Kahler-Einstein metrics on Fano manifolds

爱因斯坦流形在数学和物理学中都是重要的几何对象。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，它们是更复杂几何图形的基本组成部分。因此，对爱因斯坦流形的研究是一个基本的几何问题。构造爱因斯坦流形的一种有效方法是要求底层流形具有复杂的代数结构。换句话说，这种流形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。在70年代末，Aubin和Yau构建了具有负里奇曲率的Kaehler-Einstein度量。Yau还构造了零里奇曲率的Kaehler-Einstein度规，现在称为Calabi-Yau度规，在物理学的弦理论中起着重要作用。另一方面，直到最近，人们才确定了一类称为范诺流形的代数流形存在具有正Ricci曲率的Kaehler-Einstein度量的充分性和必要条件，称为k稳定性。这一结果依赖于许多人的工作，其中最重要的是Tian和Donaldson。在这些发现之后，我们希望进一步了解这种凯勒-爱因斯坦度量以及它们存在的障碍。这些问题是提案的主要关注点。对这些Kaehler-Einstein度量的研究将大大提高我们对爱因斯坦流形的理解，这在物理学和数学中都很重要。在本提案中，PI将研究以下密切相关的问题。1.用不同的偏微分方程连续性方法求解Kaehler-Einstein方程。最近的突破给出了这些连续性方法的爆炸行为和收敛性的定性图像。然而，需要对爆炸行为或奇点形成现象有更深入的定量认识。PI详细研究了环形法诺流形的这种定量性质。PI将研究更广泛的一类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，研究在低维中形成的奇点的分类。2.PI将研究Kaehler-Einstein度量和相关规范度量的具体结构。一方面，PI喜欢将环形Kaehler-Einstein度规的构造扩展到其他具有大对称性的Kaehler-Einstein度规，例如，在球面上。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-爱因斯坦度量的分类。相关方法也将应用于构造Kaehler- ricci孤子和极值Kaehler度量。3.PI将研究包括Kaehler- einstein度量和Kaehler- ricci孤子在内的典型Kaehler度量的变形，并了解这些典型Kaehler度量的模空间。他还将研究这些模空间边界上的奇点。4.PI和他的合作者将基于他们之前关于k稳定性的工作，使用代数几何来研究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凯勒-爱因斯坦度量。

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.