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.

这是代数几何区域的一个项目，这是对多项式方程定义的形状的研究。该领域的一个基本话题是模量理论，该理论旨在构建和研究参数所有这些形状的空间。对于低维形状，这种模量空间是充分理解的，但在较高的维度中却鲜为人知。最近，来自差异几何形状的新观点导致了某些Fano品种的模量空间的构建，这些空间是一类正面弯曲的形状。该项目的目的是将新开发的技术从后一种结构应用到其他形状类别的模量理论。该项目为研究生提供了培训机会。K-STABILTY是差分几何图形引入的代数概念，以表征复杂的投影品种某些规范指标的存在。虽然可以为任何投射品种定义该概念，但最近的研究集中在Fano品种的情况下，并在代数几何形状的应用中达到顶峰：构造投射模量空间参数化K-Stable Fano品种。该项目的目的是使用来自Birational几何形状和K稳定性的工具来构建其他类别品种的模量空间。特别是，PI将构建K-Unstable Fano品种的某些模量，这是由研究Kahler-Icki solitons在差异几何形状中的研究所激发的。接下来，PI将开发用于研究K-琐事品种模量的新方法。最后，PI将将从Bibational几何形状应用于两极化品种的K稳定性的工具。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。