CHOW-GROUP OF CODIMENSION 2 OF DEFINED VARIETIES OVER A NUMBER-FIELD - A FINITENESS THEOREM FOR TORSION

Let X be a smooth, projective variety defined over a number field and let CH2(X) denote the Chow group of codimension two cycles modulo rational equivalence. We show that if the cohomology group H-2(X, O(x)) vanishes then the torsion subgroup of CH2(X) is a finite group. This result covers all previous results in this direction. The hypothesis H-2(X, O(x)) = 0 is used to lift line bundles.