Seminario de Geometría Algebraica

We will begin by reviewing the algebraic geometry of toric varieties and the structure of their derived categories of coherent sheaves. (L1)

Next, we will introduce constructible sheaves. We will define singular supports and sheaf categories with singular support conditions, and briefly review its relation with the wrapped Fukaya categories. (L2)

Next, we will specify to toric varieties and introduce the main ideas in the works of Bondal, Fang-Liu-Treumann-Zaslow and Kuwagaki, as well as the correspondence between line bundles and twisted polytope sheaves studied by Zhou. We will review the approaches taken by these authors to prove the Coherent-Constructible Correspondence through explicit examples. In particular, we will calculate some examples by matching the Cech resolution of line bundles with the twisted polytope sheaves. We will also explain the correspondence between tensoring with a line bundle and convolution by the corresponding twisted polytope sheaf studied recently by Bose-Williams. (L3-L4)

Finally, we will place the coherent–constructible correspondence in the broader context of homological mirror symmetry and explain the connection to the approaches taken by Abouzaid and Hanlon-Hicks. We will discuss a few recent advances and open problems as an invitation. (L5)

We started in L1 by reviewing the algebraic geometry of toric varieties and the structure of their derived categories of coherent sheaves.

In this L2, we will introduce constructible sheaves. We will define singular supports and sheaf categories with singular support conditions, and briefly review its relation with the wrapped Fukaya categories.

In this series of 5 lectures, we will develop the coherent–constructible correspondence for toric varieties, which in combination with a result of Ganatra-Pardon-Shende establishes Homological Mirror Symmetry for toric varieties. The goal is to explain the mechanism of this correspondence so that participants can use it as a practical tool for studying toric mirror symmetry and its potential applications.

We will begin by reviewing the algebraic geometry of toric varieties and the structure of their derived categories of coherent sheaves. (L1)

Next, we will introduce constructible sheaves. We will define singular supports and sheaf categories with singular support conditions, and briefly review its relation with the wrapped Fukaya categories. (L2)

Next, we will specify to toric varieties and introduce the main ideas in the works of Bondal, Fang-Liu-Treumann-Zaslow and Kuwagaki, as well as the correspondence between line bundles and twisted polytope sheaves studied by Zhou. We will review the approaches taken by these authors to prove the Coherent-Constructible Correspondence through explicit examples. In particular, we will calculate some examples by matching the Cech resolution of line bundles with the twisted polytope sheaves. We will also explain the correspondence between tensoring with a line bundle and convolution by the corresponding twisted polytope sheaf studied recently by Bose-Williams. (L3-L4)

Finally, we will place the coherent–constructible correspondence in the broader context of homological mirror symmetry and explain the connection to the approaches taken by Abouzaid and Hanlon-Hicks. We will discuss a few recent advances and open problems as an invitation. (L5)