CAREER: Birational Geometry and K-stability of Algebraic Varieties

代数几何研究的是由多项式方程定义的几何对象--代数簇。它在微分几何等邻近领域中起着重要的作用。连接这些领域的一个自然而重要的问题是找到并分类具有良好几何结构的代数簇，例如具有常曲率的度量。近年来，随着高维代数几何的发展，当代数簇是正曲的时候，在寻找这种度量方面取得了一些突破。这个项目的目标是推进这些想法，以进一步理解一般代数簇的几何结构。此外，该项目还通过研讨会、暑期学校和其他活动为学生和相关领域的早期职业研究人员提供培训和研究机会。更详细地说，该项目的动机是有影响力的Yau-Tian-唐纳森猜想，即正则度量的存在应该等价于称为K-稳定性的代数稳定性条件。PI将为一般的Kawamata日志终端奇点开发一个局部K稳定性理论，以期了解它们的双有理几何和模量。PI还将研究一些有趣的新问题，在双有理几何的启发，在最近的进展K-稳定性。最后，该项目将进一步推进代数理论的K-稳定性，从法诺的情况下，一般极化的情况下，和攻击一些开放的问题有关的姚田-唐纳森猜想。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

Algebraic geometry studies algebraic varieties which are geometric objects defined by polynomial equations. It plays an important role in neighboring fields such as differential geometry. A natural and important question connecting these areas is to find and classify algebraic varieties with nice geometric structures, such as metrics with constant curvature. In recent years, the development of higher dimensional algebraic geometry has led to a number of breakthroughs in the search for such metrics when the algebraic varieties are positively curved. The goal of this project is to advance these ideas to further understand the geometric structure of general algebraic varieties. In addition the project provides training and research opportunities for students and early-career researchers in related areas, through seminars, summer schools, and other activities.In more detail, the project is motivated by the influential Yau-Tian-Donaldson Conjecture that existence of canonical metrics should be equivalent to some algebraic stability condition known as K-stability. The PI will to develop a local K-stability theory for general Kawamata log terminal singularities, with a view towards the understanding of their birational geometry and moduli. The PI will also investigate some interesting new questions in birational geometry that are inspired by recent progress in K-stability. Finally the project will further advance the algebraic theory of K-stability from the Fano case to the general polarized case, and attack some open problems related to the Yau-Tian-Donaldson conjecture.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.