代数多様体の変形に付随する幾何学と解析学

• Project/Area Number: 20540086
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kumamoto University
• Principal Investigator: 田邊 晉 (田邊 晋) 熊本大学, 大學院・自然科学研究科, 教授 (90432997)
• Co-Investigator(Kenkyū-buntansha): 福井 敏純 埼玉大学, 理工学研究科, 教授 (90218892) ; 植田 一石 大阪大学, 理学研究科, 助教 (60432465)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 幾何学 / 解析学

Research Abstract

今年度は代数的超曲面から出発して空間内をホイヘンスの原理に従って伝播する波面の幾何学に関して研究を行った。こうした波面が自由因子乃至は概自由因子を与えるための十分条件を書きだした。具体的に概自由因子を与えるような波面を構成するのはかなり困難なことではあるが、こうした因子の計算例もいくつか示した。
これら波面の幾何学の研究過程において従順多項式といわれる多項式からなる族の変形理論を考察する必要が生じ、変形多項式に対する大域的なガウス・マニン系や判別集合の構造なども調べた。
重み付き射影空間内の完全交叉多様体に対するミラー対称の研究において、超幾何方程式の解のうち周期積分としての意味づけが必ずしも出来ないようなものがミラー対称多様体の量子コホモロジーとして解釈されることが判明した。これはGivental'-Coates, Corti-Golyshevなど先行する研究と我々の結果の関係を明らかにする重要な一歩であるといえる。
上記の研究結果のうち波面の幾何学に関するものはすでに雑誌に投稿し、ミラー対称に関わるものも近日中に論文を完成させて投稿する予定である.

Research Products

• [Journal Article] (概)自由因子を定める大波面について (2010)
  • Author(s): 田邊晋
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1707 Pages: 76-93
• [Journal Article] Systems of uniformization equations along Saito free divisors and related topics. (2010)
  • Author(s): 関口次郎
  • Journal Title: Proceeding of Centre for Mathematics and its Applications, Australian National University Pages: 102-125
  • Peer Reviewed
• [Journal Article] Systems of uniformization equations related with dihedral groups. (2010)
  • Author(s): 関口次郎
  • Journal Title: Kumamoto J.Math. Volume: 23 Pages: 7-26
  • Peer Reviewed
• [Journal Article] Reflection subgroups of the monodromy groups of Appell's F4. (2010)
  • Author(s): 関口次郎
  • Journal Title: Kyushu J.Math. Volume: 64 Pages: 281-296
  • Peer Reviewed
• [Journal Article] Tame nonsmooth inverse mapping theorem (2010)
  • Author(s): 福井敏純
  • Journal Title: SIAM J.Optim. 20(3) Pages: 1573-1590
  • Peer Reviewed
• [Journal Article] Hypergeometric groups for complete intersections associated to Calabi-Yau varieties in weighted projective spaces (2009)
  • Author(s): 田邊晋
  • Journal Title: 京都大学数理解析研究所講究録 1662 Pages: 158-175
• [Journal Article] Wave front propagation and the discriminant of a tame polynomial (2009)
  • Author(s): 田邊晋
  • Journal Title: 京都大学数理解析研究所講究録 1664 Pages: 1-19
• [Journal Article] Logarithmic vector fields along smooth divisors in projective spaces (2009)
  • Author(s): 植田一石
  • Journal Title: Hokkaido Math.J. 38 Pages: 409-415
  • Peer Reviewed
• [Journal Article] Saito free divisors and integrable connections (2009)
  • Author(s): 関口次郎
  • Journal Title: 京都大学数理解析研究所講究録 1662 Pages: 102-135
  • Peer Reviewed
• [Journal Article] A classification of weighted homogeneous Saito free divisors (2009)
  • Author(s): 関口次郎
  • Journal Title: J.Math.Soc.Japan 61 Pages: 1071-1095
  • Peer Reviewed
• [Journal Article] Reflection groups of rank three and systems of uniformization equations (2009)
  • Author(s): 関口次郎
  • Journal Title: Belgorod State University, Scientific Bulletin, Mathematics and Physics 13(68) Pages: 77-88
  • Peer Reviewed
• [Journal Article] Triangulated categories of Gorenstein cyclic quotient singularities (2008)
  • Author(s): 植田一石
  • Journal Title: Proc. of the Amer. Math. Soc. 136 Pages: 2745-2747
  • Peer Reviewed
• [Journal Article] On moduli spaces of quiver representations associated with dimer models (2008)
  • Author(s): 植田一石
  • Journal Title: 京都大学数理解析研究所講究録別冊 B9 Pages: 127-141
  • Peer Reviewed
• [Journal Article] Logarithmic vector fields along smooth plane cubic curves (2008)
  • Author(s): 植田一石
  • Journal Title: Kutnamoto Journal of Mathematics 21 Pages: 11-20
  • NAID: 110009930082
  • Peer Reviewed
• [Journal Article] Three dimensional Saito free divisors and deformations of singular curves (2008)
  • Author(s): 関口次郎
  • Journal Title: J. Siberian Federal Univ., Mathematicas and Physics 1 Pages: 33-41
  • Peer Reviewed
• [Journal Article] 特異点論の入り口ご紹介 (2008)
  • Author(s): 福井 敏純
  • Journal Title: 数学セミナー 5 Pages: 26-27
• [Presentation] Saito free divisors, uniformization equations and related topics (2010)
  • Author(s): 関口次郎
  • Organizer: Commutative algebra and algebraic geometry.
  • Place of Presentation: Feza Gursey Institute, Istanbul, Turkey
  • Year and Date: 2010-09-24
• [Presentation] Dimer models, exceptional collections, and non-commutative crepant resolutions (2010)
  • Author(s): 植田一石
  • Organizer: Test problems for the theory of finite dimensional algebras
  • Place of Presentation: Banff International Research Station, Canada
  • Year and Date: 2010-09-15
• [Presentation] (概)自由因子を定める大波面について (2010)
  • Author(s): 田邊晋
  • Organizer: 埼玉大学数学教室談話会
  • Place of Presentation: 埼玉大学理学部
  • Year and Date: 2010-07-02
• [Presentation] Homological mirror symmetry for singularities (2010)
  • Author(s): 植田一石
  • Organizer: A mini-conference in geometry
  • Place of Presentation: Chinese University of Hong Kong
  • Year and Date: 2010-04-29
• [Presentation] 完全交叉孤立特異点に付随した Gauss-Manin 系と齋藤自由因子 (2010)
  • Author(s): 田邊晋
  • Organizer: Workshop on Accessory Parameters at Kumamoto
  • Place of Presentation: 熊本大学理学部
  • Year and Date: 2010-03-17
• [Presentation] 3次元の重み付き齋藤自由因子について (2010)
  • Author(s): 関口次郎
  • Organizer: Workshop on Accessory Parameters at Kumamoto
  • Place of Presentation: 熊本大学理学部
  • Year and Date: 2010-03-17
• [Presentation] On Horn-Kapranov uniformisation of the discriminantal loci (2010)
  • Author(s): 田邊晋
  • Organizer: Application of the singularity theory to the computer sciences
  • Place of Presentation: Univ. of Athens, Greece
  • Year and Date: 2010-03-10
• [Presentation] Lectures on dimer models and homological mirror symmetry (2010)
  • Author(s): 植田一石
  • Organizer: Summer School on Mirror Symmetry
  • Place of Presentation: Korea Institute for Advanced Study
• [Presentation] (概)自由因子を定める大波面について (2009)
  • Author(s): 田邊晋
  • Organizer: 可微分写像の特異点論とそれに関連する幾何学
  • Place of Presentation: 日本大学文理学部
  • Year and Date: 2009-12-08
• [Presentation] Systems of uniformization equations and hyperelliptic integrals (2009)
  • Author(s): 関口次郎
  • Organizer: Complex analysis and its applications
  • Place of Presentation: Institute of control sciences, Moscow, Russia
  • Year and Date: 2009-10-22
• [Presentation] 一般超幾何方程式の可約な大域モノドロミー表現 (2009)
  • Author(s): 田邊晋
  • Organizer: Workshop on Accessory Parametersat Kumamoto
  • Place of Presentation: 熊本大学理学部
  • Year and Date: 2009-02-24
• [Presentation] Global Monodromy group of an aff ine complete intersectionassociated to a Clabi-Yau variety (2009)
  • Author(s): 田邊晋
  • Organizer: 微分方程式のモノドロミーをめぐる諸問題
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-02-05
• [Presentation] Saito free divisors and integrabie connections (2009)
  • Author(s): 関口次郎
  • Organizer: 微分方程式のモノドロミーをめぐる諸問題
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-02-03
• [Presentation] Unitary reflection groups and uniformization differentialequations (2009)
  • Author(s): 関口次郎
  • Organizer: 「微分方程式と対称空間」大島利雄先生還暦記念研究集会
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2009-01-15
• [Presentation] Monodromy of isolated hypersurface singularities I, II (2009)
  • Author(s): 田辺晋
  • Organizer: Mathematik Bolumu seminer
  • Place of Presentation: Galatasaray University Istanbul, Turkey
• [Presentation] Wave front propagation and the discriminantal loci of a complete intersection singularity (2008)
  • Author(s): 田邊晋
  • Organizer: 微分方程式と微分幾何学への応用特異点論
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2008-12-08
• [Presentation] Monodromy groups for complete intersections related to Calabi Yau complete intersections in projective spaces (2008)
  • Author(s): 田邊晋
  • Organizer: Analysis and Geometry on complex algebraic varieties
  • Place of Presentation: Independent Univ. of Moscow, Moscow, Russia
  • Year and Date: 2008-09-25
• [Presentation] Unitary reflection groups of rank three and Saito Free Divisors (2008)
  • Author(s): 関口次郎
  • Organizer: Analysis and Geometry on complex algebraicvarieties
  • Place of Presentation: Independent Univ. of Moscow, Moscow, Russia
  • Year and Date: 2008-09-24
• [Presentation] 関数の特異点の判定とその微分幾何への応用 (2008)
  • Author(s): 福井 敏純
  • Organizer: 応用特異点論の現状と展望
  • Place of Presentation: 北海道大学数学連携研究センター
  • Year and Date: 2008-03-17