Categorical Kahler Geometry and Applications

分类卡勒几何及其应用

• 批准号: 2001319
• 负责人: Ludmil Katzarkov
• 金额: $ 22.5万
• 依托单位: University of Miami
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-05-01 至 2024-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001319&HistoricalAwards=false
• 关键词: Categorical; Kahler; Geometry; Applications

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.

双有理几何是一门经典数学学科，其根源可以追溯到古希腊，然而，它仍然提出了许多尚未解决的难题。该项目的核心部分是利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起源于量子场论的深层几何对偶性，已用于研究新现象并证明代数几何提出的辛几何中意想不到的结果。该项目旨在利用同调镜像对称来引入新的应用。辛拓扑到代数几何，并回答双有理几何中的经典开放问题。该项目将使用基于分类凯勒几何的方法。证明泛型四维立方的非理性，这是代数几何中备受争议的核心问题。更具体地说，对量子 D 模奇点的详细研究产生了一种全新的类型。这个新的不变量是基于同时使用同调镜像对称的两侧（代数和辛）的四维三次上同调的规范分解。四维三次的例子只是冰山一角。这种方法还有许多其他潜在的应用，例如统一化问题。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持。审查标准。