Regulator Maps for Higher Chow Groups via Current Transforms
We show how to use equidimensional algebraic correspondences between complex algebraic varieties to construct pull-backs and transforms of certain classes of geometric currents. Using this construction we produce explicit formulas at the level of complexes for a regulator map from the Higher Chow groups of smooth complex quasi-projective algebraic varieties to Deligne-Beilinson cohomology with integral coefficients. A distinct aspect of our approach is the use of Suslin's complex of equidimensional cycles over $ Δ^n $ to compute Bloch's higher Chow groups. We calculate explicit examples involving the Mähler measure of Laurent polynomials.
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