Singular Kahler-Einstein Metrics: Analytic and Algebraic Aspects

复空间或复流形是带有在物理理论中自然产生的某种附加结构的空间。复几何旨在通过距离和角度来研究复流形的形状;数学上这些测量被称为凯勒度量。在物理学的背景下，带有既是凯勒度量又是爱因斯坦度量的度量的流形是有趣的;这些被称为凯勒-爱因斯坦度量。该项目考虑了复杂几何中的几个中心问题，从分析和代数的角度来看，并围绕所谓的奇异凯勒-爱因斯坦度量的概念建立。这个相对较新的概念已被引入试图延长丘的定理的决议卡拉比猜想的奇异设置直接来自代数几何，更确切地说，双有理（复）几何。当前的一个主要挑战是在各种背景下构造这些对象，并试图理解它们的行为;对奇异凯勒-爱因斯坦度量行为的良好理解将在复杂几何和理论物理领域（如弦理论）产生重要影响。此外，研究奇异凯勒-爱因斯坦度量族与代数几何的主要组成部分之一的模理论有着非常密切的联系。预计该项目关于奇异凯勒-爱因斯坦度量应用的成果将促进奇异空间理论、解析几何和代数几何以及其他数学和物理领域的知识。该项目侧重于三个中心主题：Kähler-Einstein度量在奇异Kähler-Einstein度量的簇、族和退化的奇异点附近的行为，以及凯勒-爱因斯坦理论在代数几何中的应用。第一圈的想法是关于寻找凯勒-爱因斯坦度量在奇点附近的模型。奇点可以来自于簇的奇点，也可以来自于要考虑的对的边界因子。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）对数典型对的一般情况。这些主题包括改进多能方法以获得更精确的估计，通过与奇点和完整性相关来理解奇异凯勒-爱因斯坦度量的（黎曼）几何，以及在奇点处获得更高的正则性。第二圈的思想集中在家庭的奇异凯勒-爱因斯坦度量，特别是研究模空间的规范极化品种从微分几何的观点，通过威尔-彼得森几何。其他相关的问题是关于锥Kähler-Einstein度量的Gromov-Hausdorff（或更精确的）收敛。我们希望多潜能理论的最新发展能为解决这些问题提供一个新的角度。第三圈的想法集中在各种应用的奇异凯勒-爱因斯坦度量，特别是在代数几何。在项目中出现的典型问题涉及消失/平行定理或奇异品种的切层的半稳定性。在这里，问题的一个必然部分是由最近的进展，在理论的蒙赫-安培方程。

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.