Symplectic structures and singularities

辛结构和奇点

• 批准号: 11440015
• 负责人: ONO Kaoru
• 金额: $ 7.68万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440015/
• 关键词: Floer homology; Lagrangian submanifold; simple singularity; symplectic structure; contact structure; フロアーホモロジー; シンプレクティック・フィリング; ミルナー・ファイバー; 正則曲線; A_∞-代数; Floerホモロジー; モノポール方程式; 特異点

It is not always the case that Floer homology for pairs of Lagrangian sumanifolds can be defined. We constructed the obstruction theory for defining Floer homology for pairs of Lagrangian submanifolds in order to clarify when it is defined. When all the obstruction classes vanish, Floer homology can be defined. However, it depends on a choice of so-called bounding chains. Dependence of Floer homology over bounding chains can be understood in the framework of filtered A_∞-algebra associated to Lagrangian submanifolds. This algebra controls the deformation (extended moduli) of unobstructed Lagrangian submanifolds and is important in itself. These results are presented in a preprint by Fukaya, Oh, Ohta and Ono.Ono and Ohta classified diffeomorphism types of minimal symplectic fillings of links of simple singularities and simple elliptic singularities (complex dimension 2). For an isolated singularity, the minimal resolution and the Milnor fibe, if it exists, give typical example of minimal symplectic fillings. But they a not diffeomorphic in general. In the case of simple singularity, they turn out diffeomorphic thanks to existence of the simultaneous resolution by Brieskorn. We studied this phenomenon from contact/symplectic viewpoint. Kanda also contributed in a course of this research.

并非总是可以定义拉格朗日苏流形对的弗洛尔同调性。我们构建了定义拉格朗日子流形对的弗洛尔同调性的阻碍理论，以澄清当所有阻碍类别消失时，弗洛尔同调性。然而，它取决于所谓的边界链的选择。弗洛尔同源性对边界链的依赖性可以在过滤的框架中理解。与拉格朗日子流形相关的 A_∞ 代数该代数控制无阻碍拉格朗日子流形的变形（扩展模量），其本身很重要。这些结果在 Fukaya、Oh、Ohta 和 Ono 的预印本中提出。Ono 和 Ohta 分类了微分同胚类型。简单奇点和简单椭圆奇点链接的最小辛填充（复维2) 对于孤立奇点，最小分辨率和 Milnor 纤维（如果存在）给出了最小辛填充的典型例子，但它们通常不是微分同胚的。我们从接触/辛的角度研究了这种现象，Kanda 也在这项研究过程中做出了贡献。

项目成果

Kenji Fukaya: "Floer homology over integer of general simplistic manifolds, -summary-"Advanced Studies in Pure Mathematics. 31. 75-91 (2001)

Kenji Fukaya：“一般简单流形整数上的弗洛尔同调，-摘要-”纯数学高级研究。

Yutaka Kanda: "The monopole eqation and J-holomorphic curves on weakly convex almost Kahler 4-manifolds"Transactions of American Mathematical Society. Vol. 353. 2215-2243 (2001)

Yutaka Kanda：“弱凸几乎 Kahler 4-流形上的单极方程和 J-全纯曲线”美国数学会汇刊。

Go-o Ishikawa: "Topological classification of the tangent developable pf space cirves"Journal of London mathematical Society. Vol. 62. 583-598 (2000)

Go-o Ishikawa：“切线可展空间环路的拓扑分类”伦敦数学会杂志。

T. Ono: "Simple singularities and topology of symplectically filling 4-manifolds"Commentarii Mathematici Helvetici. 74. 575-590 (1999)

T. Ono：“辛填充 4 流形的简单奇点和拓扑”Commentarii Mathematici Helvetici。

Kenji Fukaya: "Floer homology over integer of general symplectic manifolds -summary-"Advanced Studies in Pure Mathematics. (印刷中).

Kenji Fukaya：“一般辛流形整数上的弗洛尔同调 - 摘要 -”纯数学高级研究（正在出版）。