Transformation groups and geometry

1.辛流形与群作用。主要研究了S^1群作用下的允许矩映射的辛流形。(1)在辛结构确定复线丛的情况下，证明了线丛在每个定点所确定的权重与在同一点所取的矩图的值(Hattori)有密切的关系。(2)如果所有不动点都是孤立的，且作用是半自由的，则S^1-流形本质上与2-球面的乘积重合(Hattori)。(3)得到了允许矩映射的4维辛S^1-流形的完全分类(Hattori)。自对偶连通模空间。(1)确定了瞬时子数为2的SU(2)-瞬时子模空间的拓扑(Hattori)。(2)计算了瞬时子数为1的SU(2)-瞬时子模空间上的自然黎曼度量的截面曲率(MatSumoto)。(3)Furuta研究了瞬时子模空间上的群作用。作为推广，Furuta证明了瞬子数L的模空间的欧拉数是L的正因子数。(4)作为模空间拓扑的应用，Furuta证明了同调3-球面的同调余边群包含无限个循环群的乘积。其他的。(1)利用射影变换群(Ochiai)刻画球面和射影空间。(2)关于3-折叠极小模型存在性的重要结果(Kawamata)。(3)椭圆曲面的微分同胚分类研究(MatSumoto和UE)。这里有效地使用了松本提出的圆环纤化的概念。(4)具有零数量曲率和不可服从基本群(ONO)的Span流形上的A-亏格的消失定理。

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 semi-free then the S^1-manifold coincides essentially with a product of 2-spheres (Hattori).(3) A complete classification of 4-dimensional symplectic S^1-manifolds admitting moment map was derived (Hattori).2. Moduli spaces of self-dual 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 3-spheres 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 3-folds (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 A-genus on span manifolds with zero scalar curvature and non-amenable fundamental group (Ono).