Moduli theory and singularities

模理论和奇点

• 批准号: 1301888
• 负责人: Sandor Kovacs
• 金额: $ 23.85万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301888&HistoricalAwards=false
• 关键词: Moduli; theory; singularities

The investigator will work on several problems in higher dimensional algebraic geometry, especially moduli theory and singularities. In several projects joint with Kollár, the PI plans to work on various problems related to the existence of a coarse moduli space of stable log varieties, an analog of the moduli space of stable pointed curves. These include the study of rational pairs and thrifty resolutions and their connections with Du Bois singularities and other singularities of the minimal model program. In another project, also motivated by the moduli project, jointly with Patakfalvi the PI will work on proving a logarithmic version of Kollár's Ampleness Lemma and use it to prove the projectivity of the moduli space of stable log varieties. The PI will also continue his work on the refined Viehweg conjecture regarding subvarieties of moduli stacks of canonically polarized smooth projective varieties. This conjecture evolved from a landmark conjecture of Shafarevich, and its solution by Arakelov and Parshin, which played an important role in Faltings' proof of the Mordell Conjecture. This project is joint work with Kebekus. This research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one that blossomed to the point where it has solved problems that have stood for centuries. Originally, and still in its simplest form it treats figures defined in the plane by polynomials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics. A central problem in algebraic geometry is the classification of all geometric objects. In turn, an important part of classification theory is the theory of moduli. The latter's core idea is that one does not only want to understand these objects, but also understand the way they can be deformed. Moduli spaces play a very important role in theoretical physics. Studying curves on moduli spaces provides information on how an object is changing in space-time. One of the focuses of this project is on compact moduli spaces. Those are extensions of moduli spaces in general and they give additional information about singular deformations, ones that are essentially different from others.

调查员将在高维代数几何，特别是模理论和奇点的几个问题的工作。 在与Kollár合作的几个项目中，PI计划研究与稳定对数品种的粗模空间的存在相关的各种问题，这是稳定尖曲线的模空间的模拟。 这些包括研究合理的对和节俭决议和他们的连接与杜波依斯奇异性和其他奇异性的最小模型计划。 在另一个项目中，也是受到模项目的启发，PI将与Patakfalvi一起证明Kollár的Ampleness引理的对数版本，并使用它来证明稳定对数品种的模空间的投影性。 PI还将继续他的工作，精制Viehweg猜想关于子品种的模堆叠规范极化光滑投影品种。 这个猜想是从Shafarevich的一个里程碑式的猜想发展而来的，Arakelov和Parshin对这个猜想的解答在Faltings证明Mordell猜想的过程中扮演了重要的角色。 该项目是与Kebekus的联合工作。 这项研究是在代数几何领域，现代数学的最古老的部分之一，但一个蓬勃发展的地步，它已经解决了几个世纪以来的问题。最初，仍然在其最简单的形式，它对待的数字定义在平面上的多项式。今天，该领域使用的方法不仅来自代数，而且来自分析和拓扑学，相反，它被广泛用于这些领域。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人学等不同领域都很有用。代数几何中的一个中心问题是所有几何对象的分类。反过来，分类理论的一个重要部分是模理论。后者的核心思想是，人们不仅要理解这些物体，还要理解它们变形的方式。模空间在理论物理中起着非常重要的作用。研究模空间上的曲线提供了关于物体在时空中如何变化的信息。这个项目的重点之一是紧模空间。这些是模空间的一般扩展，它们提供了关于奇异变形的额外信息，这些信息与其他信息有本质的不同。