Kahler geometry and canonical metrics

卡勒几何和规范度量

• 批准号: 1306298
• 负责人: Gabor Szekelyhidi
• 金额: $ 13.14万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-06-01 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1306298&HistoricalAwards=false
• 关键词: Kahler; geometry; canonical; metrics

AbstractAward: DMS 1306298, Principal Investigator: Gabor SzekelyhidiThe PI's proposed research is concerned with the study of canonical Kahler metrics on algebraic varieties, in particular the extremal metrics introduced by Calabi in the 80's. The main conjecture in the field relates the existence of such metrics to the stability of the variety in the sense of geometric invariant theory. The proposed research focuses on situations when no extremal metric exists. On the algebro-geometric side the goal is to construct canonical degenerations of the variety and the PI's proposal is to use filtrations of the homogeneous coordinate ring, in analogy with Harder-Narasimhan filtrations of unstable vector bundles. On the differential geometric side one needs to understand the possible limiting behavior of families of extremal metrics. In general this is much more intricate than the much more thoroughly understood case of Kahler-Einstein metrics, and the PI proposes to first restrict attention to metrics with bounded curvature, and to relate the limiting behavior to filtrations. In the proposal a special emphasis is placed on the construction of new examples of extremal metrics, and the applications of these ideas to other problems in Kahler geometry.Geometric partial differential equations govern much of the physical world. For example solutions of Einstein's equations are intimately related to our understanding of the universe. The proposed research studies differential equations related to Einstein's equations and the key question is how the global structure of a space influences the local, analytic properties, such as singularities of the solutions of such equations. Understanding this phenomenon will have applications in physics and the sciences in general.

Abstractaward：DMS 1306298，主要研究者：Gabor Szekelyhidithe Pi的拟议研究与对代数品种的规范Kahler指标的研究有关，尤其是Calabi在80年代引入的极端指标。该领域的主要猜想将这种指标的存在与几何不变理论意义上的稳定性联系起来。拟议的研究重点是不存在极端指标的情况。在代数几何方面，目标是构建该品种的规范变性，而PI的建议是使用均匀坐标环的过滤，类似于较难的纳拉西姆汉（Narasimhan）的较硬矢量捆绑包。在差异几何方面，人们需要了解极端指标家族的可能限制行为。通常，这比Kahler-Einstein指标的案例更加彻底地理解要复杂得多，并且PI提议首先限制对指标的关注，并将限制行为与过滤相关联。在该提案中，特别重点是建造新的极端指标示例，以及这些思想在Kahler几何学中的其他问题中的应用。几何部分微分方程控制了物理世界的大部分。例如，爱因斯坦方程的解决方案与我们对宇宙的理解密切相关。 拟议的研究研究了与爱因斯坦方程相关的微分方程，关键问题是空间的全球结构如何影响局部分析特性，例如这种方程式的解决方案的奇异性。了解这一现象将在物理学和科学中有应用。