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

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