The study of differential geometry of Kahler-fibrations and its applications.

在本研究中，我们研究了2003-2004年度Kahler纤维的复杂微分几何及其应用。特别是，我们已经构建了Kahler纤维的基础研究，将其应用于复杂的Finsler几何。该项目的主要课题是(1)研究等距纤维的Kahler纤维，(2)研究紧致黎曼曲面上最小直纹曲面上的Kahler度规。本研究的主要内容是研究纤维总空间上的连接，这自然与芬斯勒连接理论有关。特别地，在空间是紧致黎曼曲面上的最小直纹曲面P(E)的情况下，我们得到了关于含标量曲率的Kahler度量的一些结果。特别是研究了束E的射影平坦性，并且由于在这种情况下束的射影平坦性与束的稳定性是等价的，在某些假设下，得到了常标量曲率Kahler度量的存在性与束E的稳定性之间的一些关系。项目负责人已在德国德布勒森（匈牙利，2003年）、天津（中国，2004年）和日本那津岛（仙台，2004年）举行的国际会议上报告了该项目的一些成果，并正在准备发表题为《论复杂芬斯勒束上的Chenr-Finsler连接》的论文。

In this research, we have investigated the complex differential geometry of Kahler fibrations and its applications in the term 2003-2004. In particular, we have constructed the fundamental research of Kahler fibrations to apply it to complex Finsler geometry. The main subjects of this project are (1)the study of Kahler fibartion with isometric fibres, (2)the study of Kahler metric on a minimal ruled surface over a compact Riemann surface.The main content of this research is the investigation of connections on the total space of the fibration, which is naturally related to the theory of Finsler connections. In particular, in the case where the space is a minimal ruled surafce P(E) over a compact Riemann surface, we have obtained some results on Kahler metrics of contant scalr curvature. Especially we have investigated the projective flatness of the bundle E. Moreover, since the projective flatness and the stability of bundles are equivalent in this case, we have obtained some relations between the existence of Kahler metrics of constant scalar curvature and the stability of the bundle E under the some assumptions.The head investigator have reported some results in this project at the international conference held at Debrecen(Hungary,1003), Tianjin(China,2004) and Natsushima(Sendai, Japan,2004), and he is preparing a paper entitled "On the Chenr-Finsler connection on complex Finsler bundles" for publishing.