K-stability in Algebraic Geometry

一个著名的猜想，由于丘指出，一个卡勒流形承认一个常数标量曲率卡勒度量当且仅当它是“稳定”的意义上的几何不变理论。 这一猜想已扩大和澄清田和唐纳森，并通过他们的工作和其他人的工作，这一猜想的一个方向已基本上得到证明：存在这样一个度量意味着稳定性。匡威的是在很大程度上是开放的，应该被认为是一个中心问题，在卡勒几何。 PI将从代数几何的角度研究流形和orbifolds的稳定性，并作为常数标量曲率度量存在的障碍。数学中的一个基本概念是度量，它定义了距离的概念，从而定义了几何对象的“形状”。 在某些情况下，可以找到具有特定属性的良好度量。例如，在一个球上存在圆度规，使球成为一个球体（而不是椭圆体）。 在高维中类似的度量被称为Kahler-Einstein度量和极值度量。 这些度量在几何和数学物理中有应用，重要的是要了解它们何时存在。 一个值得注意的猜想将这种度量的存在与代数几何中关于稳定性的问题联系起来。 PI将研究这种稳定性的概念，并将猜想扩展到更一般的设置。

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.