Project/Area Number  08640105 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Geometry

Research Institution  Nagoya University 
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)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1996 – 1997

Project Status 
Completed(Fiscal Year 1997)

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)

Keywords  Sympectic geometry / contact geometry / Floer homology / gauge theory / Arnald conjecture / Low dimensional manifold / シンプレクティック幾何 / 接触幾何 / Floerホモロジー / ゲージ理論 / Arnold予想 / 低次元多様体 / 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 
Research Abstract 
Using monopole equations by Witten, we studied and obtained some results on 4dimensional symplectic geometry. We studied the fundamental group and numerical invariants of a symplectic 4manifold with posive c_1 (TX). As an application, we proved easily that the underlying 4manifold 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 4manifold 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 4manifold of (S^3/GAMMA, xi_0) is negative definite. For the proof, we showed a vanishing theorem on the SeibergWitten invariants. In the case GAMMA=E_8, we proved the intersection form of any minimal symplectically filling 4manifold is equivalent to E_8. These are joint works with K.Ono.
