K-Stability, Moduli Spaces, and Singularities

K-稳定性、模空间和奇点

• 批准号: 2148266
• 负责人: Yuchen Liu
• 金额: $ 16万
• 依托单位: Northwestern 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: 2021-07-01 至 2023-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2148266&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.

簇是多元多项式方程的公共解的几何对象。变量可以通过改变多项式的系数而变化，而变量的几何形状何时保持良好是一个重要的问题。当簇是正弯曲的称为法诺簇时，一个自然的方法是观察那些具有爱因斯坦结构的簇，这是一个起源于广义相对论的概念。爱因斯坦结构与代数几何中的稳定性理论K-稳定性密切相关。PI将研究具有爱因斯坦结构的Fano簇，以及它们如何退化和发展奇点。这些问题是基础性的，并且在不同的数学领域之间有着深刻的联系，例如代数几何，微分几何，交换代数和数学物理。PI将研究Fano簇的K-稳定性，它们的模空间和奇点几何。PI旨在证明K-多稳定Fano簇的模空间是紧的，这与找到K-不稳定Fano簇上的Harder-Narasimhan滤子有关。PI将研究对数Fano对及其壁交叉的显式模空间，并建立与曲线和K3曲面的模空间的连接。PI将研究Kawamata日志终端奇点的局部体积分布，他的目标是证明0是唯一的聚集点。上述项目将使用最小模型程序和几何不变理论的经典技术，以及PI和合作者开发的K稳定性代数理论、K模空间和归一化体积的新工具。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

K-stability of cubic fourfolds

• DOI: 10.1515/crelle-2022-0002
• 发表时间: 2020-07
• 期刊: Journal für die reine und angewandte Mathematik (Crelles Journal)
• 作者: Yuchen Liu

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

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-stability of Fano threefolds of rank 2 and degree 14 as double covers

Fano 的 K 稳定性是双覆盖的 2 级和 14 级的三倍

• DOI: 10.1007/s00209-022-03192-4
• 发表时间: 2023
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Liu, Yuchen