Topological Study on the Degeneration Phenomena of Riemann Surfaces

黎曼曲面退化现象的拓扑研究

• 批准号: 12440013
• 负责人: MATSUMOTO Yukio
• 金额: $ 4.67万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 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/en/grant/KAKENHI-PROJECT-12440013/
• 关键词: Riemann surfaces; degeneration; singular fiber; splitting; monodromy; 大域的モノドロミー; クアンドル; 分裂変形; Seiberg-witten方程式; 大域モノドロミー; カンドル; 同伴不等式; 擬周期写像

During the term of the project, we have made quite a good progress on the problem of splitting a given complicated singular fiber into a number of simpler singular fibers. Among others, we would like to point out Arakawa and Ashikaga's proof of the splittability of hyperelliptic singular fibers and Takamura's research on the construction of splitting families for not necessarily hyperelliptic singular fibers, Takamura being a cooperative researcher. An experimental method is also very important to the study of splitting of singular fibers. In fact Ahara developed a software "Splitica" by which we can observe the splitting motion of "star-shaped" singular fibers on a computer display. As for the global study of monodromy, Imayoshi, Ito, and Yamamoto made a research from function theoretic viewpoint. Also the quandle of cords which might be important to the study of Lefschetz fibrations, Kamada and Matsumoto studied their algebraic representation. There are obtained many other related results.

在项目期间，我们在将给定的复杂奇异纤维分裂成许多更简单的奇异纤维的问题上取得了相当好的进展。其中，我们想指出荒川和足利对超椭圆奇异纤维可分裂性的证明，以及高村对不一定是超椭圆奇异纤维的分裂族构建的研究，高村是合作研究员。实验方法对于奇异纤维分裂的研究也非常重要。事实上，Ahara开发了一款软件“Splitica”，通过它我们可以在计算机显示器上观察“星形”奇异纤维的分裂运动。对于单一性的整体研究，今义、伊藤、山本从函数论的角度进行了研究。此外，对于莱夫谢茨纤维的研究可能很重要的绳索问题，镰田和松本还研究了它们的代数表示。还获得了许多其他相关结果。

项目成果

T.Ashikaga: "Classification of degenerations of curves of genus 3 via Matsumoto-Montesinos' theorem"Tohoku Math.J.. 54. 195-226 (2002)

T.Ashikaga：“通过 Matsumoto-Montesinos 定理对 3 型曲线的退化进行分类”Tohoku Math.J.. 54. 195-226 (2002)

S.Kamada: "Certain racks associated with the braid groups"Proc.Int.Conf.on Knot Theory "Knots in Hellas". 118-130 (2000)

S.Kamada：“与辫子组相关的某些机架”Proc.Int.Conf.on 结理论“希腊的结”。

K.Ahara: "Conjugacy classes of hyperbolic mapping class group of genus 2 and 3"Experimental math.. 9. 383-396 (2000)

K.Ahara：“属 2 和 3 的双曲映射类群的共轭类”实验数学.. 9. 383-396 (2000)

S.Kamada: "Wirtinger presentations for higher dimensional manifold knots"Fund. Math. 168. 105-112 (2001)

S.Kamada：“Wirtinger 演示高维流形结”基金。

今吉洋一(訳): "ヴィジュアル複素解析"培風館. 662 (2002)

Yoichi Imayoshi（译者）：“视觉复杂分析”Baifukan 662（2002）。