2002 Fiscal Year Final Research Report Summary
Algebraic vaieties with Kodaira dimansion O and A generalization of the Bogomolor-decomposition
| Project/Area Number |
12440007
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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 | Osaka University |
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Principal Investigator |
NAMIKAWA Yoshinori Osaka Univ. Dept. Math. Associate Prof., 大学院・理学研究科, 助教授 (80228080)
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| Co-Investigator(Kenkyū-buntansha) |
GOTO Ryoshi Osaka Univ. Dept. Math. Associate Prof., 大学院・理学研究科, 助教授 (30252571)
MIYANISHI Masayoshi Osaka Univ. Dept. Math. Prof., 大学院・理学研究科, 教授 (80025311)
FUJIKI Akira Osaka Univ. Dept. Math. Prof., 大学院・理学研究科, 教授 (80027383)
SATAKE Ikuo Osaka Univ. Dept. Math. Assistant, 大学院・理学研究科, 助手 (80243161)
OHNO Koji Osaka Univ. Dept. Math. Assistant, 大学院・理学研究科, 助手 (20252570)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Complex symplectic variety / derived category / deformation theory / singularity / Period / Torelli problem / higher dimmsional algebraic varieries |
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| Research Abstract |
Does the period of the second cohomology determine an isomorphism class of a complex irreducible symplectic manifold? This is the Torelli problem. It is true for a K3 surface, but there is an counter-example of Debarre for dim 【greater than or equal】 4. Mukai, Huybrechts and others have posed the birational Torelli problem modifying the original one. We construeted a counter-example even for this problem.
On the other hand, for a complex irreducible symplectic manifold, which kind of information on the original variety can be recoverd from the derived category of coherent sheaves ? When do we have an equivalence of derived categories of two complex irreducible manifolds ? Such questions are quite interesting. We proved that the derived categories are equivalent if two smooth projective varieties are connected by a Mukai flop. As an application, one knows that birationally equivalent, complex, projective symplectic 4-folds have equivalent derived categories.
We also studied the deformation theory of complex symplectic varieties and generalized several important facts on a complex symplectic manifold to a singular symplectic variety.
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