Singular Kahler-Einstein Metrics: Analytic and Algebraic Aspects

复数空间或复数流形是承载在物理理论中自然产生的某种附加结构的空间。复几何的目的是通过距离和角度来研究复杂流形的形状；在数学上，这些度量被称为Kähler度量。在物理学的背景下，具有既是Kähler度量又是爱因斯坦度量的流形是有意义的；这被称为Kähler-Einstein度量。这个项目从解析和代数的角度考虑复杂几何中的几个中心问题，并建立在所谓的奇异Kähler-Einstein度规的概念上。这一相对较新的概念被引入，试图将游迪关于Calabi猜想分解的定理推广到直接来自代数几何的奇异环境，更准确地说，是二次(复数)几何。目前的一个主要挑战是在不同的背景下构建这些物体，并试图理解它们的行为；很好地理解奇异Kähler-Einstein度规的行为将在复杂几何以及理论物理领域(如弦理论)中产生重要的结果。此外，研究奇异Kähler-Einstein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Kähler-Einstein度量应用的结果将促进奇异空间理论、解析和代数几何以及其他数学和物理领域的知识。该项目集中在三个中心主题上：Kähler-Einstein度量在奇点附近的行为，奇异Kähler-Einstein度量的族和退化，以及Kähler-Einstein理论在代数几何中的应用。第一个想法圈是关于找到接近奇点的卡勒-爱因斯坦度规的模型。奇点可以产生于种类的奇点，也可以产生于要考虑的对的边界因子。该项目通过PI和其他人的过去结果以及在这些工作中出现的许多问题来研究(半)对数正则对的一般情况。这些主题包括改进多势方法以获得更精确的估计，通过将奇点和完备性联系起来来理解奇异Kähler-Einstein度量的(Riemannian)几何，以及在奇点获得更高的正则性。第二圈的思想集中在奇异Kähler-Einstein度量族上，特别是通过Weil-Petersson几何从微分几何的角度研究正则极化簇的模空间。其他相关问题是关于二次Kähler-Einstein度量的Gromov-Hausdorff(或更精细的)收敛。希望多势理论的最新发展能为解决这些问题提供一个新的视角。第三个思想圈集中于奇异Kähler-Einstein度规的各种应用，特别是在代数几何中。该项目中出现的典型问题涉及奇异变种的切线束的消失/平行定理或半稳定性质。在这里，随后的问题的一部分是由蒙格-安普雷方程理论的最新进展引起的。

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.