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.

几何形状可以光滑或单数。微积分对光滑的几何形状有效地研究了，而一般的几何形状在我们周围的世界中大量出现。一种理解奇异形状的方法是测量其点之间的距离。为此，最好的方法是使用所谓的爱因斯坦结构，该结构源自一般相对论。在这个项目中，研究者计划在一类称为代数品种的几何形状上研究爱因斯坦的结构，它们是数学许多分支中的中心对象。这项研究将使我们能够测量距离并揭示某些代数品种的神秘结构。该项目需要许多技术的组合，并将带来来自不同领域的专家进行交互。 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.这需要新的策略来克服由于存在奇异性而遇到困难。研究者引入了一个新的过程，即归一化体积的最小化，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形状的最小模型程序的深度技术来继续他对FANO品种的K稳定性的研究。这可能会导致新品种K稳定性的新标准。 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.这些技术的组合将有效地解决Kahler-Einstein方程上的奇异品种。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。