Regulator maps for higher Chow groups via current transforms

Abstract

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 n ↦ Z Δ , eq p ( X , n ) $n \mapsto \mathcal {Z}^p_{\Delta , \text{eq}}(X,n)$ of equidimensional cycles over Δ n $ \Delta ^n$ to compute Bloch's higher Chow groups. We calculate explicit examples involving the Mähler measure of Laurent polynomials.