Mirror Symmetry and the Microlocal Theory of Sheaves

镜像对称和滑轮的微局域理论

• 批准号: 1314010
• 负责人: David Treumann
• 金额: $ 10.55万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1314010&HistoricalAwards=false
• 关键词: Mirror; Symmetry; Microlocal; Theory; Sheaves

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.

拟议的研究计划提供了一种方法Kontsevich？的同调镜像对称结构（HMS）的大类镜对的环面超曲面的Batyrev和Borisov发现。该方法使用可构造层，特别是由Kashiwara和Schapira开发的层的微局部理论，在一个基本的方式，并导致新的结构层理论。 在HMS的辛侧的大体积限制下，该项目的目标是展示一个明确的组合骨架，在该骨架上，福谷类别有望局部化-即形成一个dg类别层。 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勒让德空间，都带有一个dg范畴的规范层，即“柏原-夏皮拉层”。“它是由微局部层技术定义的，并且适合于计算。 因此，即使在严格的辛几何构造之前，也可以定义和计算拓扑的“福谷范畴层”，这被认为是等价的。 第三个目的是表明，全球范畴的柏原Schapira层是等价的完美的复杂的大复杂的结构限制HMS的复杂的一面的范畴。拟议的研究计划提供了一种方法Kontsevich？对Batyrev和Borisov发现的复曲面镜像对的同调镜像对称结构（HMS）进行了研究。该方法使用可构造层，特别是由Kashiwara和Schapira开发的层的微局部理论，在一个基本的方式，并导致新的结构层理论。 在HMS的辛侧的大体积限制下，该项目的目标是展示一个明确的组合骨架，在该骨架上，福谷类别有望局部化-即形成一个dg类别层。 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勒让德空间，都带有一个dg范畴的规范层，即“柏原-夏皮拉层”。“它是由微局部层技术定义的，并且适合于计算。 因此，即使在严格的辛几何构造之前，也可以定义和计算拓扑的“福谷范畴层”，这被认为是等价的。 第三个目的是证明柏原-夏皮拉层的全局范畴等价于HMS复杂侧的大复杂结构极限上的完美复合物范畴。