We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring , provides the right theory to obtain, using λ-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.

Let U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, and

clr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))

the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

Let X be a reduced and projective singular surface over ℂ and let → X be a resolution of singularities of X. We show that CH2(X) ≅ CH2() if and only if for i = 0, 1. This verifies a conjecture of Srinivas.

We also verify Bloch's conjecture for singular surfaces assuming it holds for smooth surfaces. As a byproduct, we give an application to projective modules on certain singular affine surfaces.

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel–Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

Let X/C be a projective algebraic manifold, and further let CHk(X)Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F[ell ]}[ell ] [ges ] 0 on CHk(X)Q of ‘Bloch-Beilinson’ type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that Fk+1 = 0.