Mirror Symmetry and the Microlocal Theory of Sheaves

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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.