Study of singularities and geometry by means of representation theory

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We studied two-dimensional McKay correspondence and a canonical compactification of the moduli of abelian varieties.(I) For any finite subgroup of SL(2) we can construct a minimal resolution of a simple surface singularity CィイD12ィエD1/G as the Hilbert scheme of G-orbits. Via this construction we are able to provide a new explanation of two-dimensional McKay correspondence.(ii) We proved that for any finite abelian subgroup G of SL(3) the Hilbert scheme of G-orbits is a crepant resolution of the singularity CィイD13ィエD1/G. For simple finite subgroup G of SL(3) (there are only two such) we determined the structure of the Hilbert scheme of G-orbits.(iii) We constructed a canonical compactification SQィイD1g,NィエD1 of the moduli of abelian varieties over Z[ζィイD2NィエD2, 1/N]. Any point of SQィイD2g,NィエD2 is represented by an isomorphism class of a possibly singular abelian variety with certain level structure.