Singular Kahler-Einstein Metrics: Analytic and Algebraic Aspects

复空间或复流形是带有物理理论中自然产生的某种附加结构的空间。复杂几何旨在通过距离和角度研究复杂流形的形状；数学上这些度量称为Kähler度量。在物理环境中，带有Kähler度规和爱因斯坦度规的流形很有趣；这些被称为Kähler-Einstein指标。该项目从解析和代数的角度考虑了复杂几何中的几个核心问题，并围绕所谓的奇异Kähler-Einstein度量的概念进行了构建。这个相对较新的概念是为了将Yau关于Calabi猜想的解的定理扩展到直接来自代数几何，更准确地说是双对数（复数）几何的奇异设置而引入的。当前的主要挑战包括在各种环境中构建这些对象并试图理解它们的行为；对奇异度规Kähler-Einstein行为的良好理解将在复杂几何以及理论物理领域（如弦理论）中产生重要影响。此外，研究奇异Kähler-Einstein度量族与模理论密切相关，模理论是代数几何的主要组成部分之一。预计有关奇异Kähler-Einstein度量应用的项目结果将促进奇异空间理论、解析几何和代数几何以及数学和物理其他领域的知识。该项目侧重于三个中心主题：Kähler-Einstein度量在奇异度Kähler-Einstein的变化、族和退化的奇点附近的行为，以及Kähler-Einstein理论在代数几何中的应用。第一个想法是为Kähler-Einstein度量在奇点附近找到模型。奇点可以产生于变量的奇点，也可以产生于要考虑的对的边界除数。该项目通过PI和其他人过去的结果，以及这些作品中出现的许多问题，研究了（半）对数正则对的一般情况。主题包括改进多能方法以获得更清晰的估计，通过关联奇点和完备性来理解奇异Kähler-Einstein度量的（黎曼）几何，以及在奇点处推导更高的正则性。第二圈思想集中于奇异Kähler-Einstein度量族，特别是通过Weil-Petersson几何从微分几何的角度研究正则极化变体的模空间。其他相关的问题是关于经济Kähler-Einstein指标的Gromov-Hausdorff（或更精确的）收敛性。希望近年来多能理论的发展能为解决这些问题提供一个新的角度。第三种思想集中在奇异Kähler-Einstein度量的各种应用上，特别是在代数几何中。项目中出现的典型问题涉及奇变切线束的消失/并行性定理或半稳定性性质。在这里，问题的后续部分是由蒙日-安培方程理论的最新进展所激发的。

Complex spaces or complex manifolds are spaces that carry a certain additional structure that arises naturally in physical theories. Complex geometry aims to study the shapes of complex manifolds through distances and angles; mathematically these measurements are called Kähler metrics. In the physics context, manifolds that carry a metric that is both a Kähler metric and an Einstein metric are of interest; these are called Kähler-Einstein metrics. This project considers several central problems in complex geometry, both from analytic and algebraic points of view, and builds around the notion of a so-called singular Kähler-Einstein metric. This relatively recent notion has been introduced in an attempt to extendi Yau's theorem on the resolution of Calabi's conjecture to singular settings directly coming from algebraic geometry, and more precisely birational (complex) geometry. A major current challenge consists of constructing these objects in various contexts and trying to understand their behavior; a good understanding of the behavior of singular Kähler-Einstein metrics would have important consequences both in complex geometry as well as in areas of theoretical physics, such as string theory. Moreover, studying families of singular Kähler-Einstein metrics is very closely linked with the moduli theory, one of the main components of algebraic geometry. It is expected that results of the project concerning applications of singular Kähler-Einstein metrics will advance knowledge in the theory of singular spaces, in analytic and algebraic geometry, and in other areas of mathematics and physics.The project focuses on three central topics: behavior of the Kähler-Einstein metrics near the singularities of the variety, families and degenerations of singular Kähler-Einstein metrics, and applications of the Kähler-Einstein theory to algebraic geometry. The first circle of ideas is about finding models for Kähler-Einstein metrics near their singularities. The singularities can arise from those of the variety or from the boundary divisor of the pair to consider. The project examines the general case of (semi) log canonical pairs, through past results of the PI and others, as well as numerous questions that arise in the these works. The topics include improving the pluripotential methods to get sharper estimates, understanding the (Riemannian) geometry of the singular Kähler-Einstein metrics by relating singularities and completeness, and deriving higher regularity at the singular points. The second circle of ideas centers on families of singular Kähler-Einstein metrics, and in particular studying the moduli space of canonically polarized varieties from a differential geometric point of view through Weil-Petersson geometry. Other related questions are about Gromov-Hausdorff (or more refined) convergence of conic Kähler-Einstein metrics. The hope is that the recent developments in pluripotential theory should give a new angle to tackle these problems. The third circle of ideas focuses on various applications of singular Kähler-Einstein metrics, particularly in algebraic geometry. Typical questions that arise in the project concern vanishing/parallelism theorems or semi-stability properties of the tangent sheaf of singular varieties. Here again, a consequent part of the questions is motivated by recent progress in the theory of Monge-Ampère equations.