Derived equivalences and autoequivalences in algebraic geometry

代数几何中的导出等价和自等价

• 批准号: EP/X01066X/1
• 负责人: Federico Barbacovi
• 金额: $ 44.44万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX01066X%2F1
• 关键词: Derived; equivalences; autoequivalences; algebraic; geometry

The fundamental belief upon which algebraic geometry is founded is that geometric questions can often be answered using algebraic techniques. This idea has shaped itself in countless forms during the years, but the fundamental strategy can be summarised thus: to every geometric object we attach an algebraic gadget that encodes the information we are interested in; then, we study the algebraic object rather than the geometric one. In the transformation process, some information will be inevitably lost, but this is not major issue: having less variables means having a simpler problem, and some of the information might have been useless to us anyway.Depending on the question we want to answer, we might need to discard more or less information. Indeed, to distinguish between a line and a plane we just need to consider the number of directions in which we can move, that is, their dimension. However, if we wanted to tell apart a sphere from a doughnut, we would need a more refined invariant.The central object of study in algebraic geometry are algebraic varieties. These objects are locally modelled by zero loci of polynomials functions and thus one might think that they are easily studied. However, it is their global structure that matters. Compare: a sphere and a doughnut are locally (topologically) the same, it is the hole, which one sees only when zooming out enough, that distinguishes them.The invariant we consider to study algebraic varieties is their bounded derived category of coherent sheaves, and we look at it from from three different perspectives.1. The flexibility of the derived categoryThe derived category is a more flexible object than the variety it comes from, and it is interesting to ask to what extent is this true. Namely, when does it happen that two different varieties have the same derived category? And, what is the relation between two varieties with the same derived category? There is a conjecture that answers this question, and one of the aims of the research project is to work towards a better understanding of this picture.2. Symmetries of the derived categoryA strategy that has proved to be extremely useful in studying geometric and algebraic objects is to look at their symmetries. The symmetries of the derived category are called autoequivalences and appear naturally in many different contexts. With the aim of broadening our knowledge regarding autoequivalences, in this research project we study those symmetries that arise as compositions of spherical twists around spherical objects, both from a group-theoretic and a dynamical system point-of-view.3. Structures built from the derived categoryStarting from the derived category, new invariants have been constructed. For example, Bridgeland's stability conditions and Hochschild cohomology. One of the aims of this research project is to widen our understanding of these invariants: we plan to use Bridgeland's stability conditions in the study of categorical dynamical systems, and to compute the Hochschild cohomology of some explicit varieties, so to enlarge the list of available examples.The beauty of this whole story is that not only the three perspectives above are linked with each other - studying the symmetries of an object helps understanding the object itself - but they are also influenced by, and in turn influence, neighbouring areas of mathematical studies. Thus, interdisciplinarity is at the core of the proposed research project, and its completion will produce sensible advancements in many different research areas.

代数几何形状建立的基本信念是，通常可以使用代数技术回答几何问题。多年来，这个想法以无数形式形式塑造，但是可以总结基本策略：对于每个几何对象，我们附加了一个代数小工具，它编码了我们感兴趣的信息；然后，我们研究代数对象而不是几何对象。在转换过程中，某些信息将不可避免地丢失，但这不是主要问题：更少的变量意味着要有一个简单的问题，无论如何，某些信息可能对我们没有用。在我们要回答的问题上，我们可能需要丢弃更多或更少的信息。实际上，要区分一条线和平面，我们只需要考虑我们可以移动的方向数，即它们的维度。但是，如果我们想从甜甜圈分开一个球体，我们将需要一个更精致的不变性。代数几何学研究的中心对象是代数品种。这些对象是由多项式函数的零基因座局部建模的，因此人们可能会认为它们很容易研究。但是，重要的是他们的全球结构。比较：一个球体和甜甜圈在本地（拓扑）是相同的，这是一个孔，只有在放大足够的缩小时才能看到它。我们认为不变式研究代数品种是它们有界的相干吊杆类别，我们从三个不同的角度来看一下。1。派生类别的灵活性比衍生类别比其来自的品种更灵活，并且询问这一程度在多大程度上是真实的。也就是说，什么时候发生两个不同的品种具有相同的派生类别？而且，两个品种与同一派生类别之间的关系是什么？有一个猜想可以回答这个问题，研究项目的目的之一是致力于更好地理解这张照片2。被证明在研究几何和代数对象中非常有用的类别策略的对称性是查看它们的对称性。派生类别的对称性称为自动等量，在许多不同的情况下自然而然地出现。为了扩大我们对自动等量的知识，在该研究项目中，我们研究了那些作为球形曲折围绕球形对象的组成而产生的对称性，无论是从组理论和动力学系统观察点）。3。从派生类别中构建的结构构建，已经构建了新的不变性。例如，布里奇兰（Bridgeland）的稳定条件和霍奇柴尔德（Hochschild）的共同体学。该研究项目的目的之一是扩大我们对这些不变的人的理解：我们计划在对分类动态系统的研究中使用Bridgeland的稳定条件，并计算出一些明确品种的Hochschild共同体，因此要扩大该列表的列表。整个故事的美感都不只有三个观察者的对象 - 在其他方面的依据 - 在其他方面的依据 - 某个观点的对象链接 - 链接的链接是链接的。 - 但它们也受数学研究的邻近领域的影响，进而影响了它们。因此，跨学科性是拟议的研究项目的核心，其完成将在许多不同的研究领域产生明智的进步。