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.

研究者将在更高维代数几何形状，尤其是模量理论和奇异性方面处理几个问题。在与Kollár的几个项目中，PI计划处理与存在稳定的对数变化的粗制模量空间有关的各种问题，这是稳定尖曲线的调节空间的类似物。其中包括对理性对和节俭的分辨率的研究及其与Du Bois奇异性和最小模型计划的其他奇异性的联系。在另一个项目（也与Moduli项目融合在一起）的另一个项目中，PI将与Patakfalvi共同证明Kollár的Gloganme gomplementes Lemma的对数版本，并使用它来证明稳定日志变化的模量空间的预测性。 PI还将继续他在精致的ViewWeg猜想上，涉及规范两极分化的光滑投射品种的模量堆栈。这个猜想是从沙法雷维奇（Shafarevich）的里程碑式的猜想以及阿拉克洛夫（Arakelov）和帕辛（Parshin）的解决方案演变而来的，这在福尔丁斯（Faltings）的莫德尔（Mordell）猜想的证明中发挥了重要作用。该项目是与Kebekus的联合合作。这项研究是在代数几何的领域，这是现代数学的最古老的部分之一，但它绽放到已经解决了几个世纪停滞的问题的地步。最初，它仍然以最简单的形式处理，它处理由多项式在平面上定义的数字。如今，该领域不仅使用代数的方法，而且使用分析和拓扑的方法，相反，它在这些领域中广泛使用。此外，它已经证明了自己在物理，理论计算机科学，密码学，编码理论和机器人技术等多样化的领域中有用。代数几何形状中的一个核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量理论。后来的核心想法是，一个人不仅想了解这些对象，而且还了解它们可以变形的方式。模量空间在理论物理学中起着非常重要的作用。在模量空间上研究曲线提供了有关对象在时空如何变化的信息。该项目的重点之一是紧凑型模量空间。这些通常是模量空间的扩展，它们提供了有关奇异变形的其他信息，这些变形与其他变形基本不同。