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

Sabin Cautis, Alastair Craw, Timothy Logvinenko

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For any finite subgroup G ⊂ SL3(ℂ), work of Bridgeland–King–Reid constructs an equivalence between the G-equivariant derived category of ℂ3 and the derived category of the crepant resolution Y = G-Hilb(ℂ3) of ℂ3/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 [J. reine angew. Math. 636 (2009), 193–236] and [J. Algebra 324 (2010), no. 8, 2064–2087] which dealt only with the case when ℂ3/G has one isolated singularity.
Original languageEnglish
Pages (from-to)1-48
Number of pages48
JournalJournal Fur Die Reine Und Angewandte Mathematik
Issue number727
Early online date18 Nov 2014
Publication statusPublished - 1 Jun 2017


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