# Semi-algebraic chains on projective varieties and the Abel-Jacobi map for higher Chow cycles

## TL;DR

This paper introduces delta-admissible chains to describe the cohomology of smooth quasi-projective varieties and applies this framework to elucidate the Abel-Jacobi map for higher Chow cycles, connecting algebraic cycles with Hodge theory.

## Contribution

It develops a new geometric approach using delta-admissible chains to understand cohomology and the Abel-Jacobi map for higher Chow cycles, linking algebraic and Hodge-theoretic aspects.

## Key findings

- Cohomology groups described via delta-admissible chains
- Abel-Jacobi map characterized through delta-admissible chains
- Hodge realization of polylog cycles linked to Abel-Jacobi map

## Abstract

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to a normal crossing divisor can be described in terms of delta-admissible chains. Roughly speaking, a delta-admissible chain is a simplicial semi-algebraic chain meeting the "faces" properly. As an application, we show that the Abel-Jacobi map for higher Chow cycles can be described via delta-admissible chains. As an example, we will describe the Hodge realization of the polylog cycles constructed by Bloch-Kriz in terms of the Abel-Jacobi map.