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Research on Periods of Algebraic Varieties and Hypergeometric Functions

Research Project

Project/Area Number 09440015
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKobe University

Principal Investigator

SAITO Masa-hiko  Kobe University, Faculty of Science, Professor, 理学部, 教授 (80183044)

Co-Investigator(Kenkyū-buntansha) TAKANO Kyoichi  Kobe University, Faculty of Science, Professor, 理学部, 教授 (10011678)
SASAKI Takeshi  Kobe University, Faculty of Science, Professor, 理学部, 教授 (00022682)
NOUMI Masatoshi  Kobe University, Graduate School of Science and Technology, Professor, 自然科学研究科, 教授 (80164672)
YOSHIOKA Kota  Kobe University, Faculty of Science, Associate Professor, 理学部, 助教授 (40274047)
TAKAYAMA Nobuki  Kobe University, Faculty of Science, Professor, 理学部, 教授 (30188099)
山田 泰彦  神戸大学, 理学部, 助教授 (00202383)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥12,900,000 (Direct Cost: ¥12,900,000)
Fiscal Year 1999: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 1998: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 1997: ¥5,800,000 (Direct Cost: ¥5,800,000)
KeywordsMirror Symmetry / GKZ Hypergeometric System / Calabi-Yau manifolds / Gromov-Witten invariants / Rational Elliptic Surfaces / Painleve equations / Spaces of Initial Values / Moduli spaces of Vector Bundles / Mirror 対称性 / GKZ 超幾何系 / カラビ・ヤウ多様体 / グロモフ・ウイッテン不変量 / プレポテンシャル / ベクトル波 / 化積分系 / ミラー対称性 / 湯川カップリング / GK2型超幾何微分方程式系 / カラビ=ヤウ多様体
Research Abstract

During the period of the project, we have investigated the following subjects and obtained the following results. (i)Mirror Symmetry Conjecture for Calabi-Yau manifolds, (ii)Counting Curves of higher genus in Rational Elliptic Surfaces, (iii)Lie theoretic aspects on Painleve equations, Algebro-geometric aspects on Painleve equations, GKZ hypergeometric systems and their Grobner deformations. As for (i), we have been studying Gromov-Witteninvariants for certain Calabi-Yau 3-folds and computed a part of A-model prepotentials by means of the theta function of the EィイD28ィエD2-lattice while their B-model prepotential had already calculated by GKZ hypergometric series. As a result, we have checked MSC mathematically for those cases. Developing further, in (ii)we have investigated counting problems of curves of higher genus in rational elliptic surfaces. We propose the holomorphic anomaly equation(HAE), which the prepotential should satisfy. By using the Jacobi's triple product formula and the relative Lefschetz decomposition, we have checked the prepotential satisfies the HAE.
As for the studies of Painleve equations ((iii)), our group have been developing Lie theoretic approach and algeblo-geometric approach, which clarify the relations among Painleve equations, the symmetry of Affine Weyl groups and the geometry of rational surfaces.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (39 results)

All Other

All Publications (39 results)

  • [Publications] M. -H. Saito: "Prepotentials of Yukawa couplings of certain Calabi-Ysu 3-folds and Mirror Symmetry"to appear in the Proceedings of Nato Advance Study Institute, CRM Summer School, Banff. (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S. Hosono, Saito & Takahashi: "Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface"Adv. Theor. Math. Phys.. 3. 177-208 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Noumi and Y. Yamada: "Affine Weyl groups, discrete dynamical systems and Painleve equations"Commun. Math Phys.. 199. 281-295 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Noumi and Y. Yamada: "Symmetries in the fourth Painleve equation and Okamoto polynomials"Nagoya Math. J.. 153. 53-86 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sasaki and M. Yoshida: "The uniformizing differential equitation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc. Japan Acad. Ser. A.. 75. 129-133 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takano, K.: "Defining manifolds for Painleve equations""Toward the exact WKB analysis of differential equations, linear and nonlinear", (2000). 261-133 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K. Yoshida: "Euler characteristics of SU (2) instanton moduli spaces on ration elliptic surfaces"Commun. Math. Physics. 205. 501-517 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Saito, B. Sturmfels and N. Takayama: "Grobner deformations of hypergeometric differential equations"Springer Werlag. (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.-H.Saito: "Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and Mirror Symmetry"Proceedings of Nato Advanced Study Institute, CRM Summer School, Banff. (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] S.Hosono, Saito & Takahashi: "Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface"Adv. Theor. Math. Phys.. 3. 177-208 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Noumi and Y.Yamada: "Affine Weyl groups, discrete dynamical systems and Painleve equations"Commun. Math. Phys.. 199. 281-295 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Noumi and Y.Yamada: "Symmetries in the fourth Painleve equation and Okamoto polynomials"Nagoya Math. J.. 153. 53-86 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Sasaki and M.Yoshida: "The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc. Japan Acad. Ser. A. 75. 129-133 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Takano,K.: "Defining manifolds for Painleve equations"Toward the exact WKB analysis of differential equations, linear and nonlinear. 261-269 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] K.Yoshioka: "Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces"Commun. Math. Physics. 205. 501-517 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] B.Sturmfels & N.Takayama: "Grobner bases and hypergeometric functions"London Mathematical Society Lecture Notes Series. 251. 246-258 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Saito & N.Takayama et al.: "Hypergeometric polynomials and Integer Programming"Compositio Math.. 115. 185-204 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Saito, B.Sturmfels and N.Takayama: "Grobner deformations of hypergeometric differential equations"Springer Verlag. (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.-H.Saito: "Prepotentials of Yukawa couplings of certain Calabi-Yau 3-folds and Mirror Symmetry"to appear in the Proceedings of Nato Advanced Study Institute,CRM Summer School,Banff. (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Hosono: "Holomorphic Anomaly Equation and BPS State Counting of Rational Elliptic Surface"Adv.Theor.Math.Phys.. 3. 177-208 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Noumi: "Affine Weyl groups,discrete dynamical systems and Painleve equations,"Commun.Math.Phys.. 199. 281-295 (1998)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Noumi: "Symmetries in the fourth Painleve equation and Okamoto polynomials"Nagoya Math.J.. 153. 53-86 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Sasaki: "The uniformizing diffferential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces"Proc.Japan Acad.Ser.A. 75. 129-133 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Takano,: "Defining manifolds for Painleve equations""Toward the exact WKB analysis of differential equations,linear and nonlinear",(2000). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Yoshida: "Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces"Commun.Math.Physics. 205. 501-517 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Saito,B.Sturmfels and N.Takayama: "Grobner deformations of hypergeometric differential equations"Springer Verlag. (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S.Hosono: "On the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-foids" Proceedings of Taniguchi Symposium 97,“Integrable Systems あんd Algebraic Geometry",Kyoto,World Scientific,eds.by M.-H.Saito,Y.Shimizu,K.Ueno. 194-235 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Ogata: "Signature defects and eta functicons of degenerations of abelian varieties" Japan.J.Math. 23. 319-364 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Nounmi: "Affine Weyl groups,discrete dynamical systems and Painleve equations" Comm.Math.Phys.199. 281-295 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Noumi: "Affine Hecke algebras and Macdonald Polynomials" Progress in Mathematics,Birkhauser. 160. 365-377 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] F.Dillen: "The Classification of projectively homogeneous surfaces II" Osaka J.Math. 35. 117-146 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Yoshioka: "Euler characteristics of SU (2) instanton moduli spaces on rational elliptic surfaces" Comm.Math.Phys.(発表予定). (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Saito: "Grobner deformations of hypergeometric differential equations" Springer verlag (印刷中), (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Shinobu Hosono Masa-Hiko Saito,Jan Stionstra: "On the Mirror Symmetry conjecture for Schoen′s Calabi-Yan 3-folds" Proceedings of Taniguchi Symposium, Integrable Systems and Aljelraie Geometory. (発表予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] K. Mimachi, M. Noumi: "An integral representation of eigenfunctions for Macdonald′s q-difference operator" Tohoku Math. J.49. 517-525 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Takeshi Sasaki, et al: "On the rigidity of differential systems modelled on hermition symmetory spaces and disproofs of a conjecture concerning modular interpretation of config space" Advanced studies in Pure. Math.25. 318-354 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kyoichi Takano, et al: "On some Hamiltonian Structures of Painleve Systems I" Funkcial. Ekvac.40. 271-291 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K. Nakamura, Y. Nakanishi, Y. Uchida: "Delta-unknotting numbers for knots" Knot Theory Its. Ram.(発売予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Yoshioka Kota: "A note on universal family of moduli of stable sheaves" J. reine. angew. Math.(発売予定). (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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