Kahler-Einstein metrics on Fano manifolds

Fano 流形上的卡勒-爱因斯坦度量

• 批准号: 1636488
• 负责人: Chi Li
• 金额: $ 10.43万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-07 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1636488&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.

爱因斯坦歧管是在数学和物理学中都很重要的几何对象。在物理学中，它们用于描述爱因斯坦的一般相对论理论中的时空。在数学中，它们是更复杂的几何形状的基本基础。因此，爱因斯坦歧管的研究是几何学中的基本问题。构建爱因斯坦歧管的一种有效方法是要求基础歧管具有复杂的代数结构。换句话说，这种歧管的点是多项式方程的复杂值解决方案。这种代数歧管上的爱因斯坦指标称为Kaehler-Einstein指标。在70年代后期，Aubin和Yau构建了带有RICCI曲率负曲率的Kaehler-Einstein指标。 Yau还构建了具有零RICCI曲率的Kaehler-Einstein指标，现在称为Calabi-yau指标，并在物理弦理论中起重要作用。另一方面，直到最近，人们才钉住了一个充分和必要的条件，称为K稳定性，因为存在具有正ricci曲率的Kaehler-Einstein指标，用于一类称为Fano歧管的代数歧管。该结果取决于许多人的工作，最重要的是天和唐纳森。在这些发现之后，我们希望进一步了解这种Kaehler-Einstein指标及其存在的障碍。这些问题是该提案的主要问题。对这些Kaehler-Einstein指标的研究将大大提高我们对爱因斯坦流形在物理和数学中都很重要的理解。在此提案中，PI将研究以下密切相关的问题。 1.各个差分方程的各种连续性方法用于求解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融合的定性图片。但是，需要对爆炸行为或奇异性形成现象的更深入的定量理解。 PI详细研究了用于复曲面的Fano歧管的定量特性。 PI将研究形成更广泛的Fano歧管的奇异性过程。 PI还将通过结合Riemannian几何形状和代数几何形状的方法来研究在低维中形成的奇点的分类。 2. PI将研究Kaehler-Einstein指标和相关规范指标的混凝土结构。一方面，PI喜欢将复曲面的Kaehler-Einstein指标扩展到其他具有较大对称性的Kaehler-Einstein指标，例如，在球形品种上。另一方面，PI将根据他的重要例子的计算，研究具有较大对称性的Sasaki-Einstein指标的分类。相关方法还将应用于构建Kaehler-Ricci孤子和极端Kaehler指标。 3. PI将研究包括Kaehler-Einstein指标和Kaehler-Ricci孤子的规范Kaehler指标的变形，并了解这些规范Kaehler指标的模量空间。他还将研究这些模量空间边界的奇异性。 4. PI和他的合作者将根据其先前关于K稳定性的工作使用代数几何形状来研究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。这将使我们能够使用代数几何方法获得Kaehler-Einstein指标。