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.

研究员将研究高维代数几何中的几个问题，特别是模理论和奇点，在与 Kollár 合作的几个项目中，PI 计划研究与稳定对数簇（一种模拟）的粗模空间的存在相关的各种问题。这些包括有理数对和节俭分辨率的研究及其与杜波依斯奇点和最小模型程序的其他奇点的联系在另一个项目中，也是由模项目推动的。 PI 将与 Patakfalvi 合作，致力于证明 Kollár 的充足性引理的对数版本，并用它来证明稳定对数簇的模空间的射影性。 PI 还将继续研究关于模堆栈子品种的改进 Viehweg 猜想。这个猜想是从 Shafarevich 的一个里程碑猜想及其 Arakelov 和 Parshin 的解决方案演变而来的，该项目在法尔廷斯证明莫德尔猜想的过程中发挥了重要作用。这项研究属于代数几何领域，它是现代数学最古老的部分之一，但已经发展到了成熟的程度。最初，它仍然以最简单的形式解决了由多项式定义的平面问题，如今，该领域不仅使用代数方法，还使用分析和拓扑方法。相反，它也被用于这些领域。此外，它在物理学、理论计算机科学、密码学、编码理论和机器人学等领域也被证明是有用的。分类理论的重要组成部分是模理论，后者的核心思想是人们不仅要理解这些对象，还要了解它们的变形方式在理论物理中发挥着非常重要的作用。在模空间提供了有关对象在时空中如何变化的信息，这些是一般模空间的扩展，它们提供了有关奇异变形的附加信息，这些变形本质上是不同的。来自其他人。