# Regulator Maps for Higher Chow Groups via Current Transforms

## TL;DR

This paper develops a new method to construct regulator maps from higher Chow groups to Deligne-Beilinson cohomology using algebraic correspondences and current transforms, with explicit formulas and examples.

## Contribution

It introduces a novel approach employing equidimensional correspondences and Suslin's complex to explicitly compute regulator maps for higher Chow groups.

## Key findings

- Explicit formulas for regulator maps are derived.
- The method applies to smooth complex quasi-projective varieties.
- Examples include calculations involving M"{a}hler measures.

## 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 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.