The study of Kahler fibrations and its applications to Finsler geometry

卡勒纤维的研究及其在芬斯勒几何中的应用

• 批准号: 17540086
• 负责人: AIKOU Tadashi
• 金额: $ 2.18万
• 依托单位: Kagoshima University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540086/
• 关键词: Kahler-fibration; Complex Finsler metrics; Chern-Finsler connection; Complex Finsler manifolds; Dual connections; Finsler manifold; Chern-Finsler connection; Finsler-Kahler manifolds; Ampleベクトル束; 複素Finsler計量; Finsler-Kahler多様体; Chern-Finsler接続

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.

在本研究中，我们研究了Kahler纤维的微分几何及其在复杂Finsler几何中的应用。本课题的主要课题是：(1)从复解析几何的角度研究Kahler纤维；(2)研究芬斯勒几何中Weil-Petersson型度规；(3)研究基基底的复结构与给定复芬斯勒度规的关系。在这个项目中，我们研究了纤维总空间的连接理论，这与Finsler连接理论自然相关。设E是紧化Kahler流形m上的全纯向量束，则E的射影束的总空间ep (E)也是紧化Kahler流形。P (E)上的Kahler度规决定了E上的强伪凸Finsler度规，这样的Finsler度规自然与E的丰度有关，这种性质的特征是chen - fisnler连接的曲率。该项目的主要成果是在chen -Finsler连接和m上给定的复杂结构方面对Finsler- kahler流形的表征。首席研究员在布达佩斯（匈牙利，2005），札幌（日本，2006），仙台（日本，2007），Balatonfoldvar（匈牙利，2007）举行的国际会议上报告了该项目的一些结果，并发表了两篇关于复杂Finsler几何的论文。此外，一篇题为“芬斯勒几何中的对偶连接”的论文即将发表。本文讨论了在该方案中得到的芬斯勒连接的“双重连接”概念。

项目成果

Grafting hyperbolic metrics and Eisenstein series

嫁接双曲度量和爱森斯坦级数

• 发表时间: 2008
• 期刊: Mathematische Annalen 341
• 作者: 砂田利一;橋本義武;ほか;S.Yokura;S. Yokura;S. Yokura;K.Obitsu;K. Miyajima;S. Tsuboi;S. Tsuboi;T.Nagano and T.Aikou;T.Aikou and L.Kozma;K.Obitsu and S.A.Wolpert
• 通讯作者: K.Obitsu and S.A.Wolpert

Dual connections in Finder geometry

Finder 几何结构中的双连接

• 发表时间: 2008
• 期刊: to appear in Acta Math.Acad.Paedagog.Nyhazi.(N.S.) 24
• 作者: T.Kobayashi;D.Heath;Tsuyoshi Kobayashi;Tsuyoshi Kobayashi;T.Aikou and L.Kozma;T.Nagano and T.Aikou
• 通讯作者: T.Nagano and T.Aikou

Analytic approach of deformation of normal isolated singularities

法向孤立奇点变形的解析方法

• 发表时间: 2007
• 期刊: In "Real and Complex Singularities, Proceeding of the Australian -Japanese workshop on real and complex singularities(ed. L. Paunescuet al))," World Scientific
• 作者: Z.Habib;M.Sakai;Tadashi Aikou;Kimio Miyajima
• 通讯作者: Kimio Miyajima

Deligne parings over Moduli spaces of punctured Riemann sufaces

穿孔黎曼曲面模空间上的德利涅配对

• 发表时间: 2006
• 期刊: Arithmetic Geometry and Number Theory(World Scientific)
• 作者: K.Obitsu;W.-K.To;L.Weng
• 通讯作者: L.Weng

The asymptotic behavior of the Takhtajan-Zograf metric

Takhtajan-Zograf 度量的渐近行为

• 发表时间: 2008
• 期刊: Communications in Mathematical Physics 284
• 作者: K. Obitsu;W. -K. To;L. Weng
• 通讯作者: L. Weng