Signature divisor on the moduli of curves, signature defect and its applications

• Project/Area Number: 21540047
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tohoku Gakuin University
• Principal Investigator: ASHIKAGA Tadshi 東北学院大学, 工学部, 教授 (90125203)
• Co-Investigator(Renkei-kenkyūsha): YOSHIKAWA Ken-ichi 京部大学, 大学院・理学研究科, 教授 (20242810) ; KONNO Kazuhiro 大阪大学, 大学院・理学研究科, 教授 (10186869) ; ISHIDA Hirotaka 宇部工業高等専門学校, 一般科, 准教授 (30435458)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 符号数 / モジュライ空間 / モノドロミー / 退化 / 局所化 / 複素曲面 / ファイバー空間 / 安定還元 / 特異点 / デデキンド和 / 連分数 / モジュライ写像 / ファイバー芽 / エータ不変量 / モジュライ

Research Abstract

We have a new formulation of the local signature of a fiber germ of degeneration of curves as a sum of "moduli term" and "monodromy term" in the following sense.
The former is defined as the pull back by the moduli map of the signature divisor on the moduli space of stable curves(joint work with Ren-ichi Yoshikawa).
The latter is called local signature defect, which is obtained via the orbifold signature theorem by interpreting the topological monodromy data as a group action to the stable reduction.

Research Products

• [Journal Article] Maximal ideal cycles over normal surface singularities of Brieskorn type (2012)
  • Author(s): K. Konno and D. Nagashima
  • Journal Title: Osaka J. Math. Volume: 49 Pages: 225-245
  • NAID: 120004838543
• [Journal Article] Local signature defect of fibered complex surfaces via monodromy and stable reduction (2010)
  • Author(s): T. Ashikaga
  • Journal Title: Comment. Math. Helv. Volume: 85 Pages: 417-461
• [Journal Article] Local signature defect of fibered complex surfaces via moduli and monodromy (2010)
  • Author(s): T. Ashikaga
  • Journal Title: Demonstratio Math Volume: XLIII Pages: 263-276
• [Journal Article] Chain-connected component decomposition of curves on surfaces (2010)
  • Author(s): K. Konno
  • Journal Title: J. Math. Soc. Japan Volume: 62, no.2 Pages: 467-486
  • NAID: 10026999112
• [Journal Article] Canonical fixed parts of fibredalgebraic surfaces (2010)
  • Author(s): K. Konno
  • Journal Title: Tohoku Math. J. Volume: 62, no.l Pages: 117-136
• [Journal Article] The existence of hyper-elliptic fibrations with slope four and high relative Euler-Poincare character istic (2010)
  • Author(s): H. Ishida
  • Journal Title: Proc. AMS Volume: 139 Pages: 1221-1235
• [Journal Article] Local signature defect of fibered complex surfaces via monodromy and stable reduction (2010)
  • Author(s): T.Ashikaga
  • Journal Title: Comment.Math.Helv. Volume: 85 Pages: 417-461
  • Peer Reviewed
• [Journal Article] Local signature defect of fibered complex surfaces via moduli and monodromy (2010)
  • Author(s): T.Ashikaga
  • Journal Title: Demonstratin Math. Volume: XLIII Pages: 263-276
  • Peer Reviewed
• [Journal Article] Local signature defect of fibered complex surfaces via monodromy and stable reduction (2010)
  • Author(s): T.Ashikaga
  • Journal Title: Comment.Math.Helv. 85 Pages: 417-461
  • Peer Reviewed
• [Journal Article] A divisor on the moduli spaces of curves associated to the signature of fibered complex surfaces(with an Appendix by Kazuhiro Konno) (2009)
  • Author(s): T. Ashikaga and K.-I. Yoshikawa
  • Journal Title: Adv. St. Pure Math. Volume: 56 Pages: 1-34
