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

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.

2020. May. 20: Calculation in Section 4.2 is made clearer

References in corpus (2)

• The Abel-Jacobi Map for Higher Chow Groups, II
• Integrals of logarithmic forms on semi-algebraic sets and a generalized Cauchy formula, Part I: convergence theorems