Counting resolutions of symplectic quotient singularities

Let Γ be a finite subgroup of Sp(V ). In this article we count the number of symplectic resolutions admitted by the quotient singularity V/Γ. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space. In this way, we give a simple formula for the number of Q-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik-Solomon algebra naturally associated to the Calogero-Moser deformation. This dimension is explicitly calculated for all groups Γ for which it is known that V/Γ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.