Zero cycles on rationally connected varieties over Laurent fields

TL;DR

This paper investigates zero cycles on rationally connected varieties over Laurent fields, establishing conditions under which the degree map is an isomorphism, and introduces new techniques from the minimal model program for this purpose.

Contribution

It proves the degree map is an isomorphism for rationally connected threefolds over Laurent fields and links this to the validity of the integral Hodge/Tate conjecture and Tate conjecture in specific cases.

Findings

Degree map is an isomorphism for rationally connected threefolds over Laurent fields.

Conditions involving the integral Hodge/Tate conjecture and Tate conjecture ensure the isomorphism.

Introduces minimal model program techniques to study homology of complexes related to zero cycles.

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 homology of certain complexes defined by Kato/Bloch-Ogus.