Dynkin diagrams and crepant resolutions of quotient singularities

Abstract

Let $V$ be a complex vector space on which a finite group $G$ acts by linear transformations. Let $W=V\oplus V^{\ast }$ be the sum of $V$ with its dual $V^{\ast }$. We prove that if the quotient $W/G$ admits a smooth crepant resolution, then the subgroup $G\subset AutV$ is generated by complex reflections. We also obtain some results on the structure of smooth crepant resolutions of the quotients $W/G$, where $W$ is a symplectic vector space, and $G\subset AutW$ is a finite group of symplectic linear transformations of the vector space $W$.

30 pages, LaTeX2e