Let X be a smooth projective variety over the complex number field C. Let 7/{X) he the group of algebraic cycles of codimension ' and let CHr(X) (resp. CHm(X)) be the Chow group of cycles of codimension r (resp. dimension m) modulo rational equivalence. UtAr{X) C CHr(X) (resp. jJjC) C CHm(X)) be the subgroup of cycle classes which are algebraically equivalent to zero. By the classical theory of Abel-Jacobi A](X) has in the natural way the, structure of an abelian variety. One naturally asks if we are still in the same peaceful sit-uation even for ^(Xywith r S; 2. The naive anticipation was blown out by the following seminal result of Mumford on Hon-represeniahility of Chow groups of surfaces (cf. [Mu] and also its generalizations [Bl-1, Sect. 1. Appendix]; [Ro], [M]).
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