K-Stability and Birational Geometry of Fano Varieties

代数几何是研究由多项式方程组定义的几何对象-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。当变量是正弯曲的时候，很自然地要从广义相对论中寻找满足爱因斯坦方程的度量。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研究爱因斯坦度量，并研究它们与代数几何，微分几何，交换代数和数学物理交叉路口的其他结构的相互作用。首席研究员计划举办活动，以加强不同组别研究人员（包括代表性不足的组别）之间的沟通。第一个是证明显式Fano簇的K-稳定性，如完全交及其轻度奇异退化。在双有理几何的帮助下，PI的目的是证明这些Fano簇中的大多数是K-稳定的，并明确描述了它们相应的紧致K-模。第二部分是利用K-稳定性研究Fano簇和奇点的模理论。我们的目标是构建紧凑的模空间，参数化某些最佳退化的品种或奇点。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，特别是它们的双理性刚度和合理性。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。

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.