K-Stability and Birational Geometry of Fano Varieties

Fano 品种的 K 稳定性和双有理几何结构

• 批准号: 2240926
• 负责人: Ziquan Zhuang
• 金额: $ 15.47万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2240926&HistoricalAwards=false
• 关键词: Stability; Birational; Geometry; Fano; Varieties

Algebraic geometry is the study of the geometric objects – called algebraic varieties – defined by systems of polynomial equations. A fundamental problem is to find and classify algebraic varieties with nice geometric structures. When the varieties are positively curved, it is natural to search for metrics that satisfy Einstein's equation from general relativity. The main theme of this project is to study such Einstein metrics using an algebro-geometric stability theory called K-stability, and to investigate their interaction with other structures that lie at the crossroads of algebraic geometry, differential geometry, commutative algebra and mathematical physics. The Principal Investigator plans to organize activities to enhance communication between different groups of researchers (including under-represented groups).The project has three closely related parts. The first is to prove K-stability of explicit Fano varieties, such as complete intersections and their mildly singular degenerations. With the help of birational geometry, the PI aims to show that most of these Fano varieties are K-stable and explicitly describe some of their corresponding compact K-moduli. The second part is to study the moduli theory of Fano varieties and singularities using K-stability. The goal is to construct compact moduli spaces that parametrize certain optimal degenerations of the varieties or singularities. The last part is to explore the connection of K-stability with other geometric properties of Fano varieties, especially their birational rigidity and rationality.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.

代数几何是对由多项式方程组定义的几何对象（称为代数簇）的研究，一个基本问题是找到具有良好几何结构的代数簇并进行分类，当簇呈正曲线时，很自然地需要寻找度量。满足广义相对论中的爱因斯坦方程 该项目的主题是使用称为 K 稳定性的代数几何稳定性理论来研究此类爱因斯坦度量，并研究它们与其他结构的相互作用。在代数几何、微分几何、交换代数和数学物理学的十字路口，首席研究员计划组织活动以加强不同研究群体（包括代表性不足的群体）之间的交流。该项目分为三个密切相关的部分。为了证明显式 Fano 簇的 K 稳定性，例如完全交集及其轻微奇异退化，PI 旨在证明这些 Fano 簇中的大多数都是 K 稳定的，并明确描述了一些。第二部分是使用 K 稳定性研究 Fano 簇和奇点的模理论，目标是构造参数化簇或奇点的某些最优退化的紧致模空间。探索 K 稳定性与 Fano 簇的其他几何特性的联系，特别是它们的双有理刚性和合理性。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评估进行评估，被认为值得支持影响审查标准。