K-Stability in Higher Dimensional Geometry

代数几何的课题是研究由多项式方程定义的几何空间的种类。在所有几何空间中，簇的优点是更容易从计算的角度进行研究。此外，由于每个空间都可以用变差来逼近，所以它们是代数几何的重要构造。受对度量满足某些爱因斯坦场方程式的空间研究的启发，该提议的主要目的是提供一个新的框架，以理解在全球和局部环境中正曲线的变化。一个焦点将是这些变种如何在家族中变化，并退化到具有更特殊结构的其他变种。这个项目有几个研究方向。首先，PI的目的是解决局部高阶有限生成猜想，从而通过建立任意Kawamata对数终端奇点的稳定退化猜想来完善局部稳定性理论。此外，该项目还包括对没有K-稳定性假设的一般Fano簇的模理论的研究和对Fano簇的显式例子的K-稳定性的研究。还有一个目的是将二元几何和非阿基米德几何结合在一起，以了解Calabi-Yau流形的退化。研究生将参与研究项目。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。

The project in algebraic geometry is a study of varieties, which are geometric spaces defined by polynomial equations. Among all geometric spaces, varieties have the advantage of being easier to study from a computational viewpoint. Moreover, since every space can be approximated by varieties, they are crucial construct for algebraic geometers. Inspired by the study of spaces with a metric satisfying certain Einstein field equations, the main aim of the proposal is to provide a new framework to understand varieties which are positively curved, in both global and local settings. One focus will be how these varieties vary in families, and degenerate to others with more special structures. There are several research thrusts to this project. First, the PI aims to solve the local higher rank finite generation conjecture, and thus, complete the local stability theory by establishing the stable degeneration conjecture for any Kawamata log terminal singularity. In addition, the project includes investigation into the moduli theory for general Fano varieties without a K-stability assumption and a study of K-stability for explicit examples of Fano varieties. There are also aims to combine birational and non-archimedean geometry together to understand degenerations of Calabi-Yau manifolds. Graduate students will participate in the research project.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.