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Number theory of algebraic varieties

Research Project

Project/Area Number 03452003
Research Category

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

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionUniversity of Tokyo

Principal Investigator

KAWAMATA Yujiro (1992)  Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (90126037)

加藤 和也 (1991)  東京大学, 理学部, 教授 (90111450)

Co-Investigator(Kenkyū-buntansha) KUROKAWA Nobushige  Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (70114866)
SUNADA Toshikazu  Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 教授 (20022741)
NAKAMURA Hiroaki  Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助手 (60217883)
NAKAYAMA Noboru  Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (10189079)
SAITO Takeshi  Univ.of Tokyo,Dept.of Math.Sciences, 大学院数理科学研究科, 助教授 (70201506)
川又 雄二郎  東京大学, 理学部, 教授 (90126037)
Project Period (FY) 1991 – 1992
Project Status Completed (Fiscal Year 1992)
Budget Amount *help
¥6,200,000 (Direct Cost: ¥6,200,000)
Fiscal Year 1992: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 1991: ¥3,400,000 (Direct Cost: ¥3,400,000)
Keywordsalgebraic varieties / number theory / semistable reduction / minimal model / zeta function / log rithmic structure / Hecke character / Galois representation / 数論的代数幾何 / ゼ-タ関数 / P進Hodge理論 / 岩沢理論 / 形式群 / 局所定数 / 基本群
Research Abstract

The purpose of this research was to investigate the number theory of algebraic varieties defined over a ring of algebraic integers. In order to start this investigation, it is important to replace the originalvariety by a more natural model by a birational transformation. If the given variety has relative dimension 1 over the ring of integers, then the classical minimal model theory provides us the canonical model. Kawamata tried to extend the minimal model theory to higher dimensional case, and succeeded in the case in which the relative dimension is 2 and the variety has semistable reduction.
In the course of the proof, newly developed theory of algebraic 3-folds over the complex numbers was used. The difficulty in the proof came from the fact that the vanishing theorem of Kodaira type, which was very useful in the case over the complex numbers, is false in positive characteristic.
The singular fiber of a variety with semistable reduction is a normal crossing variety. Conversely, Kawam … More ata considered the smoothing of normal crossing variety into a variety with semistable reduction, and developed the theory of logarithmic deformations with Yoshinori Namikawa at Sophia University. In particular, they proved the existence of a smoothing of a degenerate Calabi-Yau variety.
The cohomology theory is an important tool in the investigaition of algebraic varieties. Saito investigated the 1 dimensional Galois representations on the determinant of L-adic cohomology groups. In the case of constant coefficients, he obtained the description of the corresponding quadratic extensions. In the case of variable coefficients, he proved that they are described by the algebraic Hecke characters determined by the Jacobi sums.
The zeta functions an analytic object which is attached to an algebraic variety over the ring of integers. There are several mysterious conjectures connecting the zeta functions and the number theory of algebraic varieties. Kurokawa investigated multiple zeta funcitons and multiple trigonometric functions, and found formulas of the Gamma factor of the Selberg zeta functions and of the special values of the zeta functions. Less

Report

(3 results)
  • 1992 Annual Research Report   Final Research Report Summary
  • 1991 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] Y.Kawamata: "Abundance theorem for minimal threefolds" Invent.Math.108. 229-246 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Termination of log-flips for algebraic 3-folds" Intl.J.Math.3. 653-659 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Log canonical model of a log minimal model" Intl.J.Math.3. 351-357 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Vnobstructed deformations-a remark on a paper of Z.Ran" J.Alg.Geom.1. 183-190 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Rational curves and classification of algebraic varieties,en Essays on Mirror Manifolds,ed.S-T.Yau" International Press,Hong Kong. 160-167 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "On the length of an extremal rational curve" Invent.Math.105. 609-611 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Abundance theorem for Minimal threefolds" Invent. Math.108. 229-246 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Termination of log-flips for algebraic 3-folds" Intl. J. Math.3. 653-659 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Log canonical model of a Log minimal model" Intl. J. Math.3. 351-357 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Unobstructed deformations - a remark on a paper of Z. Ran" J. Alg. Geom.1. 183-190 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Rational curves and classification of algebraic varieties, in Essays on Mirror Manifolds, ed. S-T. Yau" International Press, (Hong Kong). 160-167 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "On the length of an external rational curve" Invent. Math.105. 609-611 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1992 Final Research Report Summary
  • [Publications] Y.Kawamata: "Abundance theorem for minimal threefolds" Invent.Math.108. 229-246 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Kawamata: "Termimation of log-flips for algebraic 3-folds" Intl.J.Math.3. 653-659 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Kawamata: "Log canonical model of a log minimal model" Intl.J.Math.3. 351-357 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Kawamata: "Unobstructed deformations-a remark on a paper of Z.Ran" J.Alg.Geom.1. 183-190 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Kawamata: "Rational curves and classification of algebraic varieties,in Essays on Mirror Manifolds,ed.S-T.Yau" International Press,Hong Kong. 160-167 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] Y.Kawamata: "On the length of an extremal rational curve" Invent.Math.105. 609-611 (1991)

    • Related Report
      1992 Annual Research Report
  • [Publications] 加藤 和也: "Swan conductors for characters of degree one in the imperfect residue field case" Contemporary Math. 83. 101-132 (1989)

    • Related Report
      1991 Annual Research Report
  • [Publications] 川又 雄二郎: "On the length of extremal rational curves" Invent math.105. 609-611 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 黒川 信重: "Multiple zeta functions:an example" Advanced stud in Pure Math. 21. (1992)

    • Related Report
      1991 Annual Research Report
  • [Publications] 斎藤 毅: "The Euler numbers of Iーadic sheaves of rank 1 in positive characteristic" Algebraic Geomety and Analytic Geometry 報告集. 165-181 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 中山 昇: "On smooth exceptional curves in threefolds" J.Fac.Sci.Univ.Tokyo Sec.IA. 37. 511-525 (1990)

    • Related Report
      1991 Annual Research Report
  • [Publications] 中村 博昭: "On galois automorphisms of the fundamental group of the projective line minus three points" Math.Z.617-622 (1991)

    • Related Report
      1991 Annual Research Report

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Published: 1991-04-01   Modified: 2016-04-21  

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