K-stability and Higher Dimensional Geometry

首席调查员(PI)将研究品种。簇被定义为多项式方程组的解的集合。它们非常容易计算，纳什证明了每个空间都可以用簇很好地逼近。该项目的主要目的是了解如果我们改变定义多项式方程的系数，特别是对于正曲线的多项式方程的系数，品种是如何变化的。这种品种被称为法诺品种。特别是，这项研究将试图理解法诺变种家族退化为具有奇点的变种家族的情况。PI的目的是证明在所有的Fano变种中，K-多稳变种可以由一个称为模空间的泛空间来参数化。作为这个项目的一部分，PI的目的是证明模空间是Hausdorff和紧的。PI的目的是通过理解具体的例子和一些一般现象来了解哪些Fano簇是K-半稳定的。PI旨在通过二元几何和非阿基米德几何之间的相互作用来理解Calabi-Yau流形的退化。该奖项反映了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 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.