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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Опубліковано в: :Symmetry, Integrability and Geometry: Methods and Applications
Дата:2016
Автори: Degeratu, A., Walpuski, T.
Формат: Стаття
Мова:Англійська
Опубліковано: Інститут математики НАН України 2016
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/147430
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Цитувати: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
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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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language English
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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