Symplectic cohomology of Hilbert schemes of points

本项目属于EPSRC“数学科学(几何与拓扑学)”研究领域。几何的具体领域是辛拓扑。其目的是研究某些代数几何中感兴趣的辛流形的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，这引发了大量的研究和发展。特别是，Floer理论描述了Kontsevich的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，近年来取得了长足的进展。这个项目将考虑的空间是点的希尔伯特方案。要研究的第一个简单例子是复平面上n个点的希尔伯特格式，模平移。当n=2时，Cp^1的余切丛为非精确辛型。除了曲面上点的Hilbert格式，另一类例子是G-Hilb(C^n)，它是仿射n-空间中G-簇的模空间，对于SL(n，C)的有限子群G。在维度n=2和3中，这些定义了奇点C^n/G的可行解，特别是这些空间在广义McKay对应中是感兴趣的。对于n=2，人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，通过Ritter的工作，n=2的情况在辛拓扑中被相对较好地理解。Ito，Nakajima，Nakamura，Miles Reid和Alastair Craw，Batyrev，Denef和Loeser等人在代数几何中对McKay对应进行了大量的研究，Bridgeland-King-Reid也做了工作。然而，这个问题在辛拓扑文献中相对较少涉及，除了McLean和Ritter最近用Floer理论证明McKay对应的工作之外。该项目的一个可能的目的是在McLean-Ritter的结果的基础上，详细研究G-Hilb的Floer理论，以推断关于拉格朗日子流形的存在的结果和关于Fukaya范畴的结构结果，特别是将这些结果与G-Hilb所解析的奇异空间联系起来。第一种方法是考虑交换群G：在这种情况下，有用Toric技巧(Reid的配方)对G-Hilb的详细描述。辛几何和代数几何都会对这项工作感兴趣。研究领域是新颖的，因为非精确辛流形的Floer理论还没有被很好地理解，尽管这些流形是在代数几何中自然产生的。尤其是在这样的非精确环境下，格罗莫夫-威腾不变量在弗洛尔理论中扮演着一个有趣的角色。这项工作的一部分可能还包括与马克·麦克莱恩(纽约州石溪)的合作。和亚历山大·里特(牛津大学)，他们正在领导一个长期项目，以了解奇点分解的弗洛尔理论。

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.