Geometric construction of regulator currents with applications to algebraic cycles

This is the first complete study of regulator maps on motivic cohomology from the standpoint of complex algebraic geometry. The classical Abel-Jacobi map is used to geometrically motivate the construction of maps from Milnor K-groups $KMnCX$ to Deligne cohomology. These maps are given in terms of some new, explicit (n − 1)-currents. We study their behavior in families X_t and prove a rigidity result for the image of the Tame kernel, using techniques from the theory of variations of Hodge structure. This leads to an astonishing vanishing theorem for very general complete intersections.; The Milnor current formulas generalize to regulator maps (defined on the level of algebraic cycle complexes) on all the cubical higher Chow groups CHp(X, n), whose projections to real Deligne cohomology are (by involved computations) shown to be compatible with the Beilinson regulator. Connections with polylogarithms and higher Bloch groups are explored in several ways: for example, (1) by means of higher residue maps arising as differentials in relevant local-global spectral sequences, and (2) by way of a new approach to computing certain relative regulators.; We generalize the Milnor currents in another direction to produce explicit integrals detecting rational inequivalence to zero, for 0-cycles in the Albanese(= AJ) kernel on a product of curves; concrete examples are provided. More generally, we combine and extend the work of Green-Griffiths and Lewis on higher Abel-Jacobi maps Ψ_i, and show that the above integrals compute (essentially) quotients of the invariants Ψ _i($Z$).