Hyperkähler Geometry, Stability Conditions, and Moduli Spaces

Hyperkühler 几何、稳定性条件和模空间

• 批准号: 2101789
• 负责人: Xiaolei Zhao
• 金额: $ 10万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101789&HistoricalAwards=false
• 关键词: Hyperk; hler; Geometry; Stability; Conditions

Algebraic geometry studies the solution sets of systems of polynomial equations. A theme of the subject is to classify different geometric shapes that naturally arise in the study of these solution sets. This has a root in classical problems, and nowadays it finds deep connections with modern tools, especially moduli spaces, stability conditions, and derived categories. This project aims at further applications of these tools to concrete geometric questions open for a long time. It also supports graduate students to explore the subject through travel opportunities to conferences and workshops.This project contains three related research goals: The first is a systematic study of the connection between Fano geometry and hyperkähler geometry, like generalized Kummer varieties and Gushel-Mukai varieties, using tools from derived categories. The second goal is to construct stability conditions in several important cases including Calabi-Yau threefolds, Fano fourfolds, and hyperkähler varieties. The main approach is via various restriction theorems to reduce to lower dimensional cases. The third goal is to revisit several long-standing questions on moduli of sheaves on surfaces, including the computation of Picard groups and linear series. The recent development in moduli theory will play a central role in this study.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.

代数几何研究多项式方程组的解集。该主题的一个主题是对这些解决方案集的研究中自然出现的不同几何形状进行分类。这源于经典问题，如今它与现代工具，特别是模空间、稳定性条件和派生范畴有着深刻的联系。该项目旨在进一步将这些工具应用于长期开放的具体几何问题。它还支持研究生通过参加会议和研讨会的机会来探索该主题。该项目包含三个相关的研究目标：第一个是使用派生类别的工具系统研究 Fano 几何和 hyperkähler 几何之间的联系，例如广义 Kummer 簇和 Gushel-Mukai 簇。第二个目标是在几个重要的情况下构建稳定性条件，包括 Calabi-Yau 三重、Fano 四重和 hyperkähler 变种。主要方法是通过各种限制定理来降低维数。第三个目标是重新审视关于表面滑轮模量的几个长期存在的问题，包括皮卡德群和线性级数的计算。模理论的最新发展将在这项研究中发挥核心作用。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。

项目成果

A refined derived Torelli theorem for Enriques surfaces

Enriques 曲面的精化派生 Torelli 定理

• DOI: 10.1007/s00208-020-02113-2
• 发表时间: 2021
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Li, Chunyi;Nuer, Howard;Stellari, Paolo;Zhao, Xiaolei
• 通讯作者: Zhao, Xiaolei

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

• DOI: 10.2140/gt.2022.26.3055
• 发表时间: 2019-12
• 期刊: Geometry & Topology
• 作者: Alexander Perry;L. Pertusi;Xiaolei Zhao

Stability manifolds of varieties with finite Albanese morphisms

具有有限阿尔巴尼态射的簇的稳定性流形

• 发表时间: 2022
• 期刊: Translations American Mathematical Society
• 作者: Fu, Lie;Li, Chunyi;Zhao, Xiaolei
• 通讯作者: Zhao, Xiaolei

A refined derived Torelli theorem for enriques surfaces, II: the non-generic case

恩里克斯曲面的精化导出托雷利定理，II：非泛型情况

• DOI: 10.1007/s00209-021-02930-4
• 发表时间: 2022
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Li, Chunyi;Stellari, Paolo;Zhao, Xiaolei
• 通讯作者: Zhao, Xiaolei

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

• DOI: 10.1016/j.aim.2022.108584
• 发表时间: 2020-07
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Chunyi Li;L. Pertusi;Xiaolei Zhao