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arXiv · 1802.01534

The McKay correspondence for isolated singularities via Floer theory

Mark McLean·Alexander F. Ritter
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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.

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Mark McLean, Alexander F. Ritter. 2018-02-05. The McKay correspondence for isolated singularities via Floer theory. https://arxiv.org/abs/1802.01534

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