TY - JOUR

T1 - Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

AU - Craw, Alastair

AU - Ishii, Akira

PY - 2004

Y1 - 2004

N2 - 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.

AB - 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.

UR - http://dx.doi.org/10.1215/S0012-7094-04-12422-4

U2 - 10.1215/S0012-7094-04-12422-4

DO - 10.1215/S0012-7094-04-12422-4

M3 - Article

VL - 124

SP - 259

EP - 307

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 2

ER -