Canonical metrics and stability in complex geometry

微分几何学家研究几何结构及其性质使用的工具类似于那些在微积分。代数几何利用来自代数的方法和工具来理解几何实体。在这个项目中，PI的目标是专注于一类被称为投影流形的几何对象，并研究与称为标量曲率的几何实体相关的问题。PI将探索这些结构的微分和代数性质以及它们的底层空间，并研究它们之间的相互作用。这项研究将微分几何和代数几何领域联系起来，并将导致具有不同背景的研究人员的合作和培训。该项目的更广泛的影响包括学生和博士后指导，创建一个垂直整合的研究小组，编写和出版讲义，以及会议和研讨会的组织。复微分几何的一个中心问题是在Kahler流形上构造标准度量结构。在射影流形上，Yau-Tian-唐纳森（YTD）猜想的目的是寻找稳定性条件，以保证典型的Kahler度量的存在性。本项目的主要目的是研究常数量曲率Kahler（cscK）度量的YTD猜想。PI将继续研究模型的一致K稳定性作为cscK指标存在的充分条件，以及该稳定性条件如何与测试配置的更标准K稳定性相关。为了实现这一点，PI将连接阿基米德和非阿基米德理论的能量泛函的斜率沿着测地线在卡勒度量空间的代数退化的非阿基米德不变量。PI将进一步扩展此类研究到更一般的加权cscK度量，并探索加权cscK度量与其他规范Kahler度量之间的新联系，包括Kahler-Ricci孤子，极值Kahler度量和共形Einstein-Maxwell Kahler度量。PI还将研究不携带cscK度量的不稳定极化流形。然后，他将找到方法来构建它们的最佳退化作为cscK指标的替代品。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Differential geometers study geometric structures and their properties by using tools similar to those employed in Calculus. Algebraic geometry utilizes methods and tools that come from algebra to understand geometric entities. In this project, the PI aims to focus on a class of geometric objects known as projective manifolds and investigate questions related to a geometric entity called the scalar curvature. The PI will explore both the differential and algebraic properties of these structures as well as their underlying spaces and study their interactions with each other. This research connects the field of differential geometry and algebraic geometry and will lead to collaborations and training of researchers with diverse backgrounds. Broader impacts of the project include student and postdoctoral mentoring, creation of a vertically integrated research group, preparation and publishing lecture notes, as well as conference and seminar organization. A central problem in complex differential geometry is to construct canonical metric structures on Kahler manifolds. Over projective manifolds, the Yau-Tian-Donaldson (YTD) conjecture aims to find stability conditions to guarantee the existence of canonical Kahler metrics. The main aim of this project is to study the YTD conjecture for constant scalar curvature Kahler (cscK) metrics. The PI will continue to study uniform K-stability for models as a sufficient condition for the existence of cscK metrics and how this stability condition is related to a more standard K-stability for test configurations. To achieve this, the PI will bridge the Archimedean and non-Archimedean theory by relating slopes of energy functionals along geodesic rays in the space of Kahler metrics to non-Archimedean invariants of algebraic degenerations. The PI will further extend such study to more general weighted cscK metrics and explore new connections between weighted cscK metrics with other canonical Kahler metrics, including Kahler-Ricci solitons, extremal Kahler metrics and conformally Einstein-Maxwell Kahler metrics. The PI will also study un-stable polarized manifolds which do not carry cscK metrics. He will then find ways to construct their optimal degenerations as substitutes of cscK metrics.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.