Study on Floer cohomology, mirror symmetry conjecture and singularity

对已有的关于Lagrange Floer上同调的障碍理论和形变理论的文章进行了修正。这是与K的合作。福谷，Y-G。哦，K。小野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。我们建立了仓西结构空间的横截性论证。这对于从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，我们在我们的-过滤A_∞代数的背景下定义了势函数，并证明了它与物理文献中Landau-Ginzburg模型的超势是一致的。这是物理学家Hori和Vafa预测的。2006年，我们完成了剩余的章节，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形。在这种情况下，我们可以建立Z或Z_2系数上的结果。我们应用的结果，例如，Arnold-Givental猜想。在第十章中，我们研究了伪全纯圆盘的模空间在拉格朗日手术下的变化。我们还描述了分析的细节。此外，我们还利用L ∞同态刻画了周围辛流形的圈对滤子A ∞代数的作用.并且发现了我们的Floer上同调的挠率部分与汉密尔顿合痕的霍费尔距离之间的一个新的关系。

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.