K-stability and Higher Dimensional Geometry

K 稳定性和高维几何

• 批准号: 1901849
• 负责人: Chenyang Xu
• 金额: $ 33万
• 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: 2019
• 资助国家: 美国
• 起止时间: 2019-06-01 至 2021-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901849&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.

首席研究员（PI）将研究品种。品种定义为多项式方程系统的解决方案集。它们很容易计算，纳什证明了每个空间都可以通过品种很好地近似。该项目的主要目的是了解，如果我们更改定义多项式方程的系数，尤其是对于呈阳性的曲线的系数，品种会如何变化。这些品种称为Fano品种。特别是，当Fano品种退化为奇异性时，这项研究将尝试了解情况。 PI打算证明，在所有FANO品种中，K-Polystable可以通过称为Moduli空间的通用空间来参数。作为该项目的一部分，PI的目的是显示模量空间是Hausdorff和紧凑的。 PI旨在通过了解具体示例以及某些一般现象来了解哪些FANO品种可以k ksemessist。 PI的目的是通过Birational和非Archimedean几何形状之间的相互作用来了解Calabi-Yau流形的退化。该奖项反映了NSF的法定任务，并被认为值得通过基金会的知识分子优点和更广泛的影响标准通过评估来进行评估。

项目成果

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-