K-Stability and Birational Geometry of Fano Varieties

代数几何是研究由多项式方程系统定义的几何对象--称为代数簇。一个基本的问题是寻找和分类具有良好几何结构的代数簇。当变数是正曲线时，从广义相对论中寻找满足爱因斯坦方程的度量是很自然的。这个项目的主要主题是使用被称为K-稳定性的代数几何稳定性理论来研究这样的爱因斯坦度规，并调查它们与位于代数几何、微分几何、交换代数和数学物理的十字路口的其他结构之间的相互作用。首席调查员计划组织活动，以加强不同研究人员群体(包括代表人数不足的群体)之间的沟通。该项目由三个密切相关的部分组成。第一个是证明显式Fano簇的K-稳定性，如完全交及其温和奇异退化。在二次几何的帮助下，PI的目的是证明大多数这些Fano簇是K-稳定的，并显式地描述了它们对应的一些紧K-模。第二部分是利用K-稳定性研究Fano簇和奇点的模理论。其目的是构造紧致模空间，它将簇或奇点的某些最佳退化参数化为参数。最后一部分是探索K-稳定性与Fano品种的其他几何性质的联系，特别是它们的混生刚性和合理性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.