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——代数变异。它在微分几何等邻近领域发挥着重要作用。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变体，例如具有常曲率的度量。近年来，随着高维代数几何的发展，在寻找代数变量为正曲线时的度量方面取得了一些突破。该项目的目标是推进这些思想，以进一步了解一般代数变种的几何结构。此外，该项目通过研讨会、暑期学校和其他活动，为相关领域的学生和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you - tian - donaldson猜想，该猜想认为规范度量的存在性应该等同于一些称为k稳定性的代数稳定性条件。PI将发展一般Kawamata日志终端奇点的局部k稳定性理论，以期理解它们的双几何和模。PI还将研究一些有趣的新问题，这些问题是由最近k稳定性的进展所启发的。最后，本课题将k -稳定性的代数理论从Fano情况进一步推进到一般极化情况，并对有关you - tian - donaldson猜想的一些开放问题进行了研究。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。

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.