2006 Fiscal Year Final Research Report Summary
Multiplier ideal sheaves in complex geometry ; the foundation and its applications.
| Project/Area Number |
15340026
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | The University of Tokyo (2004-2006)
Kyushu University (2003) |
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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)
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| 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)
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| Project Period (FY) |
2003 – 2006
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| Keywords | multiplier ideal sheaf / singular Hermitian metric / canonical divisor |
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| 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.
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