Symplectic structures and singularities

它并不总是情况下，弗洛尔同调对拉格朗日子流形可以定义。我们构造了定义拉格朗日子流形对的Floer同调的障碍理论，以澄清何时定义它。当所有的障碍类都为零时，可以定义Floer同调.然而，这取决于所谓的边界链的选择。Floer同调对有界链的依赖性可以在与Lagrange子流形相关联的滤子A ∞-代数的框架下理解。这个代数控制无障碍拉格朗日子流形的变形（扩展模），并且本身很重要。这些结果是由福谷，Oh，Ohta和Ono在预印本中给出的。Ono和Ohta对简单奇点和简单椭圆奇点（复维数为2）的链的极小辛填充的同构类型进行了分类。对于孤立奇点，最小分解和Milnor纤维（如果存在的话）给出了最小辛填充的典型例子。但它们一般不是同构的。在简单奇点的情况下，由于Brieskorn的同时解决方案的存在，它们变成了同构的。我们从接触/辛的观点研究了这一现象。神田也参与了这项研究。

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.