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.

Birational Geatry是一门经典的数学学科，其根源可以追溯到古希腊。尽管如此，它仍然提供了许多困难的未解决问题。该项目的核心部分是通过来自同源镜面对称程序的尖端现代方法来解决这些问题。同源镜对称性是一种深度几何偶性，起源于量子场理论，并已用于研究新现象，并证明了代数几何形状建议的对称几何形状的意外结果。该项目旨在使用同源镜像对称性，将对称拓扑的新应用引入代数几何形状，并在生物几何形状中回答经典的开放问题。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Kähler几何形状的方法。这种方法的最显着应用是证明通用，四维立方体的非理性性，这可以说是代数几何学中的核心问题。更具体地说，对量子D模块的奇异性的详细研究产生了一种全新的生物不变性。这个新的不变是基于同源镜子对称性的（代数和符号镜）的简单使用（代数和符号镜）边的简单使用（代数和符号镜）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。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估，被认为是珍贵的支持。