Geometry of analytic and algebraic varieties

解析几何和代数簇

• 批准号: 2301374
• 负责人: Christopher Hacon
• 金额: $ 33万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2301374&HistoricalAwards=false
• 关键词: Geometry; analytic; algebraic; varieties

This research project is in the field of algebraic geometry, one of the core areas of pure mathematics whose roots date back to the ancient Greeks. At its heart, algebraic geometry studies the geometry of the solution of polynomial equations. These are the simplest possible equations and so, not surprisingly, they play an important role in almost any scientific discipline. In particular algebraic geometry has close ties with differential and analytic geometry, commutative algebra, topology, number theory, physics, theoretical computer science, cryptography, and many areas of applied mathematics. The most fundamental problem in algebraic geometry is to classify all geometric objects defined by polynomial equations. The minimal model program is the most successful approach to this classification program and has recently had extraordinary success in classifying complex varieties, i.e. solution sets consisting of complex numbers. It is hoped that these techniques will extend to other contexts such as varieties defined over fields of positive characteristics and to complex analytic varieties. The PI will involve graduate students and post-docs in various aspects of this project.This project will generalize the results of the minimal model program over the complex numbers to the case of varieties over algebraically closed fields of positive characteristic (and to the case of mixed characteristics) as well as the case of Kahler varieties. In particular, the PI will show that the minimal model program holds for compact Kahler klt threefolds and fourfolds and will develop the theory of generalized klt Kahler varieties. The PI will investigate the minimal model program for threefolds in characteristic 3 and the pluricanonical maps of projective 3-folds over algebraically closed fields of large characteristics. The minimal model program in mixed characteristic will also be developed.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.

该研究项目位于代数几何学领域，这是纯数学的核心领域之一，其根源可以追溯到古希腊人。代数几何学研究了多项式方程溶液的几何形状。这些是最简单的方程式，因此，毫不奇怪，它们在几乎任何科学学科中都起着重要作用。尤其是代数几何形状与差分和分析几何形状，交换代数，拓扑，数字理论，物理学，理论计算机科学，密码学以及许多应用数学领域的紧密联系。代数几何形状中最基本的问题是对多项式方程定义的所有几何对象进行分类。最小模型程序是该分类程序最成功的方法，最近在分类复杂品种（即由复数组成的解决方案集）中取得了非凡的成功。希望这些技术将扩展到其他环境，例如在积极特征和复杂分析品种的领域定义的品种。 PI将在该项目的各个方面涉及研究生和培训。该项目将将最小模型计划的结果概括为复数的结果，而不是阳性特征（以及混合特征的情况）的代数封闭场上的品种，以及Kahler品种的情况。特别是，PI将表明，最小模型程序适用于紧凑的Kahler Klt三倍和四倍，并将发展为广义KLT Kahler品种的理论。 PI将调查特征3中三倍的最小模型程序，以及在大型特征的代数闭合场上投影3倍的射击图。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分子优点和更广泛的审查标准来评估，这是值得支持的。