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.

并非总是可以定义对拉格朗日sumamorphs对的漂浮同源性。我们构建了用于定义一对拉格朗日submanifolds的浮子同源性的对象理论，以阐明何时定义。当所有对象类都消失时，可以定义浮动同源性。但是，这取决于选择所谓的边界链的选择。在与Lagrangian Submanifold相关的过滤A_∞-Elgebra的框架中，可以理解漂浮物同源性对边界链的依赖性。该代数控制着未打开的拉格朗日submanifolds的变形（扩展模量），并且本身很重要。这些结果在福卡亚，OH，OHTA和ONO.ONO和OHTA的预印本中呈现，分类的差异性类型的最小象征性填充物的简单奇异性和简单椭圆形奇异性的链接（复杂维度2）。对于孤立的奇异性，最小的分辨率和Milnor纤维（如果存在）给出了最小的对称填充物的典型例子。但是它们通常不是差异的。在简单的奇异性的情况下，由于布里斯科恩（Brieskorn）的简单决议的存在，它们结果变得差异化。我们从接触/对称的观点研究了这种现象。 Kanda在这项研究的过程中也做出了贡献。

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。

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）。

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。