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Algebraic and Geometric Study on the Structure of Moduli Spaces

Research Project

Project/Area Number 05452003
Research Category

Grant-in-Aid for General Scientific Research (B)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

MARUYAMA Masaki  Fac.of Science, Kyoto Univ., Professor, 理学部, 教授 (50025459)

Co-Investigator(Kenkyū-buntansha) KONO Akira  Fac.of Science, Kyoto Univ., Professor, 理学部, 教授 (00093237)
NISHIDA Goro  Fac.of Science, Kyoto Univ., Professor, 理学部, 教授 (00027377)
YOSHIDA Hiroyuki  Fac.of Science, Kyoto Univ., Professor, 理学部, 教授 (40108973)
UENO Kenji  Fac.of Science, Kyoto Univ., Professor, 理学部, 教授 (40011655)
HIJIKATA Hiroaki  Fac.of Science, Kyoto Univ., Professor, 理学部, 教授 (00025298)
齋藤 政彦  京都大学, 理学部, 助教授 (80183044)
Project Period (FY) 1993 – 1994
Project Status Completed (Fiscal Year 1994)
Budget Amount *help
¥6,700,000 (Direct Cost: ¥6,700,000)
Fiscal Year 1994: ¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 1993: ¥4,600,000 (Direct Cost: ¥4,600,000)
KeywordsModuli / Vector Bundle / Stable Sheaf / Hermit-Einstein connection / Instanton / Parabolic Stable Sheaf / Betti Number / Projective Plane / 位相構造 / 代数多様体 / Weil予想
Research Abstract

Moduli in geometry is a set of geometric objects endowed with the universal geometric structure. It is known that not only a moduli space itself is a rich geometric object but also it is often a useful tool in studying geometry. For example the moduli space of Hermit-Nesting connections reflects strongly the differential geometric nature of the base manifold. On the other hand, as the fact that a Hermit-Nesting connection is nothing but a stable vector bundle shows us, we realize that moduli spaces constructed independently in different fields sometimes coincide with each other. This has been promoting direction of studying the theory of moduli spaces from various viewpoints. In this project we have carried out our study on classifying spaces, moduli of various connections and moduli of vector bundles, in cooperation with the specialists of topology, differential geometry, number theory, algebraic geometry and commutative algebra in our department. We have got the following results.
1.We completed the computation of Betty numbers of the moduli spaces of stable sheaves of rank 2 on the projective plane and furthermore we got a similar results on ruled surfaces.
2.We could clarify an interesting relationship between parabolic stable vector bundles on the projective plane and instantaneous. Using this we could prove that the moduli spaces of instantaneous are connected.
3.The standard of the moduli spaces of stable sheaves of rank 2 the on projective plane are dominated by those of the moduli spaces of parabolic stable vector bundles. They are related under a generalization of the elementary transformation of vector bundles.
4.We could develop deep study of reflexive sheaves on surfaces with rational double points and their deformations.
5.We applied our results on vector bundles to the theory of conformal field theory.

Report

(3 results)
  • 1994 Annual Research Report   Final Research Report Summary
  • 1993 Annual Research Report
  • Research Products

    (25 results)

All Other

All Publications (25 results)

  • [Publications] 丸山正樹: "Instantons and parabolic sheaves" Proc.Internat.Collog.on Geomerty and Analysisに掲載予定.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] 河野明: "A topological proof of Bott periodicity thaeum and characterization of BU" J.of Math.Kyoto Univ.34. 873-880 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] 齋藤政彦: "On Modell-Weil lattries of higher genus fibrations on rational surfaces" J.of Math.Kyoto Univ.34. 859-872 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] 上野健爾: "On conformal field theory" Proc.of Durham Symp in Vector Bumdlesに掲載予定.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] 吉岡康太: "The Betti numbers of the moduli spaces of stoble sheaves of rank2 on P^2" J reine angew.Math. 453. 193-220 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] 土方弘明: "Bass orders in non semisimple algelras" J.of Math.Kyoto Univ.34. 797-838 (1994)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Masaki MARUYAMA: "Instantons and parabolic sheaves" Proc.Internet. Collog on Geometry and Analysis. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Akira KONO: "A topological proof of Botl periodicity theorem and characterization of BU" J.of Math.Kyoto Univ.Vol.34. 873-880 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Masahiko SAITO: "On Modell-Weil lattices of higher genus fibrations on rational surfaces" J.of Math.Kyoto Univ.Vol.34. 859-872 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Kenji UENO: "On conformal field theory" Proc.of Durham Symp on Vector Bundles. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Kota YOSHIOKA: "The Betli numbers of moduli spaces of stable sheaves of rank 2 on IP^2" J.reine angew. Math. Vol.453. 193-220 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] Hiroaki HIJIKATA: "Bass orders in non se misimple algebras" J.of Math.Kyoto Univ.Vol.34. 797-838 (1994)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1994 Final Research Report Summary
  • [Publications] 丸山正樹: "Instanton and parabolic aheaoes" Proce Intenet.Collog on Gemety and Anoly is に掲載予定. (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 土方弘明: "Bassorcler in nn seuinsuiyle algebas" J.Math Kyoto Univ. 34. 797-838 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 河野 明: "Morse functuin and attaching wop" J.Math Kyoto univ.に掲載予定. 35. (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 上野健爾: "On confovwal field thery" Proc. of Durhau Symp an Vecter Bumcllesに掲載予定.

    • Related Report
      1994 Annual Research Report
  • [Publications] 吉岡康太: "The Bettinumolers of the woduliyracecf stable sheaves of rauk 2 in IP^2" J.reine angew.Math.453. 193-220 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 斎藤政彦: "On Moclel-Weil lattices of hiyher genus filratiun ratinel surfaus" J.Math.Kyato Univ.34. 859-872 (1994)

    • Related Report
      1994 Annual Research Report
  • [Publications] 上野健爾: "代数幾何入門" 岩波書店, 342 (1995)

    • Related Report
      1994 Annual Research Report
  • [Publications] 丸山正樹: "Instantons and parabolic sheaves" Proceedings of Intern'l Colloq.,Geometry and Analysis,TIFR. (掲載予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] 吉田敬之: "On the zeta functions of Shimura varieties and period of Hilbert modular forms" Duke Math.J.(掲載予定).

    • Related Report
      1993 Annual Research Report
  • [Publications] 齋藤政彦: "Finiteness of Modell-Weil groups of Kuga spaces of abelian varieties" Publ.RIMS. 29. 29-62 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 吉岡康太: "The Betti numbers of the moduli space of stable sheaves of rank 2 on P^2" J.fur reine und angew.Mathmatik.

    • Related Report
      1993 Annual Research Report
  • [Publications] 河野明,小島一元: "The adojoint action of a Lie group on the space of loops" J.Math.Soc.Japan. 45. 495-510 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 田邊理正: "Remarks on the elliptic cohomology of finite groups" J.of Math.of Kyoto Univ.

    • Related Report
      1993 Annual Research Report

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Published: 1993-04-01   Modified: 2016-04-21  

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