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

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

2002-11-22

arxiv.org/abs/math/0211360v3

Duke Math.J.124:259-307,2004

Mathematics

Algebraic Geometry

43 pages, 2 figures. Final version includes minor changes plus one new figure, to appear in Duke Math Journal

Scientific paper

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.

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