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Derived Reid's recipe for abelian subgroups of SL3(ℂ)

  • Sabin Cautis EMAIL logo , Alastair Craw and Timothy Logvinenko

Abstract

For any finite subgroup GSL3(), work of Bridgeland–King–Reid constructs an equivalence between the G-equivariant derived category of 3 and the derived category of the crepant resolution Y=G-Hilb3 of 3/G. When G is abelian, we show that this equivalence gives a natural correspondence between irreducible representations of G and exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid’s recipe and extends earlier work from [3] and [14] which dealt only with the case when 3/G has one isolated singularity.

Award Identifier / Grant number: DMS-1101439

Award Identifier / Grant number: EP/G004048

Award Identifier / Grant number: EP/H023267/1

Funding statement: Sabin Cautis is supported by NSF grant DMS-1101439 and the Alfred P. Sloan foundation, Alastair Craw is supported by EPSRC grant EP/G004048 and Timothy Logvinenko is supported by EPSRC grant EP/H023267/1.

Acknowledgements

We would like to thank Miles Reid, Michael Wemyss, Yukari Ito, Keisuke Takahashi and Yuhi Sekiya for useful discussions while writing this paper.

References

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Received: 2012-6-29
Revised: 2014-5-5
Published Online: 2014-11-18
Published in Print: 2017-6-1

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