Transformation groups and geometry

点击翻译按钮获取中文摘要

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).