Canonical metrics and stability in complex geometry

复杂几何中的规范度量和稳定性

• 批准号: 2305296
• 负责人: Chi Li
• 金额: $ 19.59万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-11-01 至 2026-10-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305296&HistoricalAwards=false
• 关键词: Canonical; metrics; stability; complex; geometry

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.

微分几何学家使用类似于微积分中使用的工具来研究几何结构及其性质。代数几何利用代数中的方法和工具来理解几何实体。在这个项目中，PI 的目标是关注一类称为射影流形的几何对象，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及其底层空间，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，并将促进具有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学生和博士后指导、创建垂直整合的研究小组、准备和出版讲稿以及组织会议和研讨会。复杂微分几何的一个中心问题是在卡勒流形上构造规范度量结构。在射影流形上，Yau-Tian-Donaldson (YTD) 猜想旨在找到稳定性条件来保证规范卡勒度量的存在。该项目的主要目的是研究恒定标量曲率卡勒 (cscK) 度量的 YTD 猜想。 PI 将继续研究模型的统一 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。为了实现这一目标，PI 将通过将卡勒度量空间中沿着测地射线的能量泛函斜率与代数简并的非阿基米德不变量相关联，在阿基米德和非阿基米德理论之间架起桥梁。 PI 将进一步将此类研究扩展到更通用的加权 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 还将研究不带有 cscK 度量的不稳定极化流形。然后，他将找到构建最佳退化的方法来替代 cscK 指标。该奖项反映了 NSF 的法定使命，并且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，被认为值得支持。