Singularities and Moduli Theory

奇点和模理论

• 批准号: 1565352
• 负责人: Sandor Kovacs
• 金额: $ 49万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1565352&HistoricalAwards=false
• 关键词: Singularities; Moduli; Theory

This research project 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. 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 foci of this project is on compact moduli spaces, which give additional information about singular deformations, ones that are essentially different from others. The investigator is also involved in promoting mathematics to high school students; he and fellow directors of the Summer Institute for Mathematics at the University of Washington make special efforts to involve women at all levels of the program, from students through teaching assistants to instructors, to reinforce their leadership roles in mathematics.This project concerns several topics in higher dimensional algebraic geometry, especially moduli theory and singularities. The overarching theme of the research is centered on compact moduli spaces of stable log varieties, an important area that is still in the developmental stage. In particular, even the correct moduli functor needs to be identified, and most of the research is motivated by understanding the basic properties of these moduli spaces. This involves understanding the singularities that can occur on stable log varieties. Important classes of singularities in this regard are that of rational singularities and other singularities of the minimal model program. The investigator will work on advancing our currently very limited understanding of these singularities in arbitrary characteristic. The project also aims to develop cohomological methods to deal with rational pairs and thrifty resolutions in arbitrary characteristic. In particular, the investigator will study logarithmic versions of Hodge cohomology and of Grothendieck's fundamental class. Also stemming from the moduli project, the investigator plans to study properties of Du Bois singularities and Du Bois pairs. The main objectives of this part of the project are to develop a definition of these singularities that makes sense in arbitrary characteristic and to prove a subadjunction theorem for rational and Du Bois pairs.

这个研究项目属于代数几何领域，代数几何是现代数学最古老的部分之一，但它已经发展到解决了几个世纪以来的问题。它以最简单的形式处理由多项式在平面中定义的图形。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域中得到了广泛的应用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人技术等各个领域都很有用。代数几何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组成部分是模理论。后者的核心思想是，人们不仅要了解这些物体，还要了解它们变形的方式。模空间在理论物理中发挥着非常重要的作用：研究模空间上的曲线可以提供有关物体在时空中如何变化的信息。该项目的重点之一是紧致模空间，它提供了有关奇异变形的附加信息，这些变形与其他变形有本质上的不同。 研究者还致力于向高中生推广数学；他和华盛顿大学夏季数学研究所的其他主任特别努力让女性参与该项目的各个级别，从学生到助教再到讲师，以加强她们在数学方面的领导作用。该项目涉及高维代数几何中的几个主题，特别是模理论和奇点。该研究的总体主题集中在稳定原木品种的紧模空间上，这是一个仍处于发展阶段的重要领域。特别是，即使是正确的模函子也需要被识别，并且大多数研究都是通过理解这些模空间的基本属性来激发的。这涉及了解稳定原木品种可能出现的奇点。在这方面，重要的奇点类别是最小模型程序的理性奇点和其他奇点。研究人员将致力于推进我们目前对这些任意特征奇点的非常有限的理解。该项目还旨在开发上同调方法来处理任意特征的有理对和节俭分辨率。特别是，研究人员将研究霍奇上同调和格洛腾迪克基本类的对数版本。同样源于模量项目，研究人员计划研究杜波依斯奇点和杜波依斯对的性质。该项目这一部分的主要目标是制定这些奇点的定义，该定义在任意特征中都有意义，并证明有理数和杜波依斯对的从属定理。

项目成果

The minimal model program for b-log canonical divisors and applications

• DOI: 10.1007/s00209-023-03205-w
• 发表时间: 2017-07
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: D. Chan;K. Chan;Louis de Thanhoffer de Völcsey-Louis-de-Thanhoffer-de-Völcsey-2210870146;C. Ingalls;K. Jabbusch;S'andor J. Kov'acs;Rajesh S. Kulkarni;Boris Lerner;Basil Nanayakkara;Shinnosuke Okawa;Michel van den Bergh

Singularities of low degree complete intersections

低度完全交集的奇点

• DOI: 10.4310/maa.2017.v24.n1.a7
• 发表时间: 2017
• 期刊: Methods and Applications of Analysis
• 影响因子: 0.3
• 作者: Kovács, Sándor J.

Projectivity of the moduli space of stable log-varieties and subadditivity of log-Kodaira dimension

稳定对数簇模空间的投影性和对数小平维数的次可加性

• DOI: 10.1090/jams/871
• 发表时间: 2017
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Kovács, Sándor J;Patakfalvi, Zsolt
• 通讯作者: Patakfalvi, Zsolt