The McKay correspondence for isolated singularities via Floer theory

Abstract

We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity $\mathbb{C}^n / G$ for a finite subgroup $G \subset SL(n, \mathbb{C})$ and any crepant resolution $Y$, we prove that the rank of positive symplectic cohomology $SH^{\ast}_{+} (Y)$ is the number $\lvert \operatorname{Conj}(G) \rvert$ of conjugacy classes of $G$, and that twice the age grading on conjugacy classes is the $\mathbb{Z}$-grading on $SH^{\ast-1}_{+} (Y)$ by the Conley–Zehnder index. The generalized McKay correspondence follows as $SH^{\ast-1}_{+} (Y)$ is naturally isomorphic to ordinary cohomology $H^\ast (Y)$, 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.