K-Stability and Birational Geometry of Fano Varieties

Fano 品种的 K 稳定性和双有理几何结构

基本信息

批准号：
2055531
负责人：
Ziquan Zhuang
金额：
$ 15.47万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2055531&HistoricalAwards=false
关键词：
Stability Birational Geometry Fano Varieties

项目摘要

Algebraic geometry is the study of the geometric objects – called algebraic varieties – defined by systems of polynomial equations. A fundamental problem is to find and classify algebraic varieties with nice geometric structures. When the varieties are positively curved, it is natural to search for metrics that satisfy Einstein's equation from general relativity. The main theme of this project is to study such Einstein metrics using an algebro-geometric stability theory called K-stability, and to investigate their interaction with other structures that lie at the crossroads of algebraic geometry, differential geometry, commutative algebra and mathematical physics. The Principal Investigator plans to organize activities to enhance communication between different groups of researchers (including under-represented groups).The project has three closely related parts. The first is to prove K-stability of explicit Fano varieties, such as complete intersections and their mildly singular degenerations. With the help of birational geometry, the PI aims to show that most of these Fano varieties are K-stable and explicitly describe some of their corresponding compact K-moduli. The second part is to study the moduli theory of Fano varieties and singularities using K-stability. The goal is to construct compact moduli spaces that parametrize certain optimal degenerations of the varieties or singularities. The last part is to explore the connection of K-stability with other geometric properties of Fano varieties, especially their birational rigidity and rationality.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.

代数几何形状是对多项式方程系统定义的几何对象（称为代数品种）的研究。一个基本问题是找到并分类具有不错的几何结构的代数品种。当品种积极弯曲时，自然而然地搜索满足爱因斯坦方程式从一般相对论中的指标是很自然的。该项目的主题是使用称为K稳定性的代数几何稳定性理论来研究这种爱因斯坦指标，并研究它们与代数几何，差异几何，交换代数，代数和数学物理学的其他结构的相互作用。首席研究人员计划组织活动，以加强不同研究人员（包括代表性不足的群体）之间的沟通。该项目有三个密切相关的部分。首先是证明具有显式Fano品种的K稳定性，例如完整的交叉点及其轻微的奇异变性。在生物几何形状的帮助下，PI的目的是表明，这些Fano品种中的大多数都是K型，并且明确描述了它们相应的紧凑型K-Moduli。第二部分是使用K稳定性研究FANO品种和奇异性的模量理论。目标是构建紧凑的模量空间，以参数化品种或奇异性的某些最佳变性。最后一部分是探索K稳定性与FANO品种的其他几何特性的联系，尤其是其激进和理性的哲学。该奖项反映了NSF的法定使命，并被认为是通过基金会的智力优点评估和更广泛的影响来通过评估来支持的。