Derived equivalences and autoequivalences in algebraic geometry

代数几何建立的基本信念是，几何问题通常可以用代数技术来回答。多年来，这种想法以无数种形式出现，但其基本策略可以总结为：我们在每个几何物体上附加一个代数小工具，用于编码我们感兴趣的信息；然后，我们研究了代数对象而不是几何对象。在转换过程中，一些信息将不可避免地丢失，但这不是主要问题：变量越少意味着问题越简单，而且有些信息对我们来说可能是无用的。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，也就是它们的维度。然而，如果我们想要区分一个球体和一个甜甜圈，我们将需要一个更精细的不变量。代数几何研究的中心对象是代数变异。这些对象在局部由多项式函数的零轨迹建模，因此人们可能会认为它们很容易研究。然而，重要的是它们的全球结构。比较一下：球体和甜甜圈在局部（拓扑上）是相同的，区别它们的是洞，只有当你把它放大到足够大的时候才能看到。我们在研究代数变量时考虑的不变量是它们的相干束的有界派生范畴，我们从三个不同的角度来看待它。派生类别的灵活性派生类别是一个比它所来自的种类更灵活的对象，有趣的是，在多大程度上这是正确的。也就是说，什么时候两个不同的品种有相同的衍生类别？具有相同派生范畴的两个变量之间的关系是什么？有一个猜想可以回答这个问题，而这个研究项目的目的之一就是努力更好地理解这一图景。在研究几何和代数对象时，一个被证明是非常有用的策略是观察它们的对称性。派生范畴的对称性被称为自等价，在许多不同的语境中自然出现。为了扩大我们对自等价的认识，在本研究项目中，我们从群论和动力系统的角度研究了那些由围绕球形物体的球形扭曲组成的对称性。从派生类别构建的结构从派生类别开始，构建了新的不变量。例如，bridgeeland的稳定性条件和Hochschild上同调。本研究项目的目的之一是扩大我们对这些不变量的理解：我们计划在分类动力系统的研究中使用Bridgeland的稳定性条件，并计算一些显变的Hochschild上同，从而扩大可用的例子列表。整个故事的美妙之处在于，不仅上述三种观点相互联系——研究物体的对称性有助于理解物体本身——而且它们也受到数学研究的邻近领域的影响，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显着的进步。

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.