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Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2016, том 12
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Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three
Інший ID:
2010 Mathematics Subject Classification: 53C07; 14F05; 58J20
DOI:10.3842/SIGMA.2016.017
DOI:10.3842/SIGMA.2016.017
Посилання:
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.
Підтримка:
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].
Дата:
2016
Переглядів:
579
Завантажень:
237
Анотація:
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.
