| Project/Area Number |
08640105
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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 |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
OHTA Hiroshi Nagoya University, Graduate School of Mathematics, Asso.Professor, 大学院・多元数理科学研究科, 助教授 (50223839)
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| Co-Investigator(Kenkyū-buntansha) |
MINAMI Kazuhiko Nagoya University, Graduate School of Mathematics, Lecturer, 大学院・多元数理科学研究科, 教授 (40271530)
NAITO Hisashi Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 教授 (40211411)
TSUCHIYA Akihiro Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (90022673)
SATO Hajime Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (30011612)
KOBAYASHI Ryoichi Nagoya University, Graduate School of Mathematics, Professor, 大学院・多元数理科学研究科, 教授 (20162034)
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| Project Period (FY) |
1996 – 1997
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| Project Status |
Completed (Fiscal Year 1997)
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| Budget Amount *help |
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 1997: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1996: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Sympectic geometry / contact geometry / Floer homology / gauge theory / Arnald conjecture / Low dimensional manifold / サイバーグ・ウィッテン / モノポール方程式 / 擬射影平面 / 4次元多様体 |
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| Research Abstract |
Using monopole equations by Witten, we studied and obtained some results on 4-dimensional symplectic geometry. We studied the fundamental group and numerical invariants of a symplectic 4-manifold with posive c_1 (TX). As an application, we proved easily that the underlying 4-manifold of any rational surface can not be diffeomorphic to one of minimal surfaces of general type. As a special case, this theorem contains Hirzebruch's conjecture that there does not exist a minimal surface of general type which is diffeomorphic to S^2 * S^2. Furthermore, we proved that if (X,omega) is a symplectic 4-manifold such that c_1 (TX) [omega] [X] > 0 or X admits a metric of positive scalar curvature, then X must be diffeomorphic to a rational or ruled surface up to blow up down.
Let GAMMA be a finite subgroup of SU (2). We have a specific fillable contact structure xi_0 on S^3/GAMMA induced from the standard contact structure on S^3. When we consider S^3/GAMMA with the fixed xi_0, we proved that the intersection form of any symplectically filling 4-manifold of (S^3/GAMMA, xi_0) is negative definite. For the proof, we showed a vanishing theorem on the Seiberg-Witten invariants. In the case GAMMA=E_8, we proved the intersection form of any minimal symplectically filling 4-manifold is equivalent to E_8. These are joint works with K.Ono.
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