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CAREER: Canonical metrics and stability in complex geometry

职业：复杂几何中的规范度量和稳定性

基本信息

批准号：
1350696
负责人：
Gabor Szekelyhidi
金额：
$ 44.28万
依托单位：
University of Notre Dame
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1350696&HistoricalAwards=false
关键词：
CAREER Canonical metrics stability complex

项目摘要

AbstractAward: DMS 1350696, Principal Investigator: Gabor SzekelyhidiGeometric 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. In addition the principal investigator proposes integrated research and educational programs. These programs will raise interest in high school students towards mathematics, and it will encourage undergraduate students to pursue K-12 STEM education.The principal investigator'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 1980's, and questions related to the J-flow introduced by Donaldson. The main conjecture in the field relates the existence of extremal metrics to the stability of the variety in the sense of geometric invariant theory, and the PI proposes an analogous conjecture for the J-flow. A particular feature of the proposal is a focus 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.

摘要奖：DMS 1350696，首席研究员：Gabor Szekelyhidi几何偏微分方程支配着物理世界的大部分。例如，爱因斯坦方程的解与我们对宇宙的理解密切相关。拟议的研究研究与爱因斯坦方程相关的微分方程，关键问题是空间的全局结构如何影响局部分析性质，例如此类方程解的奇异性。理解这种现象将在物理学和一般科学中得到应用。此外，首席研究员提出综合研究和教育计划。这些项目将提高高中生对数学的兴趣，并鼓励本科生继续进行K-12 STEM教育。首席研究员的拟议研究涉及代数簇上的规范Kahler度量的研究，特别是Calabi在20世纪80年代引入的极值度量，以及与唐纳森引入的J流相关的问题。该领域的主要猜想是将极值度量的存在性与几何不变理论意义上的簇的稳定性联系起来，PI对J流提出了类似的猜想。该提案的一个特别之处是关注不存在极值度量的情况。在代数几何方面的目标是构建规范退化的品种和PI的建议是使用过滤的齐次坐标环，在类比哈德-Narasimhan过滤的不稳定向量丛。在微分几何方面，人们需要理解极值度量族的可能极限行为。一般来说，这比更透彻理解的卡勒-爱因斯坦度量的情况要复杂得多，PI建议首先将注意力限制在有界曲率的度量上，并将限制行为与过滤联系起来。在建议中特别强调的是建设新的极端度量的例子，并应用这些想法的其他问题在Kahler几何。