Davison proved that the moduli space of objects in a k-linear 2-Calabi--Yau category is formally locally a quiver variety. Bellamy--Schedler gave a classification of which quiver varieties admit symplectic resolutions of singularities, and more recently with Craw classified symplectic resolutions in most cases. It is natural to wonder to what extent these two results could be combined to classify symplectic resolutions of singularities for the moduli space of objects in a 2-Calabi--Yau category. Note that 2-Calabi--Yau categories include the bounded derived category of a K3 surface, the wrapped Fukaya category of a symplectic Liouville 4-manifold, and the category of Higgs bundles on a closed Riemann surface.
In joint work with Travis Schedler, we develop an obstruction theory to extend local resolutions of stratified spaces to global resolutions. The strategