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 cl...
| Autores principales: | , |
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| Formato: | Journal article |
| Lenguaje: | English |
| Publicado: |
International Press
2023
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| _version_ | 1826310369141850112 |
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| author | McLean, M Ritter, AF |
| author_facet | McLean, M Ritter, AF |
| author_sort | McLean, M |
| collection | OXFORD |
| description | 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 nonexact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology. |
| first_indexed | 2024-03-07T07:51:29Z |
| format | Journal article |
| id | oxford-uuid:111ab9d8-e3db-4ca2-9ed9-8d1a642dcb6f |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:51:29Z |
| publishDate | 2023 |
| publisher | International Press |
| record_format | dspace |
| spelling | oxford-uuid:111ab9d8-e3db-4ca2-9ed9-8d1a642dcb6f2023-07-20T09:20:56ZThe McKay correspondence for isolated singularities via Floer theoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:111ab9d8-e3db-4ca2-9ed9-8d1a642dcb6fEnglishSymplectic ElementsInternational Press2023McLean, MRitter, AFWe 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 nonexact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology. |
| spellingShingle | McLean, M Ritter, AF The McKay correspondence for isolated singularities via Floer theory |
| title | The McKay correspondence for isolated singularities via Floer theory |
| title_full | The McKay correspondence for isolated singularities via Floer theory |
| title_fullStr | The McKay correspondence for isolated singularities via Floer theory |
| title_full_unstemmed | The McKay correspondence for isolated singularities via Floer theory |
| title_short | The McKay correspondence for isolated singularities via Floer theory |
| title_sort | mckay correspondence for isolated singularities via floer theory |
| work_keys_str_mv | AT mcleanm themckaycorrespondenceforisolatedsingularitiesviafloertheory AT ritteraf themckaycorrespondenceforisolatedsingularitiesviafloertheory AT mcleanm mckaycorrespondenceforisolatedsingularitiesviafloertheory AT ritteraf mckaycorrespondenceforisolatedsingularitiesviafloertheory |