K-stability and Higher Dimensional Geometry

K 稳定性和高维几何

• 批准号: 2153115
• 负责人: Chenyang Xu
• 金额: $ 33万
• 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: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153115&HistoricalAwards=false
• 关键词: stability; Higher; Dimensional; Geometry

The Principal Investigator (PI) will study varieties. Varieties are defined as the set of solutions of systems of polynomial equations. They are fairly easy to compute and Nash proved every space can be well approximated by varieties. The main aim of the project is to understand how varieties vary if we change the coefficients of the defining polynomial equations, especially for the ones which are positively curved. Such varieties are called Fano varieties. In particular the research will try to understand situations when a family of Fano varieties degenerates to one with singularities. The PI intends to prove that among all Fano varieties, the K-polystable ones can be parametrised by a universal space, called moduli space. As part of the this project, the PI aims to show the moduli space is Hausdorff and compact. The PI aims to understand which Fano varieties are K-semistable by understanding concrete examples as well as some general phenomena. The PI aims to understand the degeneration of Calabi-Yau manifolds, through the interplay between birational and non-archimedean geometry.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.

主要研究者（PI）将研究品种。簇被定义为多项式方程组的解的集合。他们是相当容易计算和纳什证明了每一个空间可以很好地近似品种。该项目的主要目的是了解如果我们改变定义多项式方程的系数，特别是对于正曲线的多项式方程，变量会如何变化。这些品种被称为法诺品种。特别是研究将试图了解的情况下，一个家庭的法诺品种退化到一个奇点。PI的目的是证明在所有的Fano簇中，K-多稳定簇可以被一个称为模空间的泛空间参数化。作为该项目的一部分，PI旨在证明模空间是豪斯多夫和紧的。PI的目的是通过理解具体的例子以及一些一般现象来理解哪些Fano簇是K-半稳定的。PI旨在通过双有理和非阿基米德几何之间的相互作用来理解卡-丘流形的退化。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Algebraicity of the metric tangent cones and equivariant K-stability

• DOI: 10.1090/jams/974
• 发表时间: 2018-05
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Chi Li;Xiaowei Wang;Chenyang Xu-