Tohoku Mathematical Journal
2015
September
SECOND SERIES VOL. 67, NO. 3

Tohoku Math. J.
67 (2015), 323-347

Title TORIC MODIFICATIONS OF CYCLIC ORBIFOLDS AND AN EXTENDED ZAGIER RECIPROCITY FOR DEDEKIND SUMS

Dedicated to the memory of Professor Masayoshi Nagata
Author Tadashi Ashikaga

(Received May 7, 2013, revised April 21, 2014)
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.

Mathematics Subject Classification. Primary 11F20; Secondary 14M25, 32S45, 58J20, 11F23, 14B05, 57R18.
Key words and phrases. Singularity, toric geometry, Dedekind sum, reciprocity, signature, orbifold.

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