Topics in algebraic geometry

代数几何专题

• 批准号: RGPIN-2016-04730
• 负责人: Dhillon, Ajneet
• 金额: $ 1.31万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615268
• 关键词: Topics; algebraic; geometry

This research is centered around principal bundles on algebraic varieties. In studying bundles, tangential subjects of independent interest, such as derived categories, K-theory and Gromov-Witten invariants are touched upon. These subjects have found applications in mathematical physics and we intend to both contribute to and be guided by this literature. This research makes systematic use of algebraic stacks. These are geometric objects with added symmetries. Their virtue lies in that many problems have their natural solutions as algebraic stacks rather than spaces. Stacks show up in the work as parameter spaces and as curves with added structure. The next tool that we make use of is the derived category and its various enhancements. The derived category is a kind of abelianizing object. It takes a complicated object and replaces it with something more accessible which at the same time encapsulates all its important invariants and properties. Using these tools we aim to study three fundamental questions. The first question is of basic interest to number theorists. General theory says that certain curves and maps can be defined via equations that have coefficients in a finite extension of the rational numbers. However not much is known about which particular extension. In other words, we have an existential theorem that we wish to make constructive. The second question is about counting are calculation of a particular number. We wish to count maximal subbundles of a fixed parabolic bundle. The approach that we intend to use will make use of ideas originating in mathematical physics. We intend to generalise some important combinatorial formulas in the process of the calculation. The last question that will be addressed is about calculating parameters. We have an important universal object, the moduli stack of principal bundles that appears in many parts of mathematics and physics. A basic question that can be asked about it is, how many parameters does in generally depend on. Given the objects ubiquity, this is an important question that should be addressed.

本研究主要围绕代数簇上的主丛展开。在研究丛的过程中，涉及到了一些独立的主题，如导出范畴、K-理论和Gromov-Witten不变量。这些主题已发现应用在数学物理和我们打算都有助于和指导这方面的文献。 本研究系统地使用了代数堆栈。这些是增加了对称性的几何物体。它们的优点在于，许多问题的自然解是代数堆而不是空间。堆栈在工作中显示为参数空间和具有附加结构的曲线。 我们使用的下一个工具是派生类别及其各种增强。派生范畴是一种阿贝尔化对象。它采用一个复杂的对象，并将其替换为更容易访问的对象，同时封装了所有重要的不变量和属性。 利用这些工具，我们旨在研究三个基本问题。 第一个问题是数论家的基本兴趣。一般理论认为，某些曲线和地图可以通过具有有理数的有限扩展的系数的方程来定义。但具体是哪一种扩展，我们所知不多。换句话说，我们有一个存在定理，我们希望使其具有建设性。 第二个问题是关于计数是计算一个特定的数字。我们希望计数一个固定抛物丛的极大子丛。我们打算使用的方法将利用源于数学物理的思想。在计算过程中，我们打算推广一些重要的组合公式。 最后一个要解决的问题是计算参数。我们有一个重要的普遍对象，主丛的模栈，它出现在数学和物理的许多部分。一个基本的问题是，它通常依赖于多少个参数。考虑到对象的普遍性，这是一个应该解决的重要问题。