Study on symplectic structures and contact structures

深谷和小野引入了Kuranishi结构的概念，并构造了周期哈密顿系统和Gromov-Witten不变量的Floer同调。我们继续这一方向，并开始研究Lagrangian交集和Fukaya提出的ALPHA_*-范畴的花同源性。为了处理全纯映射模空间定义方程的横截性，我们需要使用多值摄动，并且应该使用有理系数。因此，我们需要给出各种模空间的相干取向。我们证明了拉格朗日子流形上的自旋结构在模空间上给出&自然取向。我们还有另一个问题，即有一对拉格朗日子流形的例子，使得通常的边界算子的定义不能给出Floer复形的边界算子。我们系统地考虑了全纯盘的边值，并定义了一类用于定义拉格朗日子流形对的花同调的障碍类。还证明了如果这些障碍类消失，我们可以修改边界算子的定义，并定义了Floer同调。这是由于深谷，康采维奇，吴，太和小野的结果。Ohta和Ono研究了简单奇点链路辛填充4流形的拓扑结构。根据Taubes定理，我们得到了这个方向的部分结果。受到这项工作的启发，神田将Taubes的结果推广到某些非紧致辛流形。

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.