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: 2052936
• 负责人: Daniel Halpern-Leistner
• 金额: $ 24.75万
• 依托单位: Cornell 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=2052936&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.

代数几何是研究代数变量，即由多项式方程组定义的几何对象的学科。该学科的一个驱动目标是代数变量的分类，涉及诸如如何确定何时可以使用代数函数将一个变量转换为另一个变量之类的问题，或者如何构造具有高度约束几何性质的变量。在这些经典问题和现代工具之间已经发现了惊人的联系，特别是派生范畴和它们的模空间的对象。该项目旨在进一步开发这些工具，以便在杰出的猜想上取得进展。通过会议、研讨会和指导机会，该项目还将培养这一领域的新一代数学家。该项目有三个相关的研究目标。第一个是利用奇异点的非交换解析来证明相干束的派生范畴的结构结果，这是由bond - orlov和Kuznetsov的关于这些范畴与两族几何的猜想所激发的。第二个目标是构造桥地稳定性条件，并研究它们的模空间的几何，无论是在一般情况下还是在特殊情况下。第三个目标是将上述主题的进展应用于经典问题，如超kahler变量的分类和三次四倍的合理性问题。该奖项反映了美国国家科学基金会的法定使命，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。