Symplectic structures and singularities

拉格朗日求和流形对的Floer同调并不总是可以定义的。我们构造了定义拉格朗日子流形对的Floer同调的障碍理论，以澄清它何时被定义。当所有阻挡类都消失时，可以定义Floer同调。然而，这取决于所谓的约束链的选择。Floer同调对有界链的依赖可以在与拉格朗日子流形相关的滤子A_∞-代数的框架下理解。这个代数控制着无障碍拉格朗日子流形的变形(扩展模)，并且它本身是重要的。这些结果由Fukaya，oh，Ohta和Ono.Ono和Ohta在预印本中给出。Ono和Ohta对简单奇点和简单椭圆奇点(复数维2)的链环的极小辛填充值的微分同胚类型进行了分类。对于孤立奇点，最小分解和Milnor Fibe(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。在简单奇点的情况下，由于Brieskorn同时解的存在，它们是微分同胚的。我们从接触/辛的观点研究了这一现象。神田也在这项研究过程中做出了贡献。

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.