• [Journal Article] Calabi-Yau threefolds of Borcea-Voisin, analytic torsion, and Borcherds products (2009)
  • Author(s): K.-I. Yoshikawa
  • Journal Title: Asterisque Volume: 328 Pages: 365-393
• [Journal Article] Appendix to "A divisor on the moduli space of curves associated to the signature of fihered surfaces" (2009)
  • Author(s): K. Konno
  • Journal Title: Adv. St. Pure Math. Volume: 56 Pages: 35-38
• [Journal Article] Triple covers of algebraic surfaces and a genera lization of Zariski's example (2009)
  • Author(s): H. Ishida and H. Tokunaga
  • Journal Title: Adv. Stud. Pure Math. Volume: 56 Pages: 169-185
• [Journal Article] A divisor on the moduli spaces of curves associated to the signature of fibered complex surfaces (2009)
  • Author(s): T.Ashikaga, K.Yoshikawa
  • Journal Title: Adv.St.Pure Math. 56 Pages: 1-34
  • Peer Reviewed
• [Journal Article] K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space III
  • Author(s): K.-I. Yoshikawa
  • Journal Title: Math. Zeit Volume: (to appear)
• [Journal Article] K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space II
  • Author(s): K.-I. Yoshikawa
  • Journal Title: J. Reine Angew. Math. Volume: (to appear)
• [Journal Article] A trinity of the Borcherds Phi-function, Proc. of Infinite Analysis II
  • Author(s): K.-I. Yoshikawa
  • Journal Title: Springer Proc. in Math. Volume: (to appear)
• [Presentation] 解析的捩率とBCOV予想 (2012)
  • Author(s): 吉川謙一
  • Organizer: 日本数学会企画特別講演
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-28
• [Presentation] Extended Zagier recipr ocity for Dedekind sum via cyclic quotient singularities (2012)
  • Author(s): T. Ashikaga
  • Organizer: Topology of Singularities and Related Topics III
  • Place of Presentation: Dalat. VIETNAM
  • Year and Date: 2012-03-26
• [Presentation] Extended Zagier reciprocity for Dedekind sum via cyclic quotient singularities (2012)
  • Author(s): T.Ashikaga
  • Organizer: Topology of Singularities and Related Topics III
  • Place of Presentation: Univ.Dalat, VIETNAM
  • Year and Date: 2012-03-26
• [Presentation] Normal sextic surfaces which are canonical (2012)
  • Author(s): 今野一宏
  • Organizer: 高次元代数多様体とベクトル束の代数幾何学
  • Place of Presentation: 九州大学
  • Year and Date: 2012-03-16
• [Presentation] A note on invariants of hyperelliptic fibrations (2012)
  • Author(s): H. Ishida
  • Organizer: The 4-th Branched coverings, Degenerations and Related Topics
  • Place of Presentation: 広島大学
  • Year and Date: 2012-03-07
• [Presentation] Dedekind和に関するZagier相互律の拡張と巡回商特異点間の双対性 (2012)
  • 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(兵庫県)
  • Year and Date: 2012-02-11
• [Presentation] 被覆を用いた有理曲面上の代数曲線の構成 (2011)
  • Author(s): 石田弘隆
  • Organizer: 代数幾何目白セミナー2011
  • Place of Presentation: 学習院大学
  • Year and Date: 2011-12-27
• [Presentation] An approach to the geography problem of hyperelliptic fibrations (2011)
  • Author(s): H. Ishida
  • Organizer: 第9回代数曲線論シンポジウム
  • Place of Presentation: 首都大学東京
  • Year and Date: 2011-12-11
• [Presentation] Resultants and Borcherds Phi-function, Automorphic forms and moduli spaces (2011)
  • Author(s): K. Yoshikawa
  • Organizer: Centre Internaltional de Rencontres Mathematiques
  • Place of Presentation: Luminy, FRANCE
  • Year and Date: 2011-10-11
• [Presentation] 巡回商特異点に関する高次元連分数とZagier相互律 (2011)
  • Author(s): 足利正
  • Organizer: リーマン面に関する位相幾何学研究集会
