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2006 Fiscal Year Final Research Report Summary

Multiplier ideal sheaves in complex geometry ; the foundation and its applications.

Research Project

Project/Area Number 15340026
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionThe University of Tokyo (2004-2006)
Kyushu University (2003)

Principal Investigator

TAKAYAMA Shigeharu  The University of Tokyo, Graduate school of mathematical sciences, Associate professor, 大学院数理科学研究科, 助教授 (20284333)

Co-Investigator(Kenkyū-buntansha) HIRACHI Kengo  The University of Tokyo, Graduate school of mathematical sciences, Associate professor, 大学院数理科学研究科, 助教授 (60218790)
TAKAGI Hiromichi  The University of Tokyo, Graduate school of mathematical sciences, Associate professor, 大学院数理科学研究科, 助教授 (30322150)
SATO Eiichi  Kyushu University, Graduate school of mathematical sciences, Professor, 大学院数理科学研究院, 教授 (10112278)
MABUCHI Toshiki  Osaka University, Graduate school of sciences, Professor, 大学院理学研究科, 教授 (80116102)
TSUJI Hajime  Sophia University, Faculty of sciences, Professor, 理工学部, 教授 (30172000)
Project Period (FY) 2003 – 2006
Keywordsmultiplier ideal sheaf / singular Hermitian metric / canonical divisor
Research Abstract

As in the research plan of the application form, our research result consists of two terms.
1.Fundamental research on multiplier ideal sheaves. 2.Study on the basic properties of algebraic varieties based on the theory of multiplier ideal sheaves. We will also mention 3.further developments.
1.(1)We reconstruct the so-called Iitaka fibration of algebraic varieties, by introducing an intersection theory in terms of multiplier ideal sheaves. (Trans. AMS 2003)
(2)Making a refinement on the intersection theory mentioned (1)above, we obtained a criterion for a pseudo-effective divisor to be big, as a generalization of the so-called Seshadri's criterion for ampleness. (Math. Z. 2003)
2.Combining our results in 1 above, with an algebraic theory of multiplier ideal sheaves by Lazarsfeld, and with the method of Siu, we obtained the following results on pluricanonical forms.
(1)We analyzed a behavior of plurigenera of algebraic varieties under a deformation, and gave the final answer to a well-known conjecture. (J. Alg. Geom. 2006)
(2)We showed that, for every integer n, there exists an integer m(n) depending only on n, such that for every n-dimensional algebraic variety X of general type, the m-th pluricanonical system gives a birational map for any m>m(n). (Invent. Math. 2006)
3.It is known that for a morphism f : X→Y, the direct image of the relative canonical sheaf has a positivity property. We added an analysis on the effect of singularities of the map f, and obtained a refinement and a partial answer to a conjecture of Griffiths. For further research, we would like to develop a relative version of the theory of multiplier ideal sheaves and find strong applications.

  • Research Products

    (23 results)

All 2006 2005 2004 2003 Other

All Journal Article (23 results)

  • [Journal Article] Pluricanonical systems on algebraic varieties of general type2006

    • Author(s)
      Takayama S.
    • Journal Title

      Invent. Math. 165

      Pages: 551-587

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the invariance and the lower semi-continuity of plurigenera of algebraic varieties2006

    • Author(s)
      Takayama S.
    • Journal Title

      J. Algebraic Geom. 16

      Pages: 1-18

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Pluricanonical systems of projective varieties of general type I2006

    • Author(s)
      Tsuji H.
    • Journal Title

      Osaka J. Math. 43

      Pages: 967-995

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Logarithmic singularity of the Szego kernel and a global invariant of strictly pseudoconvex domains2006

    • Author(s)
      Hirachi K.
    • Journal Title

      Ann. of Math. 163

      Pages: 499-515

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Classification of primary Q-Fano threefolds with anti-canonical Du Val K3 surfaces. I2006

    • Author(s)
      Takagi H.
    • Journal Title

      J. Algebraic Geom. 15

      Pages: 31-85

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I2005

    • Author(s)
      Mabuchi T.
    • Journal Title

      Invent. Math. 159

      Pages: 225-243

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the existence of pluricanonical forms on varieties with infinite fundamental group2004

    • Author(s)
      Takayama S.
    • Journal Title

      Amer. J. Math. 126

      Pages: 1221-1235

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Stability of extremal Kahler manifolds2004

    • Author(s)
      Mabuchi T.
    • Journal Title

      Osaka J. Math. 41

      Pages: 563-582

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An obstruction to asymptotic semistability and approximate critical metrics2004

    • Author(s)
      Mabuchi T.
    • Journal Title

      Osaka J. Math. 41

      Pages: 463-472

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Quasi-projectivity of moduli spaces of polarized varieties2004

    • Author(s)
      Schumacher G., Tsuji H.
    • Journal Title

      Quasi-projectivity of moduli spaces of polarized varieti 159

      Pages: 597-639

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Conformally invariant powers of the Laplacian -a complete nonexistence theorem2004

    • Author(s)
      Gover A.R., Hirachi K.
    • Journal Title

      J. Amer. Math. Soc. 17

      Pages: 389-405

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Conformally invariant powers of the Laplacian --- a complete nonexistence theorem2004

    • Author(s)
      Gover A.R., Hirachi K.
    • Journal Title

      J. Amer. Math. Soc. 17

      Pages: 389-405

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Quasi-projectivity of moduli spaces of polarized varieties2004

    • Author(s)
      Schumacher G., Tsuji H.
    • Journal Title

      Ann. of Math. 159

      Pages: 597-639

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Local simple connectedness of resolutions of log-terminal singularities2003

    • Author(s)
      Takayama S.
    • Journal Title

      Internat. J. Math. 14

      Pages: 825-836

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Seshadri constants and a criterion for bigness of pseudo-effective line bundles2003

    • Author(s)
      Takayama S.
    • Journal Title

      Math. Z. 243

      Pages: 179-199

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Multiplier Hermitian structures on Kahler manifolds2003

    • Author(s)
      Mabuchi T.
    • Journal Title

      Nagoya Math. J. 170

      Pages: 73-115

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Decomposition of CR-manifolds and splitting of CR-maps2003

    • Author(s)
      Hayashimoto A., Kin S.-Y., Zaitsev D.
    • Journal Title

      Ann. Sc. Norm. Super. Pisa Cl. Sci. 2

      Pages: 433-448

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Iitaka's fibrations via multiplier ideals2003

    • Author(s)
      Takayama S
    • Journal Title

      Trans. Amer. Math. Soc. 355

      Pages: 37-47

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Ambient metric construction of Q-curvature in conformal and CR geometries2003

    • Author(s)
      Fefferman Ch., Hirachi K.
    • Journal Title

      Math. Res. Lett. 10

      Pages: 819-831

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Iitaka's fibrations via multiplier ideals2003

    • Author(s)
      Takayama S.
    • Journal Title

      Trans. Amer. Math. Soc. 355

      Pages: 37-47

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Multiplier Hermitan structures on Kahler manifolds2003

    • Author(s)
      Mabuchi T.
    • Journal Title

      Nagoya Math. J. 170

      Pages: 73-115

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hodge metrics and positivity of direct images

    • Author(s)
      Mourougane Ch., Takayama S.
    • Journal Title

      Crelle's J. (to appear)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hodge metrics and positivity of direct images

    • Author(s)
      Mourougane Ch., Takayama S.
    • Journal Title

      Crelle's J. (to appear)

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

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Published: 2008-05-27  

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