The Real Regulator for a Product of K3 Surfaces

Let X be a general complex algebraic K3 surface, and CH2(X, 1) Bloch’s higher Chow group ([Blo1]). In [C-L2] it is proven that the real regulator r2,1 : CH 2(X, 1)⊗ R→ H1,1(X,R) is surjective (Hodge-D-conjecture). The question addressed in this paper is whether a similar story holds for products of K3 surfaces. We prove a general regulator result for a product of two surfaces, and then deduce, under the assumptions of [a variant of] the Bloch-Beilinson conjecture regarding the injectivity of the Abel-Jacobi map for smooth quasiprojective varieties defined over number fields, that the (induced) regulator map r3,1 : CH 3(X×Y, 1)→ { H2 tr(X,R)⊗H tr(Y,R) } ∩H2,2(X×Y ) is trivial for a general pair of K3 surfaces, where H2 tr(X,R) is the space of transcendental classes in H2(X,R).