Kahler-Einstein Metrics on Fano Varieties

几何形状可以是平滑的，也可以是奇异的。光滑的几何形状已经被微积分学有效地研究过了，而奇异的几何形状虽然在我们周围的世界中大量出现，但一般来说研究起来要困难得多。理解奇异形状的一种方法是测量它们之间的距离。要做到这一点，最好的方法是使用所谓的爱因斯坦结构，它起源于广义相对论。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上的爱因斯坦结构，代数簇是许多数学分支的中心对象。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。这个项目需要结合许多技术，并将带来来自不同领域的专家进行互动。研究结果将在微分几何中的正则度规、代数几何中的稳定性理论、数学物理中的弦理论等理论的发展中具有潜在的应用价值。研究者将研究关于奇异Fano簇上K-稳定性的等价性和Kahler-Einstein度规的存在性的Yau-Tian-唐纳森猜想。这需要新的策略来克服由于奇点的存在而造成的困难。研究人员介绍了一种新的过程，最小化的归一化体积，用于检测局部几何的代数奇点。在代数方面，研究人员将继续他的研究的K-稳定性的法诺品种通过研究最小化的归一化体积和应用深技术的最小模型程序从代数几何。这可能会导致新的标准的K-稳定性的奇异品种。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方法，奇异复Monge-Ampere方程的先验估计，来自度量几何的Cheeger-Colding-Tian正则性理论，Gromov-Hausdorff极限的代数结构，以及奇异度量的渐近分析。这些技术的结合将有效地解决Kahler-Einstein方程的奇异品种。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.