Transformation groups and geometry
Project/Area Number 
61460001

Research Category 
GrantinAid for General Scientific Research (B)

Allocation Type  Singleyear Grants 
Research Field 
代数学・幾何学

Research Institution  University of Tokyo, Faculty of Science, Department of Mathematics 
Principal Investigator 
HATTORI Akio University of Tokyo, Dept. of Mathematics, 理学部, 教授 (80011469)

CoInvestigator(Kenkyūbuntansha) 
FURUTA Mikio University of Tokyo Dept. of Mathematics, 理学部, 助手 (50181459)
UE Masaaki University of Tokyo, Dept. of Mathematics, 理学部, 助手 (80134443)
KAWAMATA Yujiro University of Tokyo, Dept. of Mathematics, 理学部, 助教授 (90126037)
MATSUMOTO Yukio University of Tokyo, Dept. of Mathematics, 理学部, 助教授 (20011637)
OCHIAI Takusiro University of Tokyo, Dept. of Mathematics, 理学部, 教授 (90028241)

Project Period (FY) 
1986 – 1987

Project Status 
Completed (Fiscal Year 1987)

Budget Amount *help 
¥6,000,000 (Direct Cost: ¥6,000,000)
Fiscal Year 1987: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 1986: ¥3,300,000 (Direct Cost: ¥3,300,000)

Keywords  Symplectic manifold / Moment map / Group action / Eixed point / Selfdual connections / Moduli space / Sain manifold / モジュライ空間 / シンプレクティック多様体 
Research Abstract 
1. Symplectic manifolds and group actions. Symplectic manifolds acted on by the group S^1 and admitting a moment map were mainly investigated. (1) In case the symplectec structure determines a complex line bundle it.was proved that there was a close relation between the weight determined by the line bundle at each fixed point and the value of moment map taken at the same point (Hattori). (2) If all the fixed points are isolated and the action is semifree then the S^1manifold coincides essentially with a product of 2spheres (Hattori).(3) A complete classification of 4dimensional symplectic S^1manifolds admitting moment map was derived (Hattori). 2. Moduli spaces of selfdual donnections. (1) The topology of the moduli space of SU(2)instantons with instanton number 2 was determined (Hattori). (2) The sectional curvature of the natural Riemannian metric on the moduli space of SU(2)instantons with instanton number 1 were calculated (Matsumoto). (3) Group actions on the moduli spaces of instantons were investigated by Furuta. As an spplication Furuta proved that the Euler number of the moduli space of instanton number l was the number of positive divisors of l (4) As an application of the topology of moduli spaces Furuta proved that homology cobordism group of homology 3spheres contained an infinite product of infinite cyclic groups. 3. Miscellaneous. (1) Characterization of spheres and projective spaces by project transformation group (Ochiai). (2) Important result on the existence of minimal models for 3folds (Kawamata). (3) Study on diffeomorphism classification of elliptic surfaces (Matsumoto and Ue). Here the notion of torus fibration introduced by Matsumoto was effectively used. (4) A vanishing theorem of Agenus on span manifolds with zero scalar curvature and nonamenable fundamental group (Ono).

Report
(2 results)
Research Products
(14 results)