Cylinder Homomorphisms and Chow Groups

Let X be a projective algebraic manifold of dimension n (over C), CH 1 (X) the Chow group of algebraic cycles of codimension l on X , modulo rational equivalence, and A 1 (X) ⊂ CH 1 (X) the subgroup of cycles algebraically equivalent to zero. We say that A 1 (X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH 1 (Γ × X) such that z * :A 1 (Γ)‐A 1 (X) is surjective. There is the well known Abel‐Jacobi map λ 1 :A 1 (X)‐J   1 a (X), where J   1 a (X) is the l th Lieberman Jacobian. It is easy to show that A 1 (X)→J   1 a (X)A 1 (X) finite dimensional. Now setwith corresponding map A * (X)→J   * a (X). Also define Level. In a recent book by the author, there was stated the following conjecture:where it was also shown that (⟹) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A * (X) finite dimensional ⇔ Level (H * (X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k ‐planes on X , where([…] = greatest integer function) and d = deg X ; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.