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.

代数几何学研究多项式方程系统的解决方案集。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。这在经典问题上具有根源，如今，它发现了与现代工具，尤其是莫德利空间，稳定条件和派生类别的深厚联系。该项目旨在进一步应用这些工具，以长期开放。它还支持研究生通过旅行机会和研讨会来探索主题。本项目包含三个相关的研究目标：第一个是对Fano Geemotry与Hyperkähler几何形状之间联系的系统研究，例如使用来自派生类别的工具，例如使用广义的Kummer品种和Gushel-Mukai品种。第二个目标是在几种重要情况下构建稳定条件，包括卡拉比（Calabi-Yau）三倍，fano四倍和超卡勒（Hyperkähller）变化。主要方法是通过各种限制定理来降低降低维度的情况。第三个目标是重新审视有关表面上带状杆子模束的几个长期问题，包括对PICARD组和线性系列的计算。 Moduli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的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。

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

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

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