Kahler-Einstein metrics on Fano manifolds

爱因斯坦流形是在数学和物理中都很重要的几何对象。在物理学中，它们被用来描述爱因斯坦广义相对论中的时空。在数学中，它们是更复杂几何的基本构件。因此，爱因斯坦流形的研究是几何学中的一个基本问题。构造爱因斯坦流形的一个有效方法是要求基础流形具有复杂的代数结构。换句话说，这样的流形上的点是多项式方程的复值解。这种代数流形上的爱因斯坦度量称为Kaehler-Einstein度量。70年代末，Aubin和Yau构造了具有负Ricci曲率的Kaehler-Einstein度规。Yau还构建了具有零Ricci曲率的Kaehler-Einstein度规，现在被称为Calabi-Yau度规，在物理学的弦理论中扮演着重要的角色。另一方面，直到最近，人们才为一类称为Fano流形的代数流形给出了具有正Ricci曲率的Kaehler-Einstein度量存在的充要条件，称为K-稳定性。这一结果取决于许多人的努力，最重要的是田和唐纳森的工作。在这些发现之后，我们想要加深我们对这样的凯勒-爱因斯坦度规以及它们存在的障碍的理解。这些问题是该提案主要关注的问题。对这些凯勒-爱因斯坦度量的研究将极大地提高我们对爱因斯坦流形的理解，爱因斯坦流形在物理和数学中都很重要。在这项建议中，国际和平研究所将研究下列密切相关的问题。1.利用偏微分方程组的各种连续性方法求解Kaehler-Einstein方程。最近的突破给出了这些连续方法的爆破行为和收敛的定性图像。然而，还需要对爆破行为或奇点形成现象有更深层次的定量理解。PI已经详细地研究了环面Fano流形的这种定量性质。PI将研究更广泛类别的Fano流形的奇点形成过程。PI还将结合黎曼几何和代数几何的方法来研究低维形成的奇点的分类。2.PI将研究Kaehler-Einstein度量和相关正则度量的具体构造。一方面，PI喜欢将Toric Kaehler-Einstein度量的构造推广到其他具有大对称性的Kaehler-Einstein度量，例如球簇上。另一方面，PI将根据他对重要例子的计算，研究具有低维大对称性的佐佐木-爱因斯坦度规的分类。相关方法也将被应用于构造Kaehler-Ricci孤子和极值Kaehler度规。3.PI将研究包括Kaehler-Einstein度规和Kaehler-Ricci孤子在内的正则Kaehler度规的变形，并了解这些正则Kaehler度规的模空间。他还将研究这些模空间边界上的奇点。4.PI和他的合作者将在他们以前关于K-稳定性的工作的基础上，利用代数几何来研究K-稳定性。他们将使用最小模型程序中的工具来测试K-稳定性。这将允许我们使用代数几何方法得到Kaehler-Einstein度规。

Einstein manifolds are geometric objects important in both mathematics and physics. In physics, they are used to describe the space-time in Einstein's theory of general relativity. In mathematics, they are basic building blocks of more complicated geometries. The study of Einstein manifolds is thus a basic problem in geometry. One effective way to construct Einstein manifolds is to require that the underlying manifold has a complex algebraic structure. In other words, the points of such a manifold are complex-valued solutions of polynomial equations. Einstein metrics on such algebraic manifolds are called Kaehler-Einstein metrics. In the late 70s, Aubin and Yau constructed Kaehler-Einstein metrics with negative Ricci curvatures. Yau also constructed Kaehler-Einstein metrics with zero Ricci curvatures, which are now called Calabi-Yau metrics and play important roles in the string theory of physics. On the other hand, only recently has people pinned down a sufficient and necessary condition, called K-stability, for the existence of Kaehler-Einstein metrics with positive Ricci curvatures for a class of algebraic manifolds called Fano manifolds. This result depends on the work of many people, most importantly by Tian and Donaldson. After these discoveries, we want to further our understandings of such Kaehler-Einstein metrics and the obstructions to their existence. These problems are the main concerns of the proposal. The study of these Kaehler-Einstein metrics will greatly improve our understanding of Einstein manifolds important in both physics and mathematics. In this proposal, the PI will study the following closely related problems. 1.Various continuity methods of partial differential equations are used to solve the Kaehler-Einstein equation. The recent breakthroughs give qualitative pictures of blow up behaviors and convergences of these continuity methods. However, deeper quantitative understandings of the blow up behaviors or singularity forming phenomena are needed. The PI has studied in detail such quantitative properties for toric Fano manifolds. The PI will study the singularities forming processes for a broader class of Fano manifolds. The PI will also study the classification of the singularities formed in low dimensions by combining the methods from Riemannian geometry and algebraic geometry. 2.The PI will study concrete constructions of Kaehler-Einstein metrics and related canonical metrics. On the one hand, the PI likes to extend the construction of toric Kaehler-Einstein metrics to other Kaehler-Einstein metrics with large symmetries, for example, on spherical varieties. On the other hand, the PI will study the classification of Sasaki-Einstein metrics with large symmetries in low dimensions based on his calculations of important examples. Related methods will also be applied to construct Kaehler-Ricci solitons and extremal Kaehler metrics. 3.The PI will study the deformations of canonical Kaehler metrics including Kaehler-Einstein metrics and Kaehler-Ricci solitons, and to understand the moduli spaces of these canonical Kaehler metrics. He will also study the singularities on the boundaries of these moduli spaces. 4.The PI and his collaborator will study the K-stability using algebraic geometry based on their previous work on K-stability. They will use tools from minimal model program to test K-stability. This will allow us to get Kaehler-Einstein metrics using algebro-geometric methods.