FRG: Collaborative Research: Algebraic Geometry and Singularities in Positive and Mixed Characteristic

FRG：合作研究：代数几何和正特征和混合特征中的奇点

• 批准号: 2139613
• 负责人: Chenyang Xu
• 金额: $ 13.67万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-05-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2139613&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Algebraic; Geometry

Algebraic Geometry studies algebraic varieties which are geometric objects defined by polynomial equations. One of the most natural problems in this area is to understand the singularities that naturally occur when considering algebraic varieties and how these singularities influence the global geometry of algebraic varieties. In recent years there have been a number of breakthroughs, especially in the case where we consider solutions over the complex numbers. At the same time new techniques and approaches have emerged for studying solutions in positive and mixed characteristics. The primary goal of this collaborative project is to advance and unify these ideas to further understand and solve some of the most challenging programs in both local and global algebraic geometry. In addition the project provides research training opportunities for graduate students. The PIs will investigate singularities in positive and mixed characteristics by using a variety of techniques including those arising from the minimal model program, from the theory of F-singularities, and from Scholze's work on perfectoid algebras and spaces. The PIs will also organize workshops, a summer school and a conference, aimed at training young researchers in this area, disseminating recent results and facilitating further advances and breakthroughs.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。该领域最自然的问题之一是了解考虑代数品种时自然发生的奇异性以及这些奇异性如何影响代数品种的全球几何形状。近年来，有许多突破，尤其是在考虑复数方面的解决方案的情况下。同时，已经出现了用于研究积极和混合特征的解决方案的新技术和方法。该协作项目的主要目标是促进和统一这些想法，以进一步理解和解决本地和全球代数几何学上最具挑战性的计划。此外，该项目为研究生提供了研究培训机会。 PI将通过使用各种技术（包括最小模型程序，F-SINGULULIS的理论）以及Scholze在Scholze在Perfectoid代数和空间上的工作，研究正面和混合特征的奇异性。 PI还将组织研讨会，暑期学校和会议，旨在培训该领域的年轻研究人员，传播最新成果并促进进一步的进步和突破性。这项奖项反映了NSF的法定任务，并被视为值得通过基金会的知识分子和广泛影响的评估来通过评估来获得支持。

项目成果

Uniqueness of the minimizer of the normalized volume function

• DOI: 10.4310/cjm.2021.v9.n1.a2
• 发表时间: 2020-05
• 期刊: Cambridge Journal of Mathematics
• 影响因子: 1.6
• 作者: Chenyang Xu-;Ziquan Zhuang

Finite generation for valuations computing stability thresholds and applications to K-stability

• DOI: 10.4007/annals.2022.196.2.2
• 发表时间: 2021-02
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Yuchen Liu;Chenyang Xu;Ziquan Zhuang

Openness of K-semistability for Fano varieties

• DOI: 10.1215/00127094-2022-0054
• 发表时间: 2019-07
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Harold Blum;Yuchen Liu;Chenyang Xu-

K-stability of Fano varieties: an algebro-geometric approach

• DOI: 10.4171/emss/51
• 发表时间: 2020-11
• 期刊: EMS Surveys in Mathematical Sciences
• 影响因子: 2.3
• 作者: Chenyang Xu