Categorical Kahler Geometry and Applications

起源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，它仍然提供了许多棘手的悬而未决的问题。这个项目的核心部分是用来自同源镜像对称程序的尖端现代方法来解决这些问题。同调镜像对称性是起源于量子场论的一种深几何对偶，已被用于研究新现象和证明由代数几何提出的辛几何中意想不到的结果。本课题旨在利用同调镜像对称性，介绍辛拓扑在代数几何中的新应用，并回答二次几何中的经典开放问题。一个博士后研究员将参与这个研究项目的各个方面。这个项目将使用一种基于分类Kähler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曲线的非合理性，这可以说是代数几何中的中心问题。更具体地说，对量子D-模的奇点的详细研究产生了一种全新的二元不变量。这种新的不变量是基于同时利用同调镜像对称性的两个边(代数和辛)的四维三次空间上同调的典型分解。四维立方体的例子只是冰山一角。这种方法还有许多其他潜在的应用，例如在统一问题上。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.