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 $\Delta^n$ to compute Bloch's higher Chow groups. We calculate explicit examples involving the M\"{a}hler measure of Laurent polynomials.