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.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI提出的研究是关于代数变量上的正则Kahler度量的研究。关于极值度量的存在与潜在代数变化的稳定性之间存在着深刻的猜想，在过去的十年中，人们对这种关系的理解进行了深入的研究。PI研究的一个主要方面是，当极值度量不存在时，人们可以说些什么。这个问题将在代数方面进行研究，以了解变异是如何失稳的，也将在度量方面进行研究，这涉及到将许多关于极值度量的现有结果扩展到具有沿因子的尖点奇点的非紧流形。PI建议的另一个方向是使用几何流来攻击规范度量的存在。这里最吸引人的一个方面是研究当极值度规不存在时，会形成什么样的奇点PI将建立在他早期关于直纹曲面上的卡拉比流和环面变化以及Kahler-Ricci流的研究基础上。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。提出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。理解这一现象将在物理学和一般科学中得到应用。