Toric modifications of cyclic orbifolds and an extended Zagier reciprocity for Dedekind sums

Abstract

We study a toric modification of Fujiki-Oka type for cyclic quotient singularities. Especially the behavior of rational Chow rings, orbifold signatures and so on are explicitly calculated. As a result, we extend Zagier's reciprocity for higher-dimensional Dedekind sums. Namely, we define Dedekind sums with weight by using Atiyah-Singer's equivariant signature with non-isolated fixed point locus, and then prove our reciprocity among them.