Application of orbifold signature to singularities and singular fibers

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540048
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tohoku Gakuin University
• Principal Investigator: Ashikaga Tadashi 東北学院大学, 工学部, 教授 (90125203)
• Co-Investigator(Renkei-kenkyūsha): Konno Kazuhiro 大阪大学, 大学院理学研究科, 教授 (10186869) ; Yoshikawa Ken-ichi 京都大学, 大学院理学研究科, 教授 (20242810) ; Matsumoto Yukio 学習院大学, 理学部, 客員研究員 (20011637) ; Ishida Hirotaka 宇部工業高等専門学校, 一般科, 准教授 (30435458)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 特異点 / 符号数 / デデキンド和 / トーリック幾何 / 連分数 / オービフォルド / 特異ファイバー / 相互律

Outline of Final Research Achievements

First we generalized Zagier’s reciprocity for higer-dimensional Dedekind sum to the case where the given integers satisfy weaker conditions. Our method is geometrical in the sense that the comparison of the invariants of several cyclic quotient singularities induce the reciprocity. The main tool of the proof is the orbifold signature theorem with non-isolated fixed points locus.
Second, we extended Myerson-Holzapfel formula which expresses the classical Dedekind sum by means of continued fraction as follows: We define the notion of higher-dimensional continued fraction whose algorithm corresponds to the toric resolution process of Fujiki-Oka for cyclic quotient singularities. Then we expressed the three-dimensional Fourier-Dedekind sum with weight zero by means of this continued fraction. As a tool of the proof, we slightly extend the observation of Pommersheim for the Chow ring of simplicial toric varieties.

Free Research Field

代数幾何