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Application of orbifold signature to singularities and singular fibers

Research Project

Project/Area Number 24540048
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionTohoku 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
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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 / 退化 / 局所不変量 / モジュライ空間 / モノドロミー / トーリック写像 / 商特異点 / 安定曲線 / リーマン面 / モジュライ / singularity / toric resolution / continued fraction / Fourier-Dedekind sum / Todd class / orbifold / 双有理変換 / 符号不足数 / オービフォールド
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.

Report

(5 results)
  • 2015 Annual Research Report   Final Research Report ( PDF )
  • 2014 Research-status Report
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (17 results)

All 2016 2015 2014 Other

All Journal Article (1 results) (of which Peer Reviewed: 1 results,  Acknowledgement Compliant: 1 results) Presentation (16 results) (of which Invited: 16 results)

  • [Journal Article] Toric modifications of cyclic orbifolds and an extended Zagier reciprocity for Dedekind sums2015

    • Author(s)
      T.Ashikaga
    • Journal Title

      Tohoku Mathematical Journal

      Volume: 67

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] A certain intersection number on moduli space of genus 3 and Horikawa index2016

    • Author(s)
      足利正
    • Organizer
      代数曲面ワークショップ
    • Place of Presentation
      首都大学東京サテライトキャンパス(東京都千代田区秋葉原)
    • Year and Date
      2016-02-06
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Dedekind 和を用いた種数3のモジュライ空間上のある交点数表示2016

    • Author(s)
      足利正
    • Organizer
      代数幾何学ミニワークショップ
    • Place of Presentation
      兵庫県多可町八千代フラザ(兵庫県多可郡)
    • Year and Date
      2016-01-09
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Automorphisms of stable curves and invariants of fiber germs2015

    • Author(s)
      足利正
    • Organizer
      代数幾何学セミナー
    • Place of Presentation
      高知工科大学(高知県高知市)
    • Year and Date
      2015-11-02
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] Another realizaion of Birman-Hilden relation via hyperellptic splitting family2015

    • Author(s)
      足利正
    • Organizer
      Mini-Symposium “Topology and singularities”
    • Place of Presentation
      学習院大学(東京都豊島区)
    • Year and Date
      2015-10-14
    • Related Report
      2015 Annual Research Report
    • Invited
  • [Presentation] 種数3のファイバー芽の堀川指数とアイヒラー型跡公式2015

    • Author(s)
      足利 正
    • Organizer
      第2回代数幾何学研究集会宇部
    • Place of Presentation
      宇部工業高等専門学校(山口県宇部市)
    • Year and Date
      2015-01-24
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] 種数3のモジュライ空間上のある交点数と堀川指数2015

    • Author(s)
      足利 正
    • Organizer
      代数幾何学ミニワークショップ
    • Place of Presentation
      八千代プラザ(兵庫県多可町)
    • Year and Date
      2015-01-10
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Horikawa index of genus 3 with periodic monodromy via Eicjler's trace formula2014

    • Author(s)
      足利 正
    • Organizer
      リーマン面に関連する位相 幾何学2014
    • Place of Presentation
      東京大学駒場キャンパス(東京都目黒区)
    • Year and Date
      2014-08-25
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Higher-dimensional continued fractions for cyclic orbifold and Dedekind sums

    • Author(s)
      足利正
    • Organizer
      研究集会「射影多様体の幾何とその周辺2013」
    • Place of Presentation
      高知大学理学部(高知県高知市曙町)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 2 次元連分数の子孫項に伴う有理 Chow 環と Fourier-Dedekind 和

    • Author(s)
      足利正
    • Organizer
      代数幾何学ミニワークショプ
    • Place of Presentation
      多可町中央公民館セミナー室(兵庫県多可郡多可町)
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Toric modification of cyclic orbifolds and Dedekind sums

    • Author(s)
      T.Ashikaga
    • Organizer
      WCU workshop on classification and construction of algebraic varieties
    • Place of Presentation
      Busan, KOREA
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Toric modification of cyclic orbifolds and extened Zagier reciprocity for Dedekin sums

    • Author(s)
      足利正
    • Organizer
      代数学シンポジューム 2012
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 特異点, 特異ファイバー, 一般型曲面, 夢

    • Author(s)
      足利正
    • Organizer
      2012年特異点と多様体の幾何学研究集会
    • Place of Presentation
      山形大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Dedekind和に関するZagier 相互律の幾何的拡張

    • Author(s)
      足利正
    • Organizer
      2012年リーマン面に関連する位相幾何学研究集会
    • Place of Presentation
      東京大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 巡回商特異点のある解消プロセスとその例外集合

    • Author(s)
      足利正
    • Organizer
      Encounter with Mathematics 第59回
    • Place of Presentation
      中央大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 巡回商特異点の解消と高次元連分数

    • Author(s)
      足利正
    • Organizer
      多変数関数論冬セミナー
    • Place of Presentation
      東北大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Cotangent sums of Atiyah-Singer and Oka toric modifications

    • Author(s)
      足利正
    • Organizer
      特異点ミニワークショップ
    • Place of Presentation
      東京理科大学
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2019-07-29  

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