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: 1952531
• 负责人: Chenyang Xu
• 金额: $ 13.67万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-06-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952531&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.

代数几何研究代数簇，代数簇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理解在考虑代数簇时自然发生的奇点以及这些奇点如何影响代数簇的全局几何。近年来，已经取得了许多突破，特别是在我们考虑复数解决方案的情况下。与此同时，用于研究正特性和混合特性解决方案的新技术和新方法也出现了。该合作项目的主要目标是推进和统一这些想法，以进一步理解和解决局部和全局代数几何中一些最具挑战性的程序。此外，该项目还为研究生提供研究培训机会。 PI 将使用各种技术来研究正特征和混合特征中的奇点，包括源自最小模型程序、F 奇点理论以及 Scholze 在完美代数和空间方面的工作的技术。 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展和突破。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。

项目成果

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

On positivity of the CM line bundle on K-moduli spaces

• DOI: 10.4007/annals.2020.192.3.7
• 发表时间: 2019-12
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 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