Studies in moduli theory and birational geometry

模量理论和双有理几何研究

• 批准号: 0901278
• 负责人: Dan Abramovich
• 金额: $ 36.64万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901278&HistoricalAwards=false
• 关键词: Studies; moduli; theory; birational; geometry

Abramovich will continue studying problems in moduli theory, including several aspects of Gromov--Witten theory, in particular (1) generalized moduli of relative stable maps;(2) comparison of virtual fundamental classes on moduli of relative and orbifold stable maps; and (3) further study of moduli of very twisted curves and their maps. Abramovich will continue studying problems in birational geometry, in particular (1) Campana constellations, their maps and arithmetic properties; (2) Birational geometry of and moduli of higher dimensional orbifolds, and their application in the compactification of moduli of higher dimensional varieties.The area of study of this project lies within algebraic geometry, the branch of mathematics devoted to geometric shapes called algebraic varieties, defined by polynomial equations. While algebraic geometry has contributed applications in coding, industrial control, and computation, the topics of this project are more closely related to applications in theoretical physics, where physicists consider algebraic varieties as components of the fine structure of our universe. This is especially true with the first topic, moduli theory. This theory studies a remarkable phenomenon in which the collection of all algebraic varieties of the same type is often manifested as an algebraic variety, called a moduli space, in its own right. Thus in algebraic geometry, the metaphor of thinking about a community of "organisms" as itself being an "organism" is not a just a metaphor but a rigorous and quite useful fact. Sometimes a collection of algebraic varieties manifests itself as a slightly more general object, called a stack, rather than a variety. Such stacks are a central object of study of this project. The other topic studied in this project is birational geometry, which is devoted to a certain abstract relationship, called birational equivalence, among algebraic varieties, which lies at the foundation of algebraic geometry.

阿布拉莫维奇将继续研究模理论中的一些问题，包括Gromov-Witten理论的几个方面，特别是(1)相对稳定映射的广义模；(2)相对稳定映射和奥比博尔德稳定映射的模的虚拟基本类的比较；(3)进一步研究非常扭曲的曲线及其映射的模。阿布拉莫维奇将继续研究二次几何中的问题，特别是(1)坎帕纳星座，它们的映射和算术性质；(2)高维orbillold的出生几何和模，以及它们在高维变体的模紧化中的应用。这个项目的研究领域是代数几何，数学的一个分支，致力于几何形状称为代数簇，由多项式定义。虽然代数几何在编码、工业控制和计算方面做出了贡献，但这个项目的主题与理论物理的应用更密切相关，在理论物理中，物理学家认为代数变体是我们宇宙精细结构的组成部分。对于第一个主题，模理论尤其如此。这一理论研究了一个值得注意的现象，在这个现象中，所有相同类型的代数簇的集合通常表现为一个代数簇，称为模空间，就其本身而言。因此，在代数几何中，把一个“有机体”群落想象成一个“有机体”本身就是一个“有机体”的比喻，不仅仅是一个比喻，而是一个严谨而又相当有用的事实。有时，代数变体的集合表现为稍微更通用的对象，称为堆栈，而不是变体。这样的堆栈是该项目研究的中心对象。这个项目中研究的另一个主题是二元几何，它致力于研究代数簇之间的某种抽象关系，称为二元等价，这是代数几何的基础。