Higher Chow cycles on a family of Kummer surfaces

TL;DR

This paper constructs higher Chow cycles on Kummer surfaces, demonstrating that for a general family member, these cycles generate a large subgroup of the indecomposable higher Chow group, using group actions and differential operators.

Contribution

It introduces a new method to construct and prove the independence of higher Chow cycles on Kummer surfaces using group actions and Picard-Fuchs operators.

Findings

Cycles generate subgroup of rank ≥ 18 in higher Chow group

Construction leverages finite group actions

Linear independence shown via differential operators

Abstract

We construct a collection of families of higher Chow cycles of type $(2,1)$ on a 2-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank $\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard-Fuchs differential operators.