K-Stability, Moduli, and Birational Geometry

这是代数几何领域的一个项目，它研究由多项式方程定义的形状。该领域的一个基本主题是模理论，其目的是构建和研究参数化所有这些形状的空间。对于低维形状，这样的模空间被很好地理解，但在高维中却知之甚少。近年来，从微分几何的新观点出发，构造了一类正弯曲形状的Fano变体的模空间。这个项目的目的是将新开发的技术从后一种构造应用到其他类型形状的模理论中。该项目为研究生提供了培训机会。k稳定性是微分几何学者为了刻画复射影变上某些正则度量的存在性而引入的一个代数概念。虽然这个概念可以定义为任何射影变量，但最近的研究主要集中在Fano变量的情况下，并最终应用于代数几何：构造参数化k稳定的Fano变量的射影模空间。本项目的目标是利用双分几何和k稳定性的工具来构造其他种类的模空间。特别是，PI将构造k -不稳定Fano变体的某些模，这是由微分几何中Kahler-Ricci孤子的研究激发的。接下来，PI将开发新的方法来研究k -平凡变量的模。最后，PI将应用双几何工具来研究极化品种的k稳定性。该奖项反映了美国国家科学基金会的法定使命，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。

This is a project in the area of algebraic geometry, which is the study of shapes defined by polynomial equations. A fundamental topic in the field is moduli theory, which aims to construct and study spaces that parametrize all such shapes. For low dimensional shapes such moduli spaces are well understood, but less is known in higher dimensions. Recently, new perspectives from differential geometry have led to the construction of moduli spaces of certain Fano varieties, which are a class of positively curved shapes. The aim of this project is to apply newly developed techniques from the latter construction to the moduli theory of other classes of shapes. The project offers training opportunities for graduate students.K-stabilty is an algebraic notion introduced by differential geometers to characterize the existence of certain canonical metrics on complex projective varieties. While the notion can be defined for any projective variety, recent research has focused on the case of Fano varieties and culminated in an application to algebraic geometry: the construction of projective moduli spaces parametrizing K-stable Fano varieties. The goal of this project is to use tools from birational geometry and K-stability to construct moduli spaces of other classes of varieties. In particular, the PI will construct certain moduli of K-unstable Fano varieties, motivated by the study of Kahler-Ricci solitons in differential geometry. Next, the PI will develop new approaches for studying moduli of K-trivial varieties. Finally, the PI will apply tools from birational geometry to the study of K-stability of polarized varieties.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.