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1993 Fiscal Year Final Research Report Summary

Research of algebro-analytic varieties of higher dimension

Research Project

Project/Area Number 03640105
Research Category

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

Allocation TypeSingle-year Grants
Research Field 代数学・幾何学
Research InstitutionTokyo Institute of Polytechnics

Principal Investigator

MAEHARA Kazuhisa  Tokyo Institute of Polytechnics, Department of Engenering, Associated Professor, 工学部, 助教授 (10103160)

Co-Investigator(Kenkyū-buntansha) UENO Yoshiaki  Tokyo Institute of Polytechnics, Department of Engenering, Lecturer, 工学部, 講師 (60184959)
NAKANE Shizuo  Tokyo Institute of Polytechnics, Department of Engenering, Associated Professor, 工学部, 助教授 (50172359)
Project Period (FY) 1991 – 1993
Keywordsalgebraic geometry / higher dimensional variety / classification theory / champ / gerbe / vanishing theorem / deformation theory / Fourier-Deligne transformation
Research Abstract

It has been known that Kummer-Kawamata covering is a key to prove Kawamata vanishing theorem. We generalize Esnault-Viehweg's result that the degeneration of Hodge spectral sequence implies vanishing thoprems through cyclic covering and desingularization. Cyclic covering (resp. Kawamata covering) takes a role of the curvature of a line bundle in differential geometry. We realize a Kummer covering as a 1-algebraic champ without singular points. Hence we can apply it to the complete non singular variety of positive characteristic which is liftable to characteristic zero, where the degeneration of Hodge spectral sequence is obtained by Deligne-Illusie. The Kummer cover as an algebraic champ enable us to take an endomorphism which satisfies the assumption of Serre's paper "Kahler analogue of Riemann conjecture". Furthermore Fourier-Deligne transformation is applicable to this endomorphism, which should induce the degeneration of Hodge spectral sequence in complex algebraic geometry. It is a pure algebraic proof. Chosen certain Grothendieck topologies, the cohomology theory of algebraic champs implies that Hodge-Kodaira decomposition and vanishing theorems are equivalent. The gerbe of the fiberd category of schemes over the ringed topos forms a relative scheme by lifting it to the classifying topos of the gerbe. Thus the gerbe of the fiberd category of schemes over the ringed topos associated to an algebraic space is algebraic. We expect to extend it to the infinite algebraic champs. It is the same as for analytic champs. The local liftings of a complete non singular variety of positive characteristic in Zariski topology to the Witt ring of length two becomes a gerbe. Taking the maximal radical extention in the classifying topos of the gerbe we obtain the Hodge decompositon. We prepare the proof of the fundamental conjecture of the birational geometry and an analogue of higher dimensional Shafarevitch conjecture over function fields. It is to be published that an analog

  • Research Products

    (16 results)

All Other

All Publications (16 results)

  • [Publications] 前原和寿: "Flat deformation theory of a family dominated by another family" The Academic Reports Tokyo Inst.of Polytech.Vol.16. 1-12 (1993)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "Remarks of Esnault-Viehweg results" Acad.Rep.T.I.P.Vol.15. 13-35 (1992)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "Diophantine geometry and Hodge theory" R.I.M.S.,Kyoto Univ.167-187 (1992)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "On the higher dimensional Mordell conjecture over function fields" Osaka J.Math.28. 255-261 (1991)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "Kawamata covering and logarithmic de Rham complex" 目白-本郷セミナー,学習院大学. 71-90 (1990)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "The Mordell-Bombieri-Noguchi conjecture over function fields" Kodai Math.J.11. 1-4 (1988)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "代数幾何学における消滅定理" 上智大学数学講究録No.34, 283 (1992)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 前原和寿: "Kaehler多様体入門" 学習院大学理学部数学教室, 246 (1990)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kasuhisa Maehara: "Flat deformation of a family dominated by another family" Acad.Rep.T.I.P.Vol.16. 1-12 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "Remarks of Esnault-Viehweg Results" Acad.Rep.T.I.P.Vol.15. 13-35 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "Diophantine geometry and Hodge theory" R.I.M.S., Kyoto Univ.167-187 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "On the higher dimensional Mordell conjecture over function fields" Osaka J.Math. Vol.28. 255-261 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "Kawamata covering and logarithmic de Rham complex" Mejiro-Hongo Seminar, Gakushuin Uni.71-90 (1990)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "The Mordell-Bombieri-Noguchi conjecture over function fields" Kodai Math.J.Vol.11. 1-4 (1988)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "Vanishing theorems in algebraic geometry ; Math.Coll" Sophia Uni.No.34. 1-283 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuhisa Maehara: "Introduction to Kaehler manifold" Math.Dept.Sci.Gakushuin Uni.91-337 (1990)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 1995-03-27  

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