Canonical metrics and stability in complex geometry

微分几何通过使用与微积分中使用的工具类似的工具来研究几何结构及其性质。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的目标是聚焦于一类被称为射影流形的几何对象，并调查与被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基本空间，并研究它们之间的相互作用。这项研究连接了微分几何和代数几何领域，并将导致不同背景的研究人员的合作和培训。该项目的更广泛影响包括对学生和博士后的指导，创建一个垂直整合的研究小组，编写和出版讲稿，以及组织会议和研讨会。复微分几何中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-Donaldson(YTD)猜想的目的是找到保证规范Kahler度量存在的稳定性条件。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度量存在的充分条件，以及该稳定性条件如何与测试配置的更标准的K-稳定性相关。为了实现这一点，PI将通过将Kahler度量空间中沿测地射线的能量泛函的斜率与代数退化的非阿基米德不变量联系起来，在阿基米德和非阿基米德理论之间架起桥梁。PI将进一步将这种研究扩展到更一般的加权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他们的最佳退化作为cscK度量的替代品。这一奖项反映了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。

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.