Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three
For G a finite subgroup of SL(3,C) acting freely on C³∖{0} a crepant resolution of the Calabi-Yau orbifold C³/G always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2016 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2016
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147430 |
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| Cite this: | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three / A. Degeratu, T. Walpuski // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862657586370707456 |
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| author | Degeratu, A. Walpuski, T. |
| author_facet | Degeratu, A. Walpuski, T. |
| citation_txt | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three / A. Degeratu, T. Walpuski // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For G a finite subgroup of SL(3,C) acting freely on C³∖{0} a crepant resolution of the Calabi-Yau orbifold C³/G always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two.
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| id | nasplib_isofts_kiev_ua-123456789-147430 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-02T06:19:23Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
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| spelling | Degeratu, A. Walpuski, T. 2019-02-14T18:31:47Z 2019-02-14T18:31:47Z 2016 Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three / A. Degeratu, T. Walpuski // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53C07; 14F05; 58J20 DOI:10.3842/SIGMA.2016.017 https://nasplib.isofts.kiev.ua/handle/123456789/147430 For G a finite subgroup of SL(3,C) acting freely on C³∖{0} a crepant resolution of the Calabi-Yau orbifold C³/G always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two. A.D. would like to thank Tom Mrowka, Tam´as Hausel, Rafe Mazzeo and Mark Stern for useful
 conversations about dif ferent aspects of this work. A.D. was supported by the DFG via
 SFB/Transregio 71 “Geometric Partial Dif ferential Equations”. Parts of this article are the
 outcome of work undertaken by T.W. while working on his PhD thesis at Imperial College London,
 supported by European Research Council Grant 247331. T.W. would like to thank his
 supervisor Simon Donaldson for his support. Both authors would like to thank the anonymous
 referee of an earlier version of this article for pointing out a way of deriving the multiplicative
 formula (1.3) from the work of Ito and Nakajima [18]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three Article published earlier |
| spellingShingle | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three Degeratu, A. Walpuski, T. |
| title | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three |
| title_full | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three |
| title_fullStr | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three |
| title_full_unstemmed | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three |
| title_short | Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three |
| title_sort | rigid hym connections on tautological bundles over ale crepant resolutions in dimension three |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147430 |
| work_keys_str_mv | AT degeratua rigidhymconnectionsontautologicalbundlesoveralecrepantresolutionsindimensionthree AT walpuskit rigidhymconnectionsontautologicalbundlesoveralecrepantresolutionsindimensionthree |