Towards a generalization of Mumford's theorem (Q2639109)

Let X be a smooth algebraic variety and \(A_ k(X)\) its Chow group of k- cycles algebraically equivalent to zero. The author gives a condition for \(A_ k(X)\) to be infinite dimensional. The condition is expressed in terms of the relationship between the filtration of \(H^*(X,{\mathbb{Q}})\) by algebraic cycles and the Hodge filtration. The author also assumes a weakened form of one of the standard conjecture of Lefschetz type. He gives a number of examples where the condition and assumption are valid. These include certain hypersurfaces and abelian varieties.