2 00 2 Noether - Lefschetz for K 1 of a Surface , Revisited

Let Z ⊂ P be a general surface of degree d ≥ 5. Using a Lefschetz pencil argument, we give a elementary new proof of the vanishing of a regulator on K1(Z). 1. Statement of result Let Z be a smooth quasiprojective variety over C, and for given nonnegative integers k,m, let CH(Z,m) be the higher Chow group as introduced in [Blo1]. In [Blo2], Bloch constructs a cycle class map into any suitable cohomology theory. In our setting, the corresponding map is: clk,m : CH (Z,m) → H D (Z,Q(k)), whereH D (Z,Q(k)) is Deligne-Beilinson cohomology, which fits in a short exact sequence 0 → H (Z,C) F kH2k−m−1(Z,C) +H2k−m−1(Z,Q(k)) → H D (Z,Q(k)) → F H(Z,C) ⋂ H(Z,Q(k)) → 0. Our primary interest is when Z is also complete, and m = 1. Thus one has the corresponding map: clk,1 : CH (Z, 1) → H (Z,C) F kH2k−2(Z,C) +H2k−2(Z,Q(k)) . Let Hg(Z) := H2k−2(Z,Q(k−1))∩F H(Z,C) be the Hodge group. Then one has an induced map clk,1 : CH (Z, 1) → H (Z,C) F kH2k−2(Z,C) + Hg(Z)⊗C +H2k−2(Z,Q(k)) . It is known that clk,1 is trivial for Z a sufficiently general complete intersection and of sufficiently high multidegree. This is an consequence of the Date: December 13, 2002. 1991 Mathematics Subject Classification. 14C25, 14C30, 14C35.