A study on dynamical system for a torus and cohomology

环面动力系统与上同调的研究

• 批准号: 12640183
• 负责人: KAZAMA Hideaki
• 金额: $ 2.11万
• 依托单位: KYUSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640183/
• 关键词: line bundle; Kodaira's lemma; toroidal group; 複素トーラス; 力学系; 解析的コモホロジー群; 擬アーベル多様体; 線形化可能性; 連分数展開; 解析的コホモロジー群; コホモロジーの非ハウスドルフ性; 擬凸性

(I) It is a problem whether one can embed a weakly 1-complete manifold with a positive line bundles into a projective space by sections of higher tensor power positive line bundles, or not. It is a very interesting result that one can embed a weakly 1-complete manifold with a positive line bundle by sections of adjoint line bundles instead of the original line bundle by Shigeharu Takayama. This result is proved by a similar method in case of compact manifolds. We apply this method by adjoint line bundle to a problem of line bundle convexity. We get some global result with respect to line bundle convexity of weakly 1-complete manifolds.(ii) Recently we find an example that Kodaira's lemma does not hold for some strongly 1-complete manifold. Now we try to characterize strongly 1-complete manifolds on which Kodaira's lemma holds alway.(iii) On troidal groups of cohomologically finite type we can apply the theory of compact Kaehler manifolds. For instance we can show the Hodge decomposition theorem for toroidal groups of cohomologically finite type. We are getting some result for them similar to the results of compact Kaehler manifolds.

(一)能否将具有正线束的弱1-完备流形通过高张量幂正线束的分段嵌入到射影空间中是一个问题。这是一个非常有趣的结果，我们可以通过一段段伴随线丛来嵌入一个带有正线丛的弱1-完全流形，而不是高山茂治的原始线丛。在紧流形的情况下用类似的方法证明了这个结果。我们将该方法通过伴随线丛应用于线丛凸性问题。我们得到了关于弱1-完全流形的线束凸性的一些全局结果。(ii)最近我们发现了一个例子，Kodaira 引理对于某些强1-完全流形不成立。现在我们尝试刻画小平引理始终成立的强1-完备流形。(iii) 在上同调有限型三环群上我们可以应用紧凯勒流形理论。例如，我们可以展示上同调有限类型环形群的霍奇分解定理。我们得到了一些与紧凑凯勒流形的结果类似的结果。

项目成果

Kazama,H.,Kim,D.K.and C.Y.Oh: "Some remarks on complex Lie groups"Nagoya Math.J.. 157. 47-57 (2000)

Kazama,H.,Kim,D.K.和C.Y.Oh：“关于复杂李群的一些评论”Nagoya Math.J.. 157. 47-57 (2000)

Jyouichi Kaneko: "Classification of Pfaffian systems of Fuchs type of a particular class"Kumamoto J. Math. 15. 17-52 (2002)

Jyouichi Kaneko：“特定类别的 Fuchs 类型的普法夫系统的分类”Kumamoto J. Math。

Hideaki Kazama: "Nonlinearizability non-Hausdorff cohomology and neighborhood structure of elliptic curves"Complex Analysis and Related Topics in Mathematics, 2001. 67-74 (2002)

风间英明：《椭圆曲线的非线性化非豪斯多夫上同调和邻域结构》数学复数分析及相关专题，2001. 67-74 (2002)

Kazama,H.and Takayama,S.: "On the dd'-equation over pseudoconvex Kaehler manifolds"Manuscripta Math., Springer Verlag. 102. 25-39 (2000)

Kazama, H. 和 Takayama, S.：“关于伪凸凯勒流形上的 dd 方程”Manuscripta Math.，Springer Verlag。

Hideaki KAZAMA: "Nonlinearizability, non-Hausdorff cohomology and neighborhood structure of elliptic curves"Complan Analysis and Related topics in Mathematics 2001. 67-74 (2002)

Hideaki KAZAMA：“椭圆曲线的非线性化、非豪斯多夫上同调和邻域结构”数学中的 Complan 分析和相关主题 2001. 67-74 (2002)