CAREER: Birational Geometry and K-stability of Algebraic Varieties

职业：双有理几何和代数簇的 K 稳定性

• 批准号: 2234736
• 负责人: Ziquan Zhuang
• 金额: $ 55万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2234736&HistoricalAwards=false
• 关键词: CAREER; Birational; Geometry; stability; Algebraic

Algebraic geometry studies algebraic varieties which are geometric objects defined by polynomial equations. It plays an important role in neighboring fields such as differential geometry. A natural and important question connecting these areas is to find and classify algebraic varieties with nice geometric structures, such as metrics with constant curvature. In recent years, the development of higher dimensional algebraic geometry has led to a number of breakthroughs in the search for such metrics when the algebraic varieties are positively curved. The goal of this project is to advance these ideas to further understand the geometric structure of general algebraic varieties. In addition the project provides training and research opportunities for students and early-career researchers in related areas, through seminars, summer schools, and other activities.In more detail, the project is motivated by the influential Yau-Tian-Donaldson Conjecture that existence of canonical metrics should be equivalent to some algebraic stability condition known as K-stability. The PI will to develop a local K-stability theory for general Kawamata log terminal singularities, with a view towards the understanding of their birational geometry and moduli. The PI will also investigate some interesting new questions in birational geometry that are inspired by recent progress in K-stability. Finally the project will further advance the algebraic theory of K-stability from the Fano case to the general polarized case, and attack some open problems related to the Yau-Tian-Donaldson conjecture.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.

代数几何研究代数簇，代数簇是由多项式方程定义的几何对象。它在微分几何等邻近领域发挥着重要作用。连接这些领域的一个自然而重要的问题是寻找和分类具有良好几何结构的代数簇，例如具有恒定曲率的度量。近年来，高维代数几何的发展使得在代数簇为正曲线时的度量寻找方面取得了一系列突破。该项目的目标是推进这些想法，以进一步理解一般代数簇的几何结构。此外，该项目通过研讨会、暑期学校和其他活动，为相关领域的学生和早期职业研究人员提供培训和研究机会。更详细地说，该项目的动机是有影响力的丘田唐纳森猜想，即存在规范度量应该等同于一些称为 K 稳定性的代数稳定性条件。 PI 将针对一般 Kawamata 对数终端奇点开发局部 K 稳定性理论，以期理解它们的双有理几何和模量。 PI 还将研究双有理几何中一些有趣的新问题，这些问题受到 K 稳定性最新进展的启发。最后，该项目将进一步将K-稳定性代数理论从Fano案例推进到一般极化案例，并解决与Yau-Tian-Donaldson猜想相关的一些开放性问题。该奖项体现了NSF的法定使命，被认为是值得的通过使用基金会的智力优势和更广泛的影响审查标准进行评估来提供支持。