FRG: Collaborative Research: Derived Categories, Moduli Spaces, and Classical Algebraic Geometry

FRG：协作研究：派生范畴、模空间和经典代数几何

• 批准号: 2052934
• 负责人: Giulia Sacca
• 金额: $ 41.63万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2052934&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Derived; Categories

Algebraic geometry is the study of algebraic varieties, the geometric objects defined by systems of polynomial equations. A driving goal of the subject is the classification of algebraic varieties, involving questions like how to determine when one variety can be transformed into another using algebraic functions, or how to construct varieties with highly constrained geometric properties. Surprising connections have been found between these classical problems and modern tools in the subject, especially derived categories and their moduli spaces of objects. This project aims to further develop these tools in order to make progress on outstanding conjectures. Through conferences, workshops, and mentoring opportunities, the project will also train a new generation of mathematicians in this area. The project has three related research goals. The first is to use noncommutative resolutions of singularities to prove structural results about derived categories of coherent sheaves, motivated by conjectures of Bondal-Orlov and Kuznetsov relating these categories to birational geometry. The second goal is to construct Bridgeland stability conditions and study the geometry of their moduli spaces, both in general settings and cases of special interest. The third goal is to apply advances on the above topics to classical problems, like the classification of hyperkahler varieties and the rationality problem for cubic fourfolds.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.

代数几何是对代数簇的研究，代数簇是由多项式方程组定义的几何对象。该主题的一个驱动目标是代数簇的分类，涉及诸如如何确定何时可以使用代数函数将一个簇转换为另一个簇，或者如何构造具有高度约束的几何性质的簇之类的问题。在这些经典问题和该学科的现代工具之间发现了令人惊讶的联系，特别是对象的派生范畴及其模空间。该项目旨在进一步开发这些工具，以便在悬而未决的猜想上取得进展。通过会议、研讨会和指导机会，该项目还将培训该领域的新一代数学家。该项目有三个相关的研究目标。第一个是使用奇点的非交换解析来证明有关相干滑轮的派生类别的结构结果，其动机是 Bondal-Orlov 和 Kuznetsov 将这些类别与双有理几何相关联的猜想。第二个目标是构建布里奇兰稳定性条件并研究其模空间的几何形状，无论是在一般情况下还是在特别感兴趣的情况下。第三个目标是将上述主题的进展应用于经典问题，例如超级卡勒簇的分类和立方四重的合理性问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。