2 00 2 Indecomposable K 1 and the Hodge - D - Conjecture for K 3 and Abelian Surfaces

Let X be a projective algebraic manifold, and CH(X, 1) the higher Chow group, with corresponding real regulator rk,1 ⊗ R : CH(X, 1) ⊗ R → H D (X,R(k)). If X is a general K3 surface or Abelian surface, and k = 2, we prove the Hodge-D-conjecture, i.e. the surjectivity of r2,1 ⊗ R. Since the Hodge-D-conjecture is not true for general surfaces in P of degree ≥ 5, the results in this paper provide an effective bound for when this conjecture is true. We then apply these results to the space of indecomposables CHkind(X, 1;Q), specifically by proving that Level ( CHkind(X, 1;Q) ) ≥ k − 2 where X is a general kfold product of elliptic curves. This leads to a hard generalization of Mumford’s famous theorem on the kernel of the Albanese map on the Chow group of zero-cycles on a surface of positive genus. 1. Statement of results Let X be a projective algebraic manifold. This paper concerns the maps, called regulators, from K1 of X to real Deligne cohomology. More specifically, in terms of Bloch’s higher Chow groups CH(X,m) [Blo1], we are interested in the case m = 1 and the map rk,1 : CH (X, 1) → H2k−1 D (X,R(k)) where R(k) = R(2π √ −1)k and H2k−1 D (X,R(k)) ≃ Hk−1,k−1(X,R(k − 1)) is Deligne cohomology. Beilinson’s Hodge-D-conjecture for real varieties would imply that rk,1 ⊗ R : CH(X, 1) ⊗ R → H2k−1 D (X,R(k)) is surjective (see [Ja]). That conjecture is now known to be false using the works of [No] and [G-S] (see [MS1]); although the corresponding conjecture Date: Oct. 27, 2002. 1991 Mathematics Subject Classification. 14C25, 14C30, 14C35.