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.

代数几何的基本信念是，几何问题往往可以用代数技巧来回答。多年来，这个想法已经以无数种形式形成，但基本策略可以概括为：对每个几何对象，我们附加一个代数小工具，编码我们感兴趣的信息;然后，我们研究代数对象，而不是几何对象。在转换过程中，一些信息不可避免地会丢失，但这并不是主要问题：变量越少，问题就越简单，而且有些信息可能对我们来说已经没用了。根据我们想要回答的问题，我们可能需要丢弃或多或少的信息。事实上，要区分直线和平面，我们只需要考虑我们可以移动的方向的数量，即它们的维度。然而，如果我们想区分一个球体和一个甜甜圈，我们需要一个更精确的不变量。代数几何的中心研究对象是代数簇。这些对象是局部建模的多项式函数的零轨迹，因此人们可能会认为，他们很容易研究。然而，重要的是它们的全球结构。比较：一个球面和一个甜甜圈在局部上（拓扑上）是相同的，只有在足够远的时候才能看到的洞，才能区分它们。我们考虑研究代数簇的不变量是它们的有界导出的相干层范畴，我们从三个不同的角度来看待它。派生范畴的灵活性派生范畴是一个比它所来自的种类更灵活的对象，有趣的是，这在多大程度上是正确的。也就是说，什么时候两个不同的变种有相同的派生范畴？那么，具有同一派生范畴的两个变种之间的关系是什么呢？有一个猜想回答了这个问题，该研究项目的目的之一是努力更好地理解这张照片。导出范畴的对称性研究几何和代数对象的对称性是一个非常有用的策略。导出范畴的对称性被称为自等价，并且自然地出现在许多不同的上下文中。为了拓宽我们关于自等价的知识，在这个研究项目中，我们从群论和动力学系统的角度研究了围绕球形物体的球形扭曲组成的对称性。从派生范畴出发，构造了新的不变量。例如，Bridgeland稳定性条件和Hochschild上同调。这个研究项目的目的之一是扩大我们对这些不变量的理解：我们计划将Bridgeland的稳定性条件用于范畴动力系统的研究，并计算某些显式簇的Hochschild上同调，因此，扩大可用的例子列表。整个故事的美妙之处在于，不仅上述三个视角相互联系-研究物体的对称性有助于理解物体本身，但它们也受到邻近的数学研究领域的影响，并反过来影响这些领域。因此，跨学科是拟议研究项目的核心，它的完成将在许多不同的研究领域产生合理的进步。