally connected varieties defined over algebraically closed fields of characteristic zero satisfy the integral Hodge/Tate conjecture for one-cycles, or if the Tate conjecture is true for divisor classes on surfaces defined over finite fields. To prove these results, we introduce techniques from the minimal model program to study the Gersten type complex defined by Kato/Bloch–Ogus. We also propose a conjecture about the Kato homology of a rationally connected fibration.