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Singularities on complex varieties

Research Project

Project/Area Number 09640016
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionTOKYO INSTITUTE OF TECHNOLOGY

Principal Investigator

ISHII Shihoko  Guraduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科数学専攻, 教授 (60202933)

Co-Investigator(Kenkyū-buntansha) SAITO Shuji  Guraduate School of Science and Engineering, Tokyo Institute of Technology, Prof, 大学院・理工学研究科, 教授 (50153804)
TSUJI Hajime  Guraduate School of Science and Engineering, Tokyo Institute of Technology, Assi, 大学院・理工学研究科, 助教授 (30172000)
KUROKAWA Nobusige  Guraduate School of Science and Engineering, Tokyo Institute of Technology, Prof, 大学院・理工学研究科, 教授 (70114866)
KOBAYASHI Masanori  Guraduate School of Science and Engineering, Tokyo Institute of Technology, Assi, 大学院・理工学研究科, 助手 (60234845)
FUJITA Takao  Guraduate School of Science and Engineering, Tokyo Institute of Technology, Prof, 大学院・理工学研究科数学専攻, 教授 (40092324)
福田 拓生  東京工業大学, 理学部, 教授 (00009599)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 1998: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1997: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordssingularities / toric variety / algebraic surfaces / 超曲面 / 極小モデル / 偏極多様体 / blow up
Research Abstract

In the Minimal Model Conjecture, it is conjectured that there exist the canonical, minimal and log-canonical models for a singularity on an algebraic variety. This is proved for 2 and 3-dimensional cases, however not proved for the higher dimensional case. In this project we proved that there exist the canonical, minimal and log-canonical models for a non-degenerate hypersurface singularity. We then obtained the globalized statement : a DELTA-regular hypersurface of a complete toric variety has the canonical and minimal models. We also got the algorithm to construct these models.
We proved that the invariant -K^2 for a normal surface singularity attains the minimum value 1/3 and these set has no accumulation point from above. This implies that the procedure to construct a new singularity from a singularity stops at finite stage, if the value of -K^2 goes strictly down by this procedure. On the other hand, we prove that there exist accumulation points from below and the accumulation poins are all rational number. And all positive integers are accumulation points.
We prove a generalization of Kawachi-Masek's result on base points of adjoint bundles on a normal surface to the case with a boundary.
We proved that if the number of branches of real analytic curve, then the curve is blow-analytically equivalent to a non-singular point.
We construct a special Lagrangian 3-torus.
We study zeta-function on categories. In particular we study the distribution of the Laplace operator on categories.

Report

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

    (34 results)

All Other

All Publications (34 results)

  • [Publications] 石井志保子: "Minimal, canonical and log^- canonical models of hypersurface singularities" Contemporary Math.207. 63-77 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Chen, 石井志保子: "On-K^2 for normal surface singularities" International J.of Math.9. 653-668 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 石井志保子: "The minimal model theorem for divisors of toic varieties" to appear in Tohoku Math.J.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 小林正典, Kuo: "Uniqueness of unibranclud curve in IR^2 up to simple blowing up" Proc.Japan Acad.73. 97-99 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 小林正典: "Real analytic algebraic singularities" Pitman Research Notes in Math.381. 30-37 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 黒川信重: "量子空間とRiemann予想" 日本物理学会誌. 53. 696-699 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 藤田隆夫: "An appendix to Kawachi-Masek's paper on global generation of adjoint bundles on normal surfaces" J.Alg.Geom.7. 251-252 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 石井志保子: "特異点入門" シュプリンガーフェアラーク東京, 263 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 黒川信重, 栗原将人, 斉藤毅: "数論 3" 岩波書店, 220 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shihoko ISHII: "The weighted blowing ups of singularities with kappa=0." Studies in Adv.Math.5. 75-82 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shihoko ISHII: "Minimal, canonical and log-canonical models of hypersurface singularities." Contemporary Math.207. 63-77 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shihoko ISHII: "Minimal model theorem for divisors of toric varieties" Tohoku J.Math.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shihoko ISHII(with Hao Chen): "On-K^2 for normal surface singularities." Int.ternational J.Math.9, No.6. 653-668 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Masanori KOBAYASHI,(with Tzee-Char Kuo): "Uniqueness of unibranched curve in R^2 up to simple blowings up" Proc .Japan Acad.Ser.A. 73. 97-99 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Masanori KOBAYASHI,(with Tzee-Char Kuo): "On blow-analytic equivalence of embedded curve singularities" Real analytic and algebraic singularitie Pitman Research Notes in A.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Masanori KOBAYASHI: "Special Lagrangian 3-torus as a real slice" Integrable Systems and Algebraic Geometry :. Proceedings of the Taniguchi Symposium, World Scientific. (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shuji SAITO: "Motives and filtrations on Chow groups." Nato ASI/1998 CRM The arithmetic and geometry of algebraic cycles.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shuji SAITO: "Motives, algebraic cycles and Hodge theory." Nato ASI/1998 CRM The arithmetic and geometry of algebraic cycles.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Shuji SAITO, (with U.Jamusen): "Class field theory for varieties over local fields." J.reine angew Math.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] 石井志保子: "Minimal, canonical and log-canonical models of hypersurface singularities" Contemporary Math.207. 63-77 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井志保子,H.Chen: "On-K^2 for normal surface singularities" International Journal of Math.9. 653-668 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井志保子: "The minimal model theorem for divisors of toric varieties" to appear in Tohoku Math.Journal.

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林正典,Kuo: "Uniqueness of unibranched curve in IR^2 up to simple blowingo up" Proc.Japan Acad.73. 97-99 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 小林正典: "Real analytic algebraic singularities" Pitman Research Notes in Math.381. 30-37 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 黒川信重: "量子空間とRiemann予想" 日本物理学会誌. 53. 696-699 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 藤田隆夫: "An appondix to Kawachi-Maseko paper on global generation of adjoint bundles on normal surfaces" J.of Alg.Geom. 7. 251-252 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井志保子: "特異点入門" シュプリンガー・フェアラーク東京, 263 (1997)

    • Related Report
      1998 Annual Research Report
  • [Publications] 黒川信重,栗原将人,斉藤毅: "数論3" 岩波書店, 220 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 石井志保子: "The Weighted blowingups of singularities with K=0" Studies in Adv.Math.5,1997,75-82. 5. 75-82 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 石井志保子: "Mirimal,caporical and log-canonical motrls of hyruasurfa or siugularitie" Cauteca porary Math.207. 63-77 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.Uren, 石井志保子: "On-K^2 fo normal surface singnlarities" Interuatinal Tornmal of Math. (携載予定).

    • Related Report
      1997 Annual Research Report
  • [Publications] 藤田隆夫: "On Kadaira energy and adjoint reduction of polarized threrldd" manuscripta math. 94. 211-229 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 小林正典: "Uniguepess of unibranched curve in lR^2 up to simple blowing ups." Proc.Japan Acad.73. 97-99 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 小林正典: "On blow-analyti egrivalence of embedded curve singularities" Pitman Research Notes in Math.Series. 381. 30-37

    • Related Report
      1997 Annual Research Report

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

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