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.

代数几何形状是代数品种的研究，这是由多项式方程系统定义的几何对象。该受试者的驱动目标是代数品种的分类，涉及诸如如何使用代数函数来确定何时可以将一种变体转化为另一种品种的问题，或者如何使用高度约束的几何特性构造品种。这些经典问题和该主题的现代工具之间已经发现了令人惊讶的连接，尤其是衍生的类别及其对象的模量空间。该项目旨在进一步开发这些工具，以取得出色的猜想。通过会议，研讨会和指导机会，该项目还将在该领域培训新一代的数学家。该项目具有三个相关的研究目标。首先是使用奇异性的非交通性分辨率来证明有关连贯滑轮类别的结构性结果，这是由邦德 - 奥洛夫（Bondal-Orolov）和库兹尼托夫（Kuznetsov）的猜想所激发的，将这些类别与Birational几何相关。第二个目标是在一般环境和特别感兴趣的情况下构建bridgeland稳定性条件并研究其模量空间的几何形状。第三个目标是将上述主题的进步应用于经典问题，例如Hyperkahler品种的分类以及立方四倍的合理性问题。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的审查标准通过评估来通过评估来支持的。