K-Stability, Moduli Spaces, and Singularities

K-稳定性、模空间和奇点

• 批准号: 2001317
• 负责人: Yuchen Liu
• 金额: $ 16万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-05-01 至 2021-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001317&HistoricalAwards=false
• 关键词: Stability; Moduli; Spaces; Singularities

Varieties are geometric objects as common solutions of multivariable polynomial equations. Varieties can vary by changing the coefficients of polynomials, and it is an important question on when their geometric shapes will remain in a nice way. When the varieties are positively curved called Fano varieties, a natural approach is to look at those with Einstein structures, a notion originated from general relativity. The Einstein structures are deeply related to K-stability, a stability theory in algebraic geometry. The PI will study Fano varieties with Einstein structures, and how they degenerate and develop singularities. The problems are foundational and have deep connections among different areas of mathematics, such as algebraic geometry, differential geometry, commutative algebra, and mathematical physics.The PI will investigate K-stability of Fano varieties, their moduli spaces, and geometry of singularities. The PI aims to show that moduli spaces of K-polystable Fano varieties are compact, which is related to finding a Harder-Narasimhan filtration on K-unstable Fano varieties. The PI will study explicit moduli spaces of log Fano pairs and their wall crossings, and estabilish connections to moduli spaces of curves and K3 surfaces. The PI will study distribution of local volumes of Kawamata log terminal singularities, where he aims to show that 0 is the only accumulation point. The proposed projects above will use classical techniques from the Minimal Model Program and Geometric Invariant Theory, as well as new tools from algebraic theory of K-stability, K-moduli spaces, and normalized volumes developed by the PI and collaborators.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.

品种是多变量多项式方程的常见解决方案。品种可能会因更改多项式的系数而有所不同，这是一个重要的问题，即它们的几何形状何时保持不错的方式。当品种被呈正弯曲称为FANO品种时，一种自然的方法是观察那些具有爱因斯坦结构的方法，一个概念起源于一般相对论。爱因斯坦结构与K稳定性密切相关，K稳定性是代数几何形状中的稳定性理论。 PI将使用爱因斯坦结构研究Fano品种，以及它们如何退化和发展奇异性。这些问题是基本的，并且在不同领域的数学领域之间建立了深厚的联系，例如代数几何，差异几何，交换代数和数学物理学。 PI的目的是表明K-Polystable Fano品种的模量是紧凑的，这与在K- Unstable Fano品种上找到更硬的Narasimhan过滤有关。 PI将研究对数扇形对的显式模量及其壁横梁的显式模量空间，并研究与曲线和K3表面模量空间的结合。 PI将研究Kawamata对数末端奇点的局部体积的分布，他的目标是表明0是唯一的积累点。上面提出的项目将使用最小模型计划和几何不变理论的经典技术，以及来自K稳定性，K-Moduli空间的代数理论和PI和协作者开发的标准化量的新工具。该奖项颁发了NSF的法定任务，并反映了通过评估的Intellia in Iffectia crit and Founlitial的支持，并反映出了该奖项。

项目成果

On K‐stability of some del Pezzo surfaces of Fano index 2

关于 Fano 指数 2 的某些 del Pezzo 曲面的 K 稳定性

• DOI: 10.1112/blms.12581
• 发表时间: 2022
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Liu, Yuchen;Petracci, Andrea
• 通讯作者: Petracci, Andrea

ACC for local volumes and boundedness of singularities

ACC 用于局部体积和奇点有界

• DOI: 10.1090/jag/799
• 发表时间: 2023
• 期刊: Journal of Algebraic Geometry
• 影响因子: 1.8
• 作者: Han, Jingjun;Liu, Yuchen;Qi, Lu
• 通讯作者: Qi, Lu

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

• DOI: 10.1017/s1474748021000384
• 发表时间: 2020-06
• 期刊: Journal of the Institute of Mathematics of Jussieu
• 影响因子: 0.9
• 作者: Kenneth Ascher;Kristin DeVleming;Yuchen Liu

On properness of K-moduli spaces and optimal degenerations of Fano varieties

• DOI: 10.1007/s00029-021-00694-7
• 发表时间: 2021-07
• 期刊: Selecta Mathematica
• 作者: Harold Blum;Daniel Halpern-Leistner;Yuchen Liu;Chenyang Xu