Symplectic cohomology of Hilbert schemes of points

希尔伯特点方案的辛上同调

• 批准号: 1941576
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 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=studentship-1941576
• 关键词: Symplectic; cohomology; Hilbert; schemes; points

This project falls within the EPSRC "Mathematical Sciences (Geometry and topology)" research area.The specific area of geometry is symplectic topology. The goal is to study the Floer cohomology of certain families of symplectic manifolds that are of interest in algebraic geometry. Floer cohomology is a geometric invariant of symplectic manifolds introduced by Floer in 1989, and this has spurred a lot of research and development over the past two decades. In particular, Floer theory describes the A-side of Kontsevich's homological mirror symmetry conjecture, which is a deep conjectural framework which relates symplectic topology and algebraic geometry, and which has seen substantial progress in recent years. The spaces this project will consider, are Hilbert schemes of points. A first simple example to investigate, will be the Hilbert scheme of n points in the complex plane, modulo translation. The case n=2 recovers the cotangent bundle of CP^1 with a non-exact symplectic form. Beyond Hilbert schemes of points on surfaces, another class of examples is G-Hilb(C^n), the moduli space of G-clusters in affine n-space, for finite subgroups G of SL(n,C). In dimensions n = 2 and 3, these define crepant resolutions of the singularity C^n/G, in particular these spaces are of interest in the generalised McKay Correspondence. For n=2 one recovers the minimal resolutions of Kleinian singularities, which are spaces that naturally arise in several disparate areas of mathematics and theoretical physics. In both classes of examples, the case n=2 is relatively well-understood in symplectic topology, by work of Ritter. Substantial research on the McKay Correspondence was carried out in algebraic geometry, by authors including Ito, Nakajima, Nakamura, Miles Reid and Alastair Craw, Batyrev, Denef and Loeser, and work by Bridgeland-King-Reid. However, the topic is relatively untouched in the symplectic topology literature, with the exception of recent work by McLean and Ritter on the proof of the McKay correspondence using Floer theory. One possible aim of the project, building upon results of McLean-Ritter, is a detailed investigation of the Floer theory of G-Hilb, to infer results about the presence of Lagrangian submanifolds and structural results about the Fukaya category, in particular relating these to the singular space that G-Hilb resolves. A first approach is to consider abelian groups G: in this case there are detailed descriptions of G-Hilb by toric techniques (Reid's recipe). This work will be of interest to both symplectic and algebraic geometers. The research area is novel, as Floer theory for non-exact symplectic manifolds is not well-understood, despite these manifolds arising quite naturally in algebraic geometry. In particular it is in such non-exact settings that Gromov-Witten invariants play an interesting role in Floer theory. Part of this work may involve also collaborations with Mark McLean (Stony Brook N.Y.) and Alexander Ritter (Oxford), who are leading a long-term program to understand the Floer theory for resolutions of singularities.

该项目福尔斯EPSRC“数学科学（几何和拓扑）”的研究领域。几何的具体领域是辛拓扑。目标是研究在代数几何中感兴趣的某些辛流形族的Floer上同调。Floer上同调是Floer在1989年提出的辛流形的一个几何不变量，在过去的二十年里，这一概念引起了大量的研究和发展。特别是，Floer理论描述了Kontsevich同调镜像对称猜想的A面，这是一个联系辛拓扑和代数几何的深层猜想框架，近年来取得了实质性进展。这个项目将考虑的空间是点的希尔伯特方案。第一个简单的例子调查，将是希尔伯特计划的n个点在复平面上，模平移。当n=2时，CP ^1的余切丛恢复为非精确辛形式。除了曲面上的点的希尔伯特方案，另一类例子是G-Hilbert（C^n），仿射n-空间中G-簇的模空间，对于SL（n，C）的有限子群G。在n = 2和3维中，这些定义了奇点C^n/G的可分辨性，特别是这些空间在广义麦凯对应中是有意义的。对于n=2，我们恢复了克莱因奇点的最小分辨率，克莱因奇点是在数学和理论物理的几个不同领域中自然出现的空间。在这两类例子中，n=2的情况在辛拓扑学中是比较好理解的，通过里特的工作。大量的研究麦凯对应进行了代数几何，作者包括伊藤，中岛，中村，迈尔斯里德和阿拉斯泰尔克劳，Batyrev，Denef和Loeser，和工作的Bridgeland国王里德。然而，这个主题在辛拓扑学文献中是相对未触及的，除了姆克林和里特最近使用弗洛尔理论证明麦凯对应的工作。该项目的一个可能的目标，建立在结果的麦克莱恩，里特，是一个详细的调查弗洛尔理论的G-Hilb，推断结果的存在拉格朗日子流形和结构的结果有关的福谷类别，特别是这些有关的奇异空间，G-Hilb解决。第一种方法是考虑阿贝尔群G：在这种情况下，有详细的描述G-Hilb的复曲面技术（里德的食谱）。这项工作将感兴趣的辛和代数几何。该研究领域是新颖的，因为非精确辛流形的Floer理论还没有得到很好的理解，尽管这些流形在代数几何中很自然地出现。特别是它是在这种非精确的设置，Gromov-Witten不变量发挥了有趣的作用，弗洛尔理论。这项工作的一部分可能还涉及与马克姆克林（斯托尼布鲁克纽约）的合作。和亚历山大里特（牛津大学），谁是领导一个长期计划，以了解弗洛尔理论的决议奇点。