Study on symplectic structures and contact structures

福谷和小野提出了Kuranishi结构的概念，并构造了周期Hamilton系统和Gromov-Witten不变量的Floer同调。我们继续这一方向，并继续研究了拉格朗日交的Floer同调和福谷提出的ALPHA_*-范畴。为了处理全纯映射模空间定义方程的横截性，需要利用多值扰动，并且必须使用有理系数。因此，我们需要给各种模空间的相干定向。我们证明了拉格朗日子流形上的自旋结构给出了模空间上的自然定向。我们也有另一个问题，即有一对拉格朗日子流形的例子，使通常的定义的边界算子的弗洛尔复杂的边界算子不给。本文系统地研究了全纯圆盘的边值问题，并定义了一类障碍类，用于定义拉格朗日子流形对的Floer同调。当障碍类为零时，我们可以修改边界算子的定义，定义Floer同调。这是由于福谷，Kontsevich，哦，太田和小野。太田和小野研究拓扑辛填充4-流形的简单奇点的联系。基于Taubes定理，我们得到了这方面的一个部分结果.受到这项工作的启发，Kanda将Taubes的结果推广到某些非紧辛流形。

Fukaya and Ono had introduced the notion of Kuranishi structure and constructed Floer homology for periodic Hamiltonian systems and Gromov-Witten invariants. We continue this direction and proceeded to investigation on Floer homology of Lagrangian intersections and ALPHA_*-category proposed by Fukaya. In order to deal with transversality for the defining equation of the moduli space of holomorphic maps, we need to use multi-valued perturbations and should work with rational coefficients. Hence we need to give coherent orientation for various moduli spaces. We showed that a spin structure on the lagrangian submanifold gives & natural orientation on moduli spaces. We also have another problem, namely there are examples of a pair of Lagrangian submanifolds so that the usual definition of the boundary operator for Floer complex does not give the boundary operator. We consider the boundary values of holomorphic disks systematically and defined a sort of obstruction classes for defining Floer homology for pairs of Lagrangian submanifolds. It is also shown that if such obstruction classes vanish, we can modify the definition of the boundary operator and define Floer homology. This is a result due to Fukaya, Kontsevich, Oh, Ohta and Ono.Ohta and Ono studied topology of symplectically filling 4-manifold of links of simple singularities. Based on Taubes' theorem, we got a partial result in this direction. Inspired by this work, Kanda extended Taubes' result to certain non-compact symplectic manifolds.