Mirror Symmetry and the Microlocal Theory of Sheaves

镜像对称和滑轮的微局域理论

• 批准号: 1206520
• 负责人: David Treumann
• 金额: $ 10.55万
• 依托单位: Northwestern University
• 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 2013-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206520&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.

所提出的研究程序为BATYREV和BURISOV发现的一大类环面超曲面的镜对提供了一种康采维奇-S同调镜对称猜想的方法。该方法从根本上使用了可构造层，特别是Kashiwara和Schapira发展的层的微域理论，并导致了层理论的新结构。在HMS辛边的大体积限制下，该项目旨在展示一个明确的组合骨架，Fukaya范畴有望在其上局部化--即形成dg范畴的一束。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里亚空间，都带有dg范畴的典范簇，即“Kashiwara-Schapira Sheaf”。它是由微局部束技术定义的，并且易于计算。因此，猜想的“Fukaya范畴层”甚至可以在严格的辛几何构造之前被定义和计算，这被认为是等价的。第三个目的是证明Kashiwara-Schapira簇的整体范畴等价于HMS复边的大复数结构极限上的完全复数范畴。该研究方案为BATYREV和BURISOV发现的一大类环面镜面对的Kontsevich？S同调镜面对称猜想提供了一种方法。该方法从根本上使用了可构造层，特别是Kashiwara和Schapira发展的层的微域理论，并导致了层理论的新结构。在HMS辛边的大体积限制下，该项目旨在展示一个明确的组合骨架，Fukaya范畴有望在其上局部化--即形成dg范畴的一束。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里亚空间，都带有dg范畴的典范簇，即“Kashiwara-Schapira Sheaf”。它是由微局部束技术定义的，并且易于计算。因此，猜想的“Fukaya范畴层”甚至可以在严格的辛几何构造之前被定义和计算，这被认为是等价的。第三个目的是证明Kashiwara-Schapira簇的整体范畴等价于HMS复边的大复数结构极限上的完全复数范畴。