Relative cycles with moduli and regulator maps

TL;DR

This paper develops a theory of higher Chow groups with modulus for schemes over a field, constructs regulator maps to relative de Rham and Deligne cohomology, and defines relative intermediate Jacobians with modulus, generalizing classical concepts.

Contribution

It introduces a cycle complex with modulus, computes its motivic complex in weight 1, and constructs regulator maps and intermediate Jacobians for pairs (X,D) with modulus, extending existing theories.

Findings

Defined higher Chow groups with modulus for schemes over a field.

Constructed regulator maps to relative de Rham and Deligne cohomology.

Established the universal property of relative intermediate Jacobians with modulus.

Abstract

Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus - generalize additive higher Chow groups of Bloch-Esnault, R\"ulling, Park and Krishna-Levine, and that sheafified on $X_{Zar}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1. When X is smooth over k and D is such that $D_{red}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction. This is used to define a natural regulator map from the relative motivic complex of (X,D) to the relative de Rham complex. When X is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups. Finally, when X is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J^r_{X|D}$ of the pair (X,D). For r= dim X, we show that $J^r_{X|D}$ is the universal regular quotient of the Chow group of 0-cycles with modulus.