  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.
