The higher Chow group is a generalization of the classical Chow group. In this talk, I will explain an explicit construction of higher Chow cycles of type (2, 1) on a certain type of Kummer surfaces. By computing their images under the regulator map, I show that for very general cases, these cycles generate a subgroup whose rank is at least 18 in the "indecomposable part" of the higher Chow group of type (2, 1), which is the quotient by the subgroup generated by the intersection products of cycles of lower codimension.
Furthermore, by deforming such cycles, we also obtain explicit higher Chow cycles on K3 surfaces with a non-symplectic involution and we can prove that such cycles are non-torsion in the indecomposable part for very general cases. The latter is a joint work with Shohei Ma.