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

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_θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter θ. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_θfor 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 M_θfor parameters lying in adjacent GIT chambers.