  • Place of Presentation: 東京大学
  • Year and Date: 2011-09-04
• [Presentation] 巡回商特異点に関する高次元連分数とZagier相互律 (2011)
  • Author(s): 足利正
  • Organizer: リーマン面に関する位相幾何学研究集会
  • Place of Presentation: 東京大学(東京都)
  • Year and Date: 2011-09-04
• [Presentation] On the value of Borcherds Phi-function, Workshop on Atithmetic and Geometry of K3 surfaces and Calabi-Yau threefolds (2011)
  • Author(s): K. Yoshikawa
  • Organizer: The Fields Institute for research in Mathematical Sciences
  • Place of Presentation: Toronto, CANADA
  • Year and Date: 2011-08-18
• [Presentation] Analytic torsion of certain Calabi-Yau threeloids and Borcherds product, Automorphic forms, Kac-Moody algebras and Strings (2011)
  • Author(s): K. Yoshikawa
  • Organizer: The Max-Planck Institut fur Mathematik
  • Place of Presentation: Bonn, GERMANY
  • Year and Date: 2011-05-11
• [Presentation] Singularities and analytic torsion, Geometric quantization in noncompact setting (2011)
  • Author(s): K. Yoshikawa
  • Organizer: Mathematisches Forschungsinstitut Oberwolfach
  • Place of Presentation: Oberwolfach, GERMANY
  • Year and Date: 2011-02-15
• [Presentation] Higher-dimensional continued linear fractions for cyclic quotient singularities (2011)
  • Author(s): T. Ashikaga
  • Organizer: Workshop on Singularity
  • Place of Presentation: 東京理科大学
  • Year and Date: 2011-01-22
• [Presentation] Higher-dimensional continued linear fractions for cyclic quotient singularities (2011)
  • Author(s): T.Ashikaga
  • Organizer: Workshop on Singularity
  • Place of Presentation: 東京理科大学(東京都)
  • Year and Date: 2011-01-22
• [Presentation] Regular canonical surfaces of degree 3n-3 in P^n, I and II (2011)
  • Author(s): K. Konno
  • Organizer: Two Weeks of Classical Algebraic Geometry
  • Place of Presentation: Universita di Padova, ITALY
• [Presentation] Planar continued linear fraction and three-dimensional quotient singularities (2010)
  • Author(s): 足利正
  • Organizer: Workshop on Lefschetz fibration and category theory
  • Place of Presentation: 大阪大学
  • Year and Date: 2010-06-27
• [Presentation] Planar continued linear fraction and three-dimensional quotient singularities (2010)
  • Author(s): 足利正
  • Organizer: Workshop on Lefschetz fibration and category theory
  • Place of Presentation: 大阪大学(大阪府)
  • Year and Date: 2010-06-27
• [Presentation] Planar continued linear fraction for three-dimensional cyclic quotient singularities (2010)
  • Author(s): T. Ashikaga
  • Organizer: 第二回東北復旦代数幾何交流シンポジューム
  • Place of Presentation: 復日大学, CHINA
  • Year and Date: 2010-06-03
• [Presentation] Planar continued linear fraction for three-dimensional cyclic quotient singularities (2010)
  • Author(s): T.Ashikaga
  • Organizer: 第二回東北復旦代数幾何交流シンボジューム
  • Place of Presentation: 復旦大学,China
  • Year and Date: 2010-06-03
• [Presentation] Towards the signature of complex surfaces and singularities (2010)
  • Author(s): T.Ashikaga
  • Organizer: Mini Workshop on Singularity Theory
  • Place of Presentation: 東京都東京理科大学
  • Year and Date: 2010-03-14
• [Presentation] Toric resolution of a certain non-normal singutarity (2010)
  • Author(s): 足利正
  • Organizer: 山形特異点研究会
  • Place of Presentation: 山形大学
  • Year and Date: 2010-03-02
• [Presentation] Toric resolution of a certain non-normal singularity, (2010)
  • Author(s): T.Ashikaga
  • Organizer: 山形特異点セミナー
  • Place of Presentation: 山形県山形大学
  • Year and Date: 2010-03-02
• [Presentation] Towards resolution of surface singularity via toroidal fiber space (2010)
  • Author(s): T.Ashikaga
  • Organizer: 代数幾何学ミニワークショップ
  • Place of Presentation: 兵庫県西脇市
  • Year and Date: 2010-02-20
• [Presentation] Another geometric proof of the reciprocity law of Dedekind sum (2009)
  • Author(s): 足利正
  • Organizer: 第一回貨東北復旦代数幾何交流シンポジューム
  • Place of Presentation: 東北大学
  • Year and Date: 2009-11-26
• [Presentation] Another geometric proof of the reciprocity law of Dedekind sum (jointly with M.Ishizaka) (2009)
  • Author(s): T.Ashikaga
  • Organizer: 第一回東北復旦代数幾何交流シンポジューム
  • Place of Presentation: 宮城県東北大学
  • Year and Date: 2009-11-26
• [Presentation] On Koyama singularities of fundamental genus 2 (2009)
  • Author(s): 今野一宏
  • Organizer: Inequalities in Algebraic Geometry
  • Place of Presentation: 東京大学
  • Year and Date: 2009-11-11
• [Presentation] Localization of signature of fibered compcx surfaces (2009)
  • Author(s): 足利正
  • Organizer: 2009年度代数幾何学シンポジューム
  • Place of Presentation: 兵庫県城崎大会議館
  • Year and Date: 2009-10-30
• [Presentation] Localization of signature of fibered compex surfaces (2009)
  • Author(s): T.Ashikaga
  • Organizer: 2009年度代数幾何学シンポジューム
  • Place of Presentation: 兵庫県城崎大会議館
  • Year and Date: 2009-10-30
• [Presentation] On Koyama singularities (2009)
  • Author(s): 今野一宏
  • Organizer: Syzygies of Projective Varieties
  • Place of Presentation: 佐賀大学理工学部
  • Year and Date: 2009-09-14
• [Presentation] Signature divisor on moduli space of curves and its application (2009)
  • Author(s): T. Ashikaga
  • Organizer: Franco-Japan symposium on Singularity
  • Place of Presentation: Strasbourg, FRANCE
  • Year and Date: 2009-08-27
• [Presentation] Local signature of fibered surfaces and its application (2009)
  • Author(s): T. Ashikaga
  • Organizer: Conference on Algebraic surfaces and related topics
  • Place of Presentation: Pohang. KOREA
  • Year and Date: 2009-08-19
• [Presentation] Local signature of fibered surfaces and its application (2009)
  • Author(s): T.Ashikaga
  • Organizer: Conference on Algebraic surfaces and related topics
  • Place of Presentation: Pohang, KOREA
  • Year and Date: 2009-08-19
• [Presentation] On local signature and Horikawa index of fibered surfaces (2009)
  • Author(s): 足利正
  • Organizer: Hodge理論と代数幾何学
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-29
• [Presentation] Certain surface singu-larities of general type (2009)
  • Author(s): 今野一宏
  • Organizer: Hodge theory and algebraic geometry
  • Place of Presentation: 京大数理解析研究所
  • Year and Date: 2009-06-29
• [Presentation] On local signature and Horikawa index of fibered surfaces (2009)
  • Author(s): T.Ashikaga
  • Organizer: Hodge理論と代数幾何学
  • Place of Presentation: 京都府京都大学数理解析研究所
  • Year and Date: 2009-06-29
• [Presentation] Analytic torsion and invarianis of K3 and Calabi-Yau manifolds(I),(II),(III), Differential Geometric Methods in Algebraic Geometry (2009)
  • Author(s): K. Yoshikawa
  • Organizer: Tata Institute of Fundamental Research
  • Place of Presentation: Mumbai INDIA
• [Presentation] An Introduction to Borcherds Products(I),(II),(III) (2009)
  • Author(s): K. Yoshikawa
  • Organizer: Holomorphically Symplectic Varleties and Moduli Spaces
  • Place of Presentation: Universit' e Lille 1, Lille, FRANCE