Categorical Kahler Geometry and Applications

双有理几何是一门经典的数学学科，其根源可以追溯到古希腊。然而，它仍然提供了许多尚未解决的难题。这个项目的核心部分是用来自同调镜像对称程序的尖端现代方法来解决这些问题。同调镜像对称是一种深刻的几何对偶性，起源于量子场论，并已被用于研究新的现象，并证明代数几何所提出的辛几何中意想不到的结果。这个计画的目的是利用同调镜像对称来介绍辛拓扑在代数几何上的新应用，并解答双有理几何中的经典开放性问题。一名博士后研究员将参与该研究项目的各个方面。该项目将使用基于分类凯勒几何的方法。这种方法最值得注意的应用是证明一般的四维三次体的非理性，这可以说是代数几何的中心问题。更具体地说，对量子D-模奇点的详细研究产生了一种全新类型的双有理不变量。这个新的不变量是一个典型的分解的上同调的四维立方的基础上同时使用双方（代数和辛）的同调镜像对称。四维立方体的例子只是冰山一角。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Birational geometry is a classical mathematical discipline whose roots go back to ancient Greece. Nevertheless, it still offers many difficult unsolved questions. The core part of this project is to tackle these questions with cutting-edge modern methods coming from the homological mirror symmetry program. Homological mirror symmetry is a deep geometric duality that originates in quantum field theory and has been used in studying novel phenomena and proving unexpected results in symplectic geometry suggested by algebraic geometry. This project aims to use homological mirror symmetry to introduce new applications of symplectic topology to algebraic geometry and to answer classical open questions in birational geometry. A postdoctoral fellow will be involved in various aspects of this research project.This project will use an approach based on categorical Kähler geometry. The most notable application of this approach is toward proving the non-rationality of generic, four-dimensional cubics, which is arguably the central problem in algebraic geometry. More specifically, a detailed study of the singularities of quantum D-modules produces a completely new type of birational invariant. This new invariant is a canonical decomposition of the cohomology of a four-dimensional cubic based on simultaneous use of both (algebraic and symplectic) sides of homological mirror symmetry. The example of four-dimensional cubics is only the tip of the iceberg. There are many other potential applications of this approach, for example to uniformization problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.