Dualities and Correspondences in Algebraic Geometry via Derived Categories and Noncommutative Methods

通过派生范畴和非交换方法的代数几何中的对偶性和对应性

基本信息

批准号：
EP/N021649/2
负责人：
Alice Rizzardo
金额：
$ 23.23万
依托单位：
University of Liverpool
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN021649%2F2
关键词：
Dualities Correspondences Algebraic Geometry via

项目摘要

The study of curves and surfaces given by the common zeroes of a set of polynomials has been pursued by humanity for thousands of years. In modern algebraic geometry, we study such sets in any dimension: these are called algebraic varieties. There are a number of questions that one can ask about one such variety: how "nice" is it? If we were standing on it, would it look to us like a curvy hill or like a rough mountain? If we are given two such varieties, can one tell if they are the same? Or if they are similar, for example if standing in most places on them they would look the same, and they only look different when looking at them from certain precise spots?Derived categories are a way to consider these geometric objects and translate much of the information about them into algebraic notions. While the derived category of a variety retains much of the information about the variety we started with, at the same time it allows us extra flexibility to work in an algebraic context. In the past two decades the field of derived categories has experienced an outpouring of activity as many classical algebraic geometry problems are solved passing through derived categories techniques.One fundamental question about derived categories is about how the derived categories of two different geometric objects are related. Some of these relations might come from relations and symmetries between the two varieties, but there are also other kinds of relations between them, which are deeper and harder to understand:1. First of all, it is important to understand what the maps (functors) between two derived categories are like. Many of these - but not all, as people used to think! - have a very pleasant and useful geometric description as "Fourier-Mukai functors". Part of my project will consist in analyzing and describing the "bad" maps that are not Fourier-Mukai functors, and how these arise naturally by deforming the "good" maps we know about. 2. Another relation between two derived categories, which will be investigated as part of my project, is given by a concept of "duality" at the categorical level. Describing this duality gives us a way to understand deeper relations between derived categories that haven't yet been discovered, and that will shed more light on the symmetries and behavior both at the level of derived categories and at the level of the geometric objects.3. Finally, in some instances the relations between derived categories turn out to be equivalences and hence representable by Fourier-Mukai functors, and the analysis on the level of derived categories gives us back a big amount of geometric information. My project will tackle one such instance, namely the investigation of some quotient singularities that are a generalization of the Kleinian singularities, and their resolutions of singularities.

数千年来，人类对一组多项式的共同零进行了对曲线和表面的研究。在现代代数几何形状中，我们在任何维度上研究了此类集合：这些集合称为代数品种。有许多问题可以问到一个这样的品种：这有多“好”？如果我们站在上面，它看起来像是弯曲的山丘还是崎山的山峰？如果我们得到了两个这样的品种，可以告诉它们是否相同？或者，如果它们是相似的，例如，如果站在它们上的大多数地方，它们看起来都一样，并且只有从某些精确的斑点查看它们时它们的外观不同？派生类别是一种考虑这些几何对象并将有关它们的大量信息转化为代数概念的一种方法。虽然派生的类别的类别保留了有关我们开头的多样性的大量信息，但与此同时，它使我们可以在代数环境中更加灵活地工作。在过去的二十年中，派生类别的领域经历了活动的大量活动，因为许多经典的代数几何问题都是通过派生类别技术解决的。关于派生类别的一个基本问题是关于两个不同几何对象的派生类别如何相关的。其中一些关系可能来自两个品种之间的关系和对称性，但是它们之间还有其他类型的关系，它们越来越难理地理解：1。首先，了解两个派生类别之间的地图（函数）是什么。其中许多 - 但不是全部，就像人们以前思考的那样！ - 具有非常愉快且有用的几何描述，例如“傅里叶 - 穆凯函子”。我的项目的一部分将包括分析和描述不是傅立叶式函数的“不良”地图，以及通过变形我们知道的“良好”地图来自然出现的“不良”地图。 2。两个派生类别之间的另一个关系，将作为我项目的一部分进行调查，是由分类层面的“双重性”概念给出的。描述这种二元性使我们有一种理解尚未发现的派生类别之间更深的关系的方法，这将在衍生类别的级别和几何对象级别上更加了解对称性和行为。3。最后，在某些情况下，派生类别之间的关系被证明是等价的，因此由傅立叶式函数代表，并且对派生类别级别的分析使我们回到了大量的几何信息。我的项目将解决一个这样的例子，即对某些商的奇异性进行调查，这些奇异性是克莱恩奇点的概括及其奇异性的决议。