Kahler-Einstein Metrics on Fano Varieties

Fano 品种的卡勒-爱因斯坦度量

• 批准号: 2109144
• 负责人: Chi Li
• 金额: $ 19.34万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-01-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2109144&HistoricalAwards=false
• 关键词: Kahler; Einstein; Metrics; Fano; Varieties

Geometric shapes can be either smooth or singular. Smooth ones have been studied effectively by the calculus, while singular geometric shapes are much more difficult to study in general although they appear with abundance in the world around us. One method to understand singular shapes is by measuring distances between their points. To do that, the best way is to use the so-called Einstein structures, which originate from general relativity. In this project, the investigator plans to study Einstein structures on a class of geometric shapes called algebraic varieties, which are central objects in many branches of mathematics. This study will allow us to measure distances and reveal certain mysterious structures of algebraic varieties. This project requires combinations of many techniques and will bring experts from different fields to interact. Its outcome will have potential applications in the development of several theories, including canonical metrics in differential geometry, stability theory in algebraic geometry, and string theory in mathematical physics.The investigator will study the Yau-Tian-Donaldson conjecture about the equivalence of K-stability and the existence of Kahler-Einstein metrics on singular Fano varieties. This requires new strategies to overcome difficulties due to the presence of singularities. The investigator has introduced a new process, the minimization of normalized volumes, for detecting local geometries of algebraic singularities. On the algebraic side, the investigator will continue his research on the K-stability of Fano varieties by studying minimization of normalized volumes and applying deep techniques of the minimal model program from algebraic geometry. This could lead to new criteria for the K-stability of singular varieties. On the analytic side, the investigator will apply various newly-developed techniques, including a variational approach via pluripotential theory, a priori estimates for singular complex Monge-Ampere equations, Cheeger-Colding-Tian's regularity theory from metric geometry, algebraic structures on Gromov-Hausdorff limits, and asymptotical analysis of singular metrics. The combination of these techniques will be effective in solving Kahler-Einstein equations on singular varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

几何形状可以是平滑的，也可以是单一的。光滑的几何形状已经通过微积分得到了有效的研究，而奇异的几何形状一般来说更难研究，尽管它们在我们周围的世界中大量出现。理解奇异形状的一种方法是测量它们的点之间的距离。要做到这一点，最好的方法是使用源自广义相对论的所谓的爱因斯坦结构。在这个项目中，研究人员计划研究一类称为代数簇的几何形状的爱因斯坦结构，代数簇是许多数学分支的核心对象。这项研究将使我们能够测量距离并揭示代数簇的某些神秘结构。该项目需要多种技术的结合，并将吸引不同领域的专家进行互动。其成果将在多种理论的发展中具有潜在的应用，包括微分几何中的正则度量、代数几何中的稳定性理论以及数学物理中的弦理论。研究人员将研究有关 K- 等价性的丘田唐纳森猜想。奇异 Fano 簇上的稳定性和卡勒-爱因斯坦度量的存在。这需要新的策略来克服由于奇点的存在而造成的困难。研究人员引入了一种新过程，即标准化体积的最小化，用于检测代数奇点的局部几何形状。在代数方面，研究人员将通过研究归一化体积的最小化并应用代数几何最小模型程序的深层技术，继续研究 Fano 簇的 K 稳定性。这可能会导致单一品种 K 稳定性的新标准。在分析方面，研究人员将应用各种新开发的技术，包括通过多能理论的变分法、奇异复数 Monge-Ampere 方程的先验估计、来自度量几何的 Cheeger-Colding-Tian 正则理论、Gromov 的代数结构豪斯多夫极限和奇异度量的渐近分析。这些技术的结合将有效解决单一品种的卡勒-爱因斯坦方程。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。