Study on Floer cohomology, mirror symmetry conjecture and singularity

对拉格朗日花上同调中的阻碍理论和变形理论进行了修正。这是与K. Fukaya, y . g .合作的作品。哦，还有小野k。2006年3月，我们完成了第1、2、3、4、5、6、7、9章的手稿（800多页），并制作了预印本，在全球发行。我们在具有Kuranishi结构的空间上建立了横向性论证。这对于在各种模空间中导出同伦代数结构具有重要的意义。在修正过程中，我们定义了自滤波A_∞代数背景下的势函数，并证明它与物理文献中Landau-Ginzburg模型的超势一致。这是物理学家Hori和Vafa预测到的。2006年，我们完成了剩余章节的编写，第8章和第10章。在第八章中，我们研究了半正拉格朗日子流形的情况。在这种情况下，我们可以建立Z或Z_2系数的结果。我们将结果应用于，例如，Arnold-Givental猜想。在第10章中，我们研究了伪全纯盘在拉格朗日手术下模空间的变化。我们还详细描述了分析的细节。此外，我们用L_∞同态描述了环境辛流形的环对滤波后的A_∞代数的作用。我们还发现了Hamilton同位素的Floer上同调的扭转部分与Hofer距离之间的新关系。

We revised our paper on obstruction theory and deformation theory of Lagrangian Floer cohomology. This is a joint work with K. Fukaya, Y-G. Oh and K. Ono. In March 2006, we wrote up the manuscript (more than 800 pages) of Chapter 1,2,3,4,5,6,7 and 9 and made the preprint version which were distributed in the world. We established the transversality argument on the spaces with Kuranishi structure. This will play very important role to derive homotopy algebraic structure from various moduli spaces. During the revision, we define the potential function in the context of our-filtered A_∞ algebra and show that it coincides with the super potential of the Landau-Ginzburg model in physics literature. This was predicted by physists Hori and Vafa. In 2006, we have finished to write up the remaining chapters, Chapter 8 and Chapter 10. In Chapter 8, we study the case of semi-positive Lagrangian submanifolds. In this case, we can establish our results over Z or Z_2 coefficients. We apply the results to, for example, the Arnold-Givental conjecture. In Chapter 10, we investigate how the moduli space of pseudo holomorphic discs changes under Lagrangian surgery. We also describe the analytic detail. Moreover, we describe the action of cycles of the ambient symplectic manifold to the filtered A_∞ algebra in terms of L_∞ homomorphism. And we discovered a new relation between the torsion part of our Floer cohomology and Hofer distance of Hamilton isotopy.