THE MCKAY CORRESPONDENCE FOR ISOLATED SINGULARITIES VIA FLOER THEORY

We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity Cⁿ/G for a finite subgroup G ⊂ SL(n, C) and any crepant resolution Y , we prove that the rank of positive symplectic cohomology SH₊^∗ (Y ) is the number |Conj(G)| of conjugacy classes of G, and that twice the age grading on conjugacy classes is the Z-grading on SH₊^∗−1(Y ) by the Conley-Zehnder index. The generalized McKay correspondence follows as SH₊^∗−1(Y ) is naturally isomorphic to ordinary cohomology H^∗(Y ), due to a vanishing result for full symplectic cohomogy. 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.