2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163063Based on a novel application of an archimedean type pairing to the geometry and deformation theory of $K3$ surfaces, we construct a regulator indecomposable $K_1$-class on a self-product of a $K3$ surface. In the Appendix, we explain how this pairing is a special instance of a general pairing on precycles in the equivalence relation defining Bloch's higher Chow groups.26 pagesAlgebraic Geometry14C25; 14C30; 14C35text