GENERALIZED POINCARE CLASSES AND CUBIC EQUIVALENCES

Let Y be a smooth projective algebraic surface over C, and T(Y) the ke rnel of the Albanese map CH0(Y)(deg 0) --> Alb(Y). It was first proven by D. MUMFORD that if the genus Pg(Y) > 0, then T(Y) is 'infinite dim ensional'. One would like to have a better idea about the structure of T(Y). For example, if Y is dominated by a product of curves E(1) x E( 2), such as an abelian or a Kummer surface, then one can easily constr uct an abelian variety B and a surjective 'regular' homomorphism B-xz2 , T(Y). A similar story holds for the case where Y is the Fano surface of lines on a smooth cubic hypersurface in P-4. This implies a sort o f boundedness result for T(Y). It is natural to ask if this is the cas e for any smooth projective algebraic surface Y? Partial results have been attained in this direction by the author [Illinois. J. Math. 35 ( 2), 1991]. In this paper, we show that the answer to this question is in general no. Furthermore, we generalize this question to the case of the Chow group of k-cycles on any projective algebraic manifold X, an d arrive at, from a conjectural standpoint, necessary and sufficient c ohomological conditions on X for which the question can be answered af firmatively.