The McKay correspondence for isolated singularities via Floer theory

TL;DR

This paper proves a generalized McKay correspondence for isolated singularities using Floer theory, linking symplectic cohomology with the topology of crepant resolutions and establishing new computational tools.

Contribution

It establishes a Floer-theoretic proof of the McKay correspondence for isolated singularities and introduces a novel filtration for symplectic chain complexes.

Findings

Rank of positive symplectic cohomology equals the number of conjugacy classes of G.

The a7-grading matches the Conley-Zehnder index up to a factor of two.

SH_+ is isomorphic to the cohomology of the resolution Y.

Abstract

We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity \C^n/G for a finite subgroup G in SL(n,\C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the \Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we construct a novel filtration on the symplectic chain complex for any non-exact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology.