K-stability in Algebraic Geometry

丘的一个著名猜想指出，一个Kahler流形当且仅当它在几何不变理论意义上是“稳定的”时，才承认一个常数标量曲率Kahler度规。这个猜想已经被Tian和Donaldson扩展和澄清，并且通过他们和其他人的工作，这个猜想的一个方向已经从本质上得到了证明：这样一个度规的存在意味着稳定性。逆在很大程度上是开放的，应该被认为是卡勒几何的中心问题。PI将从代数几何的角度研究流形和轨道的稳定性，并将其作为常数标量曲率度量存在的阻碍。数学中的一个基本概念是度量，它定义了距离的概念，从而定义了几何物体的“形状”。在某些情况下，有可能找到具有特定性质的良好度规。例如，在球上存在圆形度规，使球成为球体（而不是椭球体）。高维的类似度量称为Kahler-Einstein度量和极值度量。这些度量在几何和数学物理中都有应用，了解它们何时存在非常重要。一个引人注目的猜想将这种度量的存在与代数几何中有关稳定性的问题联系起来。PI将研究这个稳定性的概念，并将这个猜想扩展到更一般的情况。

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.