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

Craw, A. and Ishii, A. (2004) Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient. Duke Mathematical Journal, 124(2), pp. 259-307. (doi: 10.1215/S0012-7094-04-12422-4)

Abstract

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 Mθ, 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 Mθ 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 Mθ for parameters lying in adjacent GIT chambers.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Craw, Dr Alastair
• Authors: Craw, A., and Ishii, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Duke Mathematical Journal
• ISSN: 0012-7094
• ISSN (Online): 1547-7398