2001 Fiscal Year Final Research Report Summary
The structure of polarized varieties
| Project/Area Number |
12640015
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
FUJITA Takao Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (40092324)
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| Co-Investigator(Kenkyū-buntansha) |
TSUJI Hajime Graduate School of Science and Engineering, Tokyo Institute of Technology, Associate Professor, 大学院・理工学研究科, 助教授 (30172000)
ISHII Shihoko Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (60202933)
FUTAKI Akito Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (90143247)
KAWACHI Takeshi Graduate School of Science and Engineering, Tokyo Institute of Technology, Assistant, 大学院・理工学研究科, 助手 (30323778)
NAKAYAMA Chikara Graduate School of Science and Engineering, Tokyo Institute of Technology, Assistant, 大学院・理工学研究科, 助手 (70272664)
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| Project Period (FY) |
2000 – 2001
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| Keywords | polarized varieties / adjoint bundle / genus |
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| Research Abstract |
The head investigator Fujita has studied the relation between the theory of #-minimal models and the theory of adjoint bundles. He further studied the behavior of cubic surfaces under the Cremona transformations of P^3.
Investigator Ishii has studied the exceptional hypersurface singularities and showed that there are only finitely many weights which give exceptional ones. 'She further classified all the 3-dimensional exceptional singularities of Brieskorn type. She studied many interesting properties of the set of possible values of the invariant K^2 of surface singularities, and further studied the relation between the accumulation points of the set of K^2 of cyclic quotient singularities and the accumulation points, of the corresponding continued fractions. She showed also that, for a given singularity, there exists a maximal manifold through which every surjective morphism from a smooth manifold factors, and she characterized such manifold by the existence of rational curves, and studied the properties such as direct product, quotient and functoriality.
Investigator Futaki has showed that for a complex line bundle L together with its Chern class which is a Hodge class, if the action of the automorphism group lifts to L, then the Futaki character lifts to a character of the automorphism group. He gave also an explicit integral expression of this lift, and showed that it can be applied, to yield Mabuchi's K-energy and Ding's functional.
Investigator Tsuji has showed the deformation invariance of plurigenera by applying the theory of singular Hermitian metrics.
Investigator Nakayama has studied systematically mixed Hodge structures on log deformation families.
Investigator Kawachi has generalized results of Reider type on normal surfaces to cases of log algebraic surfaces.
Investigator Minagawa has studied criterions of smoothability of weak Fano 3-folds, and showed that Q-factoriality is sufficient. He also found an example of non-smoothable ones without Q-factoriality.
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