| Summary: | The aim of this thesis is to improve our understanding of when crepant resolutions exist in dimension four. In three dimensions [BKR01] proved that for any finite subgroup G ⊂ SL(3,C) the G-Hilbert scheme G-Hilb(C3) gives a crepant resolution of the quotient singularity C3/G. In four dimensions very little is known about when crepant resolutions exist. In this thesis I present several approaches to this problem. I give an algorithm which determines, for quotients by cyclic subgroups of SL(4,C) whether or not a crepant resolution exists. This algorithm seeks to find a crepant resolution by performing a tree search. In Chapter 4, building on the work of [CR02] in three dimensions, I calculate the A-Hilbert scheme for a family of abelian subgroups A ⊂ SL(4,C). I show that this can be used to find a crepant resolution of C4/A.
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