K-stability in Algebraic Geometry

代数几何中的 K 稳定性

• 批准号: 0700419
• 负责人: Michael Thaddeus
• 金额: $ 10.58万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700419&HistoricalAwards=false
• 关键词: stability; Algebraic; Geometry

A famous conjecture due to Yau states that a Kahler manifold admits a constant scalar curvature Kahler metric if and only if it is "stable" in the sense of Geometric Invariant Theory. This conjecture has been expanded and clarified by Tian and Donaldson and through their work and the work of others one direction of this conjecture has essentially be proved: the existence of such a metric implies stability. The converse is largely open and should be considered as a central problem in Kahler geometry. The PI will study stability of manifolds and orbifolds from the viewpoint of algebraic geometry and as an obstruction to the existence of constant scalar curvature metrics.A fundamental concept in mathematics is that of a metric which defines a notion of distance and thus the "shape" of a geometric object. In some cases it is possible to find a good metric with particular properties. For instance on a ball there exists the round metric, making the ball into a sphere (rather than an ellipsoid). The analogous metrics in higher dimension are called Kahler-Einstein metrics and extremal metrics. These metrics have applications in geometry and mathematical physics, and it is important to understand when they exist. A remarkable conjecture connects the existence of such metrics to a problem in algebraic geometry concerning stability. The PI will study this notion of stability, and extend the conjecture to a more general setting.

由于Yau而引起的著名猜想指出，当且仅当它在几何不变理论的意义上是“稳定”时，Kahler歧管就承认了恒定的标量曲率曲率度量。 天和唐纳森（Tian and Donaldson）以及通过其其他工作的工作实质上证明了这一猜想的扩展和澄清。匡威在很大程度上是开放的，应被视为卡勒几何形状中的核心问题。 PI将从代数几何学的角度研究流形和轨道的稳定性，并作为对恒定标量曲率度量指标的存在的障碍。数学中的基本概念是定义距离概述的度量标准，因此是距离的概念，从而定义了几何学对象的“形状”。 在某些情况下，可以找到具有特定属性的良好指标。例如，在球上存在圆形度量，将球变成球（而不是椭圆形）。 较高维度中的类似指标称为Kahler-Einstein指标和极端指标。 这些指标在几何学和数学物理学中都有应用，并且必须了解它们何时存在很重要。 一个显着的猜想将这种指标的存在与有关稳定性的代数几何形状中的问题联系在一起。 PI将研究这种稳定性的概念，并将猜想扩展到更一般的环境。