A regulator formula for Milnor K-groups.

Abstract

The classical Abel–Jacobi map is used to geometrically motivate the construction of regulator maps from Milnor K-groups K n M (C(X)) to Deligne cohomology. These maps are given in terms of some new, explicit (n – 1)-currents, higher residues of which are defined and related to polylogarithms. We study their behavior in families X s and prove a rigidity result for the regulator image of the Tame kernel, which leads to a vanishing theorem for very general complete intersections.