K-Stability, Moduli, and Birational Geometry

K 稳定性、模量和双有理几何

• 批准号: 2200690
• 负责人: Harold Blum
• 金额: $ 24.54万
• 依托单位: University of Utah
• 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=2200690&HistoricalAwards=false
• 关键词: Stability; Moduli; Birational; Geometry

This is a project in the area of algebraic geometry, which is the study of shapes defined by polynomial equations. A fundamental topic in the field is moduli theory, which aims to construct and study spaces that parametrize all such shapes. For low dimensional shapes such moduli spaces are well understood, but less is known in higher dimensions. Recently, new perspectives from differential geometry have led to the construction of moduli spaces of certain Fano varieties, which are a class of positively curved shapes. The aim of this project is to apply newly developed techniques from the latter construction to the moduli theory of other classes of shapes. The project offers training opportunities for graduate students.K-stabilty is an algebraic notion introduced by differential geometers to characterize the existence of certain canonical metrics on complex projective varieties. While the notion can be defined for any projective variety, recent research has focused on the case of Fano varieties and culminated in an application to algebraic geometry: the construction of projective moduli spaces parametrizing K-stable Fano varieties. The goal of this project is to use tools from birational geometry and K-stability to construct moduli spaces of other classes of varieties. In particular, the PI will construct certain moduli of K-unstable Fano varieties, motivated by the study of Kahler-Ricci solitons in differential geometry. Next, the PI will develop new approaches for studying moduli of K-trivial varieties. Finally, the PI will apply tools from birational geometry to the study of K-stability of polarized varieties.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.

这是代数几何领域的一个项目，它是由多项式方程定义的形状的研究。该领域的一个基本主题是模量理论，其目的是构建和研究参数化所有这些形状的空间。对于低维形状，这样的模空间是很好理解的，但在更高的维度中知之甚少。最近，微分几何的新观点导致了某些Fano簇的模空间的构建，这是一类正弯曲的形状。这个项目的目的是应用新开发的技术，从后者建设的模量理论的其他类别的形状。K-稳定性是微分几何学家提出的一个代数概念，用来刻画复射影簇上某些典型度量的存在性。虽然这个概念可以定义为任何投射簇，但最近的研究集中在Fano簇的情况下，并最终应用于代数几何：构建投射模空间参数化K-稳定Fano簇。这个项目的目标是使用双有理几何和K-稳定性的工具来构造其他类簇的模空间。特别地，PI将构造K-不稳定Fano簇的某些模，其动机是微分几何中的Kahler-Ricci孤子的研究。接下来，PI将开发新的方法来研究K-平凡簇的模。最后，PI将应用双有理几何的工具来研究极化品种的K稳定性。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。