Many symplectic resolutions of singularities (including the Hilbert–Chow morphism, the Springer resolution, and the minimal resolutions of ADE singularities) can be described by varying a stability parameter in a GIT quotient. To find such a variation of GIT description, it is useful to realize the singular space as a Nakajima quiver variety. Moreover, such a realization often results in a complete classification of symplectic resolutions by recent work of Bellamy—Craw—Schedler.
Joint work in progress with Schedler considers spaces that are everywhere formally locally quiver varieties. This is a large class of spaces thanks to Davison, who proved this property for the moduli space of objects in a (nice) 2CY category. To classify symplectic resolutions in this generality, fix a well-behaved stratification, akin to the stratification by symplectic leaves in a quiver variety. Then one can identify (global) symplectic resolutions with compatible local resolutions on each stratum. Each local resolution of a stratum in turn comes from a resolution of a neighborhood of a chosen basepoint on which the fundamental group of the stratum acts by “parallel transport” trivially.