Duke Mathematical Journal

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

Alastair Craw and Akira Ishii

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For a finite subgroup G⊂SL(3,ℂ), Bridgeland, King, and Reid [BKR] proved that the moduli space of G-clusters is a crepant resolution of the quotient ℂ3/G . This paper considers the moduli spaces $\mathcal{M}$θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of geometric invariant theory (GIT) parameter θ. For G Abelian, we prove that every projective crepant resolution of ℂ3/G is isomorphic to $\mathcal{M}$θ for some parameter θ. 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 $\mathcal{M}$θ for parameters lying in adjacent GIT chambers.

Article information

Duke Math. J., Volume 124, Number 2 (2004), 259-307.

First available in Project Euclid: 5 August 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14E15 14F05 18E30 14L24


Craw, Alastair; Ishii, Akira. Flops of G -Hilb and equivalences of derived categories by variation of GIT quotient. Duke Math. J. 124 (2004), no. 2, 259--307. doi:10.1215/S0012-7094-04-12422-4. https://projecteuclid.org/euclid.dmj/1091735976

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