Kahler-Einstein metrics on Fano manifolds

Fano 流形上的卡勒-爱因斯坦度量

• 批准号: 1405936
• 负责人: Chi Li
• 金额: $ 13.65万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405936&HistoricalAwards=false
• 关键词: Kahler; Einstein; metrics; Fano; manifolds

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.

爱因斯坦流形是数学和物理学中重要的几何对象。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建块。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，这样的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凯勒-爱因斯坦度量。 70 年代末，Aubin 和 Yau 构建了具有负 Ricci 曲率的 Kaehler-Einstein 度量。丘还构造了里奇曲率为零的凯勒-爱因斯坦度量，现在称为卡拉比-丘度量，在物理学弦理论中发挥着重要作用。另一方面，直到最近，人们才为一类称为 Fano 流形的代数流形的具有正 Ricci 曲率的 Kaehler-Einstein 度量的存在确定了一个充分必要条件，称为 K 稳定性。这个结果取决于很多人的努力，尤其是田和唐纳森的努力。在这些发现之后，我们希望进一步了解此类凯勒-爱因斯坦度量及其存在的障碍。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，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.采用偏微分方程的各种连续性方法来求解Kaehler-Einstein方程。最近的突破给出了爆炸行为的定性图像以及这些连续性方法的融合。然而，需要对爆炸行为或奇点形成现象有更深入的定量理解。 PI 详细研究了复曲面 Fano 流形的定量特性。 PI 将研究更广泛的 Fano 流形类别的奇点形成过程。 PI还将结合黎曼几何和代数几何的方法，研究低维奇点的分类。 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 喜欢将环面凯勒-爱因斯坦度量的构造扩展到其他具有大对称性的凯勒-爱因斯坦度量，例如球面簇。另一方面，PI将根据他对重要例子的计算，研究低维中具有大对称性的Sasaki-Einstein度量的分类。相关方法也将用于构造Kaehler-Ricci孤子和极值Kaehler度量。 3.PI将研究规范Kaehler度量的变形，包括Kaehler-Einstein度量和Kaehler-Ricci孤子，并了解这些规范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。这将使我们能够使用代数几何方法获得凯勒-爱因斯坦度量。