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Research of new developments in moduli spaces and integrable systems

Research Project

Project/Area Number 16340009
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) NOUMI Masatoshi  Kobe University, Grad. Sch. of Sci. and Tech, Professor, 自然科学研究科, 教授 (80164672)
YAMADA Yasuhiko  Kobe University, Faculty of Science, Professor, 理学部, 教授 (00202383)
YOSHIOKA Kota  Kobe University, Faculty of Science, Professor, 理学部, 教授 (40274047)
FUKAYA Kenji  Kyoto University, Grad. Sch. of Sci, Professor, 理学研究科, 教授 (30165261)
HOSONO Shinobu  Tokyo University, Grad. Sch. of Math. Science, Associate Professor, 数理科学研究科, 助教授 (60212198)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥16,200,000 (Direct Cost: ¥16,200,000)
Fiscal Year 2006: ¥4,900,000 (Direct Cost: ¥4,900,000)
Fiscal Year 2005: ¥5,000,000 (Direct Cost: ¥5,000,000)
Fiscal Year 2004: ¥6,300,000 (Direct Cost: ¥6,300,000)
KeywordsPainleve equations / Symplectic singularities / Moduli of parabolic connections / Moduli of representations / Riemann-Hilbert correspondences / Irregular singularities / Geometric Langlands correspondence / Mirror symmetry / シンプレクテック特異点解消 / Stokes現象 / ヒッグス場のモジュライ空間
Research Abstract

In the present research, Inaba, Iwasaki and Saito constructed the moduli space of stable parabolic connections of rank 2 over the projective line with at most n-regular singular points and its natural compactification, and showed the smoothness of the moduli space. Moreover, constructing the moduli space of the representations of the fundamental group of the complement of the n-points in projective plane, we showed that Riemann-Hilbert correspondence from the moduli space of connections to the moduli space of the representations is a surjective proper bimeromorphic morphism. This result implies that the non-linear differential equations coming from isomonodromic deformations of connections have Painleve property. Recently Inaba succeeded in generalizing this result to the case of stable parabolic connections with at most regular singularities of any rank over any smooth algebraic curve. We expect further to obtain the same result in the case of connections with irregular singular point … More s. Noumi and Yamada gave a geometric description of elliptic difference Painleve equations with Ohta, Masuda, Kajiwara, and construct the elliptic hypergeometric solutions and its degeneration.
By using the algebro-geometric construction of Painleve VI equations and applying the Riemann-Hilbert correspondence to ergotic theory of algebraic surfaces, Iwasaki and Uehara showed that the dynamics of almost loops of non-linear monodromy of Painleve VI equation are chaotic. Hosono and Doran determined the differential equation of the period integrals of certain class of Calabi-Yau hypersurfaces by oscillatory integrals. Moreover solving the Stokes phenomena associated to the integral, they observed a natural understanding from the view point of Mirror symmetry. Yoshioka and Hiraku Nakajima proved the Nekrasov conjecture for the instanton numbers. Moreover with Lotha Gottsche, they obtained a wall-crossing formula for Donaldson invariants for the rational surfaces, and extend the results. Fukaya, Ono, Ohta, Oh, are completing a big book of Floer theory of Lagrangian submanifolds. Less

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (39 results)

All 2006 2005 2004 Other

All Journal Article (36 results) Book (3 results)

  • [Journal Article] Moduli of Stable Parabolic Connections,Riemann-Hilbert correspondence and Geometry of Painleve equation of type VI, Part I2006

    • Author(s)
      M.Inaba, K.Iwasaki, M.-H.Saito
    • Journal Title

      Publ. Res. Inst. Math. Sci. 42・4

      Pages: 987-1089

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painleve equations of type VI, Part II2006

    • Author(s)
      M.Inaba, K.Iwasaki, M.-H.Saito
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 387-432

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dynamics of the Sixth Painleve Equation2006

    • Author(s)
      M.Inaba, K.Iwasaki, M.-H.Saito
    • Journal Title

      Seminaires et Congre 14

      Pages: 59-123

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Moduli of twisted sheaves on a projective variety2006

    • Author(s)
      K.Yoshioka
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 1-30

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] On deformations of Q-factorial symplectic varieties2006

    • Author(s)
      Y.Namikawa
    • Journal Title

      J. Reine Angew. Math. 599

      Pages: 97-110

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Metric Riemannian geometry.2006

    • Author(s)
      K.Fukaya
    • Journal Title

      Handbook of differential geometry. Vol. II

      Pages: 189-313

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painleve equation of type VI, Part I2006

    • Author(s)
      M.Inaba, K.Iwasaki, M.-H.Saito
    • Journal Title

      Publ. Res. Inst. Math. Sci. 42・4

      Pages: 987-1089

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painleve equation of type VI, Part II2006

    • Author(s)
      M.Inaba, K.Iwasaki, M.-H.Saito
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 387-432

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dynamics of the Sixth Painlev'e Equation2006

    • Author(s)
      M.Inaba, K.Iwasaki, M.-H.Saito
    • Journal Title

      Seminaires et Congre 14

      Pages: 59-123

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Kyoukei-Ba-Riron-nyumon2006

    • Author(s)
      Y.Yamada
    • Journal Title

      Baifu-Kan

      Pages: 265-265

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Moduli of Stable Parabolic Connections,Riemann-Hilbert correspondence and Geometry of Painlevγ'{e} equation of type $V$, Part I,2006

    • Author(s)
      M.Inaba
    • Journal Title

      Publ. Res. Inst. Math. Sci. 42・4

      Pages: 987-1089

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevγ'{e} equations of type VI, Part II2006

    • Author(s)
      M.Inaba
    • Journal Title

      Adv. Stud. Pure Math. 45

      Pages: 387-432

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Dynamics of the Sixth Painlev e Equation2006

    • Author(s)
      M.Inaba
    • Journal Title

      Seminaires et Congre 14

      Pages: 59-123

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Dynamics of the Sixth Painlev'e Equation,2006

    • Author(s)
      M.Inaba
    • Journal Title

      Proceedings of Conference in Angers, 2004, "Seminaires et Congre" of the Societe Mathematique de France(SMF) (To appear)(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Cubic pencils and Painleve Hamiltonians2005

    • Author(s)
      K.Kajiwara, T.Masuda, M.Noumi, Y, Ohta, Y.Yamada
    • Journal Title

      Funkcial. Ekvac. 48

      Pages: 147-160

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] A note on unknotting number, II2005

    • Author(s)
      Y.Nakanishi
    • Journal Title

      Journal of Knot Theory and Its Ramifications 14・1

      Pages: 3-8

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Higher direct images of dualizing sheaves of Lagrangian fibrations2005

    • Author(s)
      D.Matsushita
    • Journal Title

      Amer. J. Math. 127・2

      Pages: 243-259

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Multivalued Morse theory, asymptotic analysis and mirror symmetry.2005

    • Author(s)
      K.Fukaya
    • Journal Title

      Proc. Sympos. Pure Math. 73

      Pages: 205-278

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup. I. 4-dimensional pure gauge theory2005

    • Author(s)
      H.Nakajima, K.Yoshioka
    • Journal Title

      Invent. math. 162・2

      Pages: 313-355

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup. II. K-theoretic partition function2005

    • Author(s)
      H.Nakajima, K.Yoshioka
    • Journal Title

      Transformation Groups 10・3-4

      Pages: 489-519

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Higher direct images of dualizing sheaves of Lagrangian fibration2005

    • Author(s)
      D.Matsushita
    • Journal Title

      Amer. J. Math. 127・2

      Pages: 243-259

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Multivalued Morse theory, asymptotic analysis and mirror symmetry2005

    • Author(s)
      K.Fukaya
    • Journal Title

      Proc. Sympos. Pure Math. 73

      Pages: 205-278

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup. I. 4-dimensional pure gauge theory2005

    • Author(s)
      H.Nakajima, K.Yoshioka
    • Journal Title

      Invent. math. 162-2

      Pages: 313-355

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup. II. K-theoretic partition function2005

    • Author(s)
      H.Nakajima, K.Yoshioka
    • Journal Title

      Transformation Groups 10-3-4

      Pages: 489-519

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Instanton counting on blowup.I.$4$-dimensional pure gauge theory}2005

    • Author(s)
      H.Nakajima
    • Journal Title

      Invent.math. 162, no.2

      Pages: 313-355

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Instanton counting on blowup. II. K-theoretic partition function2005

    • Author(s)
      H.Nakajima
    • Journal Title

      Transformation Groups 10, 3-4

      Pages: 489-519

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Multivalued Morse theory, asymptotic analysis and mirror symmetry2005

    • Author(s)
      K.Fukaya
    • Journal Title

      Proc.Sympos.Pure Math., 73

      Pages: 205-278

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Construction of hypergeometric solutions to the $q$-Painleve equations2005

    • Author(s)
      K.Kajiwara
    • Journal Title

      Int.Math.Res.Notice, 24

      Pages: 1441-1463

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Autoequivalences of derived category of a K3 surface and monodromy transformations.2004

    • Author(s)
      S.Hosono, H.B.Lian, K.Oguiso, S.-T.Yau
    • Journal Title

      J. Algebraic Geom. 13・3

      Pages: 513-545

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Yau Autoequivalences of derived category of a K3 surface and monodromy transformations2004

    • Author(s)
      S.Hosono, H.B.Lian, K.Oguiso, S.-T.
    • Journal Title

      J. Algebraic Geom. 13・3

      Pages: 513-545

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Backlund transformations of the Sixth Painlev'e Equations in Terms of Riemann-Hilbert correspondences2004

    • Author(s)
      M.Inaba
    • Journal Title

      Internat.Math.Res.Notices 1

      Pages: 1-30

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nodal curves and Riccati solutions of Painleve equations2004

    • Author(s)
      M.H.Saito
    • Journal Title

      J.Math.Kyoto Univ. 44/No3

      Pages: 529-568

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Hypergeometric solutions to the q-Painleve equations2004

    • Author(s)
      K.Kajiwara
    • Journal Title

      Internat.Math.Res.Notices 47

      Pages: 2497-2521

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlev'{e} equations of type VI, Part I

    • Author(s)
      M.Inaba
    • Journal Title

      Publ.Res.Inst.Math.Sci (To appear)(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Cubic Pencils and Painlev'e Hamiltonians

    • Author(s)
      JK.Kajiwara
    • Journal Title

      Funkcial.Ekvac. (To appear)(未定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Floer homology and mirror symmetry II

    • Author(s)
      Y.Oh
    • Journal Title

      Adv.Stud.In Pure Math. (To appear)(未定)

    • Related Report
      2004 Annual Research Report
  • [Book] 共形場理論入門2006

    • Author(s)
      山田泰彦
    • Total Pages
      265
    • Publisher
      培風館
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Book] 共形場理論入門2006

    • Author(s)
      山田 泰彦
    • Total Pages
      265
    • Publisher
      培風館
    • Related Report
      2005 Annual Research Report
  • [Book] Painlev'e equations through symmetry, Translations of Mathematical Monographs2004

    • Author(s)
      M.Noumi
    • Total Pages
      156
    • Publisher
      American Mathematical Society
    • Related Report
      2004 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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