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1999 Fiscal Year Final Research Report Summary

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)
Project Period (FY) 1997 – 1999
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
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.

  • Research Products

    (18 results)

All Other

All Publications (18 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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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
  • [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

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Published: 2001-10-23  

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