Construction of higher Chow cycles on cyclic coverings of $\mathbb{P}^1 \times \mathbb{P}^1$, Part II

TL;DR

This paper constructs higher Chow cycles on certain abelian covers of   surfaces and demonstrates their contribution to the indecomposable part of the Chow group, expanding understanding of algebraic cycles on these surfaces.

Contribution

It introduces a method to construct higher Chow cycles on cyclic coverings of   surfaces and shows they generate a large subgroup of the indecomposable Chow group.

Findings

Cycles generate a subgroup of rank at least n(N) in the indecomposable Chow group.

Construction applies to very general members of the family of surfaces.

Images under the transcendental regulator map confirm the nontriviality of these cycles.

Abstract

In this paper, we construct higher Chow cycles of type $(2, 1)$ on a family of surfaces related to a product of curves, which are certain degree $N$ abelian covers of $\mathbb{P}^1$ branched over $n+2$ points. We prove that for a very general member, these cycles generate a subgroup of the indecomposable part of $\operatorname{rank} \ge n\cdot \varphi(N)$, where $\varphi(N)$ is Euler's totient function, by computing their images under the transcendental regulator map.