Representability of Algebraic CHOW Groups of Complex Projective

Let X ⊂ Pn+r be an n dimensional smooth complex projective complete intersection of type (d1, . . . , dr). CHr(X) denotes the Chow group of codimension r algebraic cycles on X modulo rational equivalence and Ar(X) denotes the subgroup of the codimension r algebraic cycles that are algebraically equivalent to zero. In 1990 James D. Lewis made a conjecture on the representability of Ar(X). We will show that his conjecture holds for smooth complex complete intersections satisfying a numerical condition and consider some applications to motives.