2002 Fiscal Year Final Research Report Summary
Complex symplectic manifolds and related topics
| Project/Area Number |
12440006
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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 |
Algebra
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| Research Institution | The University of Tokyo (2001-2002)
Kyoto University (2000) |
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Principal Investigator |
MIYAOKA Yoichi The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (50101077)
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| Co-Investigator(Kenkyū-buntansha) |
OGUISO Keiji The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (40224133)
KAWAMATA Yujiro The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (90126037)
KATSURA Toshiyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40108444)
NAKAYAMA Noboru Kyoto University, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (10189079)
MORI Shigefumi Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (00093328)
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| Project Period (FY) |
2000 – 2002
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| Keywords | complex symplectic / birational morphism / fiber space structure / projective space / hyperquadrics / Fano 3-folds |
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| Research Abstract |
The head investigator studied holomorphic maps from complex symplectic manifolds, in which he obtained a series of fundamental results on numerical characterisations of projective space and smooth hyperquadrics. One of his results asserts that a smooth Fano n-fold X is isomorphic to projective n-space of a hyperquadric if and only if the"length of X"is n + 1 or n, where the length is defined to be the minimum of (C, -Kx), C running through the set of curves on X.
Our new characterisations are strong enough to be applied to complex manifolds, enabling us to prove the following structure theorem on morphisms from complex manifolds :
・ Let Y be a projective complex symplectic manifod of dimension 2n and π : Y → Y^^^ a birational morphism onto a normal variety. Let E denote an arbitrary irreducible component and put B = π(E). Then B is a complex symplectic variety of dimension 2m 【less than or equal】 2n and, for a general point b ∈ B, the inverse image π^<-1>(b) ∩ E is projective space of dimension n + m.
・ Let f : Y → X be a nontrivial fiber space structure on a primitive ; complex symplectic manifold of dimension 2n. If f admits a holomorphic section, then X is projective n-space and the fibers of f are Lagrangian subvarieties.
Another product of his research is a joint work with J. Kollar, S. Mori and H. Takagi, which proved the boundedness of Fano 3-folds with only canonical, singularities.
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