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.

代数几何研究代数品种是由多项式方程定义的几何对象。它在相邻领域（例如差异几何形状）中起着重要作用。连接这些区域的一个自然而重要的问题是找到和分类具有良好几何结构的代数品种，例如具有恒定曲率的指标。近年来，当代数品种积极弯曲时，更高维代数的几何形状的发展导致了一些突破性的突破。该项目的目的是提高这些想法，以进一步了解一般代数品种的几何结构。此外，该项目还通过研讨会，暑期学校和其他活动为学生和早期职业研究人员提供培训和研究机会。从更详细的角度来看，该项目是由有影响力的Yau-Tian-Donaldson猜想的促进的，认为规范指标的存在应等于某些代数稳定性条件，即所知的K稳定性。 PI将为一般的Kawamata对数末端奇异性开发局部K稳定理论，以理解其生育几何形状和模量。 PI还将研究Birational几何学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稳定性的代数理论从Fano案例中提高到一般两极化的情况，并攻击与Yau-Tian-Donaldson猜想有关的一些开放问题。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识和更广泛影响的评估来通过评估来获得支持的。