Mirror Symmetry and the Microlocal Theory of Sheaves

拟议的研究计划提供了一种研究康采维奇？由Batyrev和Borisov发现的一大类环面超曲面镜像对的同调镜像对称猜想（HMS）。该方法从根本上采用了可构造的轴系，特别是Kashiwara和Schapira提出的轴系微局部理论，并在轴系理论中引入了新的结构。在HMS的大体积限制下，该项目旨在展示一个明确的组合骨架，在这个框架上，Fukaya类别有望被定位，即形成一堆dg类别。另一个目的是证明这些空间，以及更一般的奇异拉格朗日和legendris空间，都带有dg范畴的规范束，即“Kashiwara-Schapira束”。它是由微局部轴技术定义的，并且易于计算。因此，即使在严格的辛几何构造之前，也可以定义和计算推测的“Fukaya类别集”，这被认为是等效的。第三个目的是证明Kashiwara-Schapira叠层的整体类别等同于HMS复杂侧大复杂结构极限上的完美复杂类别。拟议的研究计划提供了一种研究康采维奇？由Batyrev和Borisov发现的一大类环面超曲面镜像对的同调镜像对称猜想（HMS）。该方法从根本上采用了可构造的轴系，特别是Kashiwara和Schapira提出的轴系微局部理论，并在轴系理论中引入了新的结构。在HMS的大体积限制下，该项目旨在展示一个明确的组合骨架，在这个框架上，Fukaya类别有望被定位，即形成一堆dg类别。另一个目的是证明这些空间，以及更一般的奇异拉格朗日和legendris空间，都带有dg范畴的规范束，即“Kashiwara-Schapira束”。它是由微局部轴技术定义的，并且易于计算。因此，即使在严格的辛几何构造之前，也可以定义和计算推测的“Fukaya类别集”，这被认为是等效的。第三个目的是证明Kashiwara-Schapira叠层的整体类别等同于HMS复杂侧大复杂结构极限上的完美复杂类别。

The proposed research program provides an approach to Kontsevich?s homological mirror symmetry conjectures (HMS) for the large class of mirror pairs of toric hypersurfaces discovered by Batyrev and Borisov. The approach uses constructible sheaves, especially the microlocal theory of sheaves developed by Kashiwara and Schapira, in a fundamental way, and leads to new constructions in sheaf theory. At the large volume limit of the symplectic side of HMS, the project aims to exhibit an explicit combinatorial skeleton over which the Fukaya category is expected to localize -- i.e. form a sheaf of dg categories. Another aim is to show that these spaces, and more general singular Lagrangians and Legendrians, all carry a canonical sheaf of dg categories, the "Kashiwara-Schapira sheaf." It is defined by microlocal sheaf techniques, and amenable to computations. Thus the conjectural "sheaf of Fukaya categories" can be defined and computed even in advance of a rigorous symplectic geometry construction, which is expected to be equivalent. A third aim is to show that the global category of the Kashiwara-Schapira sheaf is equivelent to the category of perfect complexes on the large complex structure limit of the complex side of HMS.The proposed research program provides an approach to Kontsevich?s homological mirror symmetry conjectures (HMS) for the large class of mirror pairs of toric ypersurfaces discovered by Batyrev and Borisov. The approach uses constructible sheaves, especially the microlocal theory of sheaves developed by Kashiwara and Schapira, in a fundamental way, and leads to new constructions in sheaf theory. At the large volume limit of the symplectic side of HMS, the project aims to exhibit an explicit combinatorial skeleton over which the Fukaya category is expected to localize -- i.e. form a sheaf of dg categories. Another aim is to show that these spaces, and more general singular Lagrangians and Legendrians, all carry a canonical sheaf of dg categories, the "Kashiwara-Schapira sheaf." It is defined by microlocal sheaf techniques, and amenable to computations. Thus the conjectural "sheaf of Fukaya categories" can be defined and computed even in advance of a rigorous symplectic geometry construction, which is expected to be equivalent. A third aim is to show that the global category of the Kashiwara-Schapira sheaf is equivelent to the category of perfect complexes on the large complex structure limit of the complex side of HMS.