Canonical metrics in complex geometry

复杂几何中的规范度量

• 批准号: 0904223
• 负责人: Gabor Szekelyhidi
• 金额: $ 11.09万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904223&HistoricalAwards=false
• 关键词: Canonical; metrics; complex; geometry

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI's proposed research is concerned with the study of canonical Kahler metrics on algebraic varieties. There are deep conjectures relating the existence of extremal metrics to stability of the underlying algebraic variety and understanding this relationship has been studied intensively in the last decade. One major aspect of the PI's research is what one can say when no extremal metric exists. This problem will be studied both on the algebraic side to understand how varieties can be destabilized, and also in terms of metrics, which involves extending many of the existing results on extremal metrics to non-compact manifolds with cusp-like singularities along divisors.Another direction in the PI's proposal is the use of geometric flows to attack the existence of canonical metrics. Here too one of the most fascinating aspects is to study what kinds of singularities can form when no extremal metric exists and the PI will build on his earlier work on the Calabi flow on ruled surfaces and toric varieties and on the Kahler-Ricci flow.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 of the solutions of such equations.Understanding this phenomenon will have applications in physics and the sciences in general.

该奖项根据 2009 年《美国复苏和再投资法案》（公法 111-5）提供资金。 PI 提议的研究涉及代数簇的规范卡勒度量研究。有一些深刻的猜想将极值度量的存在与潜在代数簇的稳定性联系起来，并且在过去的十年中对这种关系进行了深入的研究。 PI 研究的一个主要方面是当不存在极值度量时人们可以说什么。我们将在代数方面研究这个问题，以了解簇如何变得不稳定，并在度量方面进行研究，这涉及将极值度量的许多现有结果扩展到沿除数具有尖点奇点的非紧流形。 PI 提案中的方向是使用几何流来攻击规范度量的存在。这里最令人着迷的方面之一是研究当不存在极值度量时会形成什么类型​​的奇点，PI 将建立在他早期关于直纹曲面和复曲面簇上的 Calabi 流以及 Kahler-Ricci 流的工作基础上。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此类方程解的局部解析性质。理解这种现象将在物理学和一般科学中得到应用。