The study of Kahler fibrations and its applications to Finsler geometry

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In this research, we have studied the differential geometry of Kahler fibrations and its applications to complex Finsler geometry in the term 2005-2007. The main subjects of this project are (1) the study of Kahler fibration from the view point of complex analytic geometry, (2) the study of metric of Weil-Petersson type in Finsler geomrty, (3) the study of the relation of complex structure of the base maifold and the given complex Finsler metric.In this project, we have studied the theory of connections on the total space of the fibration, which is naturally related to the theory of Finsler connections. Let E be a holomorphic vector bundle over a compact Kahler manifold M. Then the total spaceP (E)of the projective bundle of E is also a compact Kahler manifold. The Kahler metric on P (E)determines a strongly pseudoconvex Finsler metric on E. Such a Finsler metric is naturally concerned with the ampleness of E. This property is characterized by the curvature of Chern-Fisnler connection.The main result in this project is the characterization of Finsler-Kahler manifold in terms of Chern-Finsler connection and the given complex structure on M. The head investigator reported some results in this project at the international conferences held at Budapest (Hungary, 2005),Sapporo (Japan, 2006), Sendai (Japan, 2007),Balatonfoldvar (Hungary, 2007 ),and he has published two paper on complex Finsler geometry. Furthermore a paper entitled " Dual connections in Finsler geometry" is in publishing. This paper is concerned with the notion of "dual connection" of Finsler connection which is obtained in this project.