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

由一组多项式的公共零点给出的曲线和曲面的研究已经被人类追求了几千年。在现代代数几何中，我们在任何维度上研究这样的集合：这些集合被称为代数簇。对于一个这样的品种，人们可以问很多问题：它有多“好”？如果我们站在上面，它会看起来像一座弯曲的小山还是一座崎岖的山？如果给我们两个这样的品种，你能说出它们是不是一样的？或者如果它们是相似的，例如，如果站在它们上面的大多数地方，它们看起来是一样的，而它们只有在从某些精确的点看它们时才看起来不同？派生类别是一种考虑这些几何对象并将关于它们的许多信息转化为代数概念的方法。虽然派生的变种类别保留了我们最初使用的变种的大部分信息，但同时它允许我们在代数环境中工作的额外灵活性。在过去的二十年里，由于许多经典的代数几何问题都是通过派生范畴技术来解决的，派生范畴领域经历了一个活跃的过程。关于派生范畴的一个基本问题是两个不同几何对象的派生范畴之间是如何联系的。这些关系中的一些可能来自这两个变种之间的关系和对称性，但也有其他类型的关系，它们之间的关系更深更难理解：1.首先，了解两个派生范畴之间的映射(函子)是什么样子是很重要的。其中的许多--但不是所有的，就像人们过去认为的那样！--有一个非常令人愉快和有用的几何描述，称为“傅立叶-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-Mukai函子的“坏”地图，以及这些地图是如何通过使我们知道的“好”地图变形而自然产生的。2.两个派生范畴之间的另一种关系，将作为我的项目的一部分进行研究，它是由范畴层面上的“二元性”概念给出的。描述这种对偶性给我们提供了一种理解尚未发现的派生范畴之间更深层次关系的方法，这将更好地揭示派生范畴水平和几何对象水平上的对称性和行为。最后，在某些情况下，派生范畴之间的关系是等价的，因此可以用Fourier-Mukai函子来表示，并且在派生范畴的水平上的分析给我们提供了大量的几何信息。我的项目将处理一个这样的例子，即对一些商奇点的研究，这些商奇点是克莱恩奇点的推广，以及它们对奇点的分解。

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.