On the (non)existence of symplectic resolutions of linear quotients

Abstract

We study the existence of symplectic resolutions of quotient singularities V/GV/G, where VV is a symplectic vector space and GG acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form K⋊S2K⋊S2 where K<SL2(C)K<SL2(C), for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for dimV≠4dim⁡V≠4, we classify all symplectically irreducible quotient singularities V/GV/G admitting a projective symplectic resolution, except for at most four explicit singularities, that occur in dimensions at most 1010, for which the question of existence remains open.