Symplectic cohomology of Hilbert schemes of 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.