Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

Alastair Craw, Akira Ishii

Research output: Contribution to journalArticle

43 Citations (Scopus)

Abstract

For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_\theta, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter \theta. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_\theta for some parameter \theta. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also uncover explicit equivalences between the derived categories of moduli M_\theta for parameters lying in adjacent GIT chambers.
Original languageEnglish
Pages (from-to)259-307
JournalDuke Mathematical Journal
Volume124
Issue number2
DOIs
Publication statusPublished - 2004

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