AbstractLetXbe a projective algebraic manifold of dimensionn(over C),CH1(X) the Chow group of algebraic cycles of codimensionlonX, modulo rational equivalence, andA1(X) ⊂CH1(X) the subgroup of cycles algebraically equivalent to zero. We say thatA1(X) isfinite dimensionalif there exists a (possibly reducible) smooth curveTand a cycle z∈CH1(Γ × X) such that z*:A1(Γ)‐A1(X) is surjective. There is the well known Abel‐Jacobi map λ1:A1(X)‐J 1a(X), whereJ 1a(X) is thelth Lieberman Jacobian. It is easy to show thatA1(X)→J 1a(X)A1(X) finite dimensional. Now setwith corresponding mapA*(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 proveA*(X) finite dimensional⇔︁ Level (H*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family ofk‐planes onX, where([…] = greatest integer function) andd= degX; 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 hypersurfacesX.Some applications to the Griffiths group, vanishing results, and (universal) algebraic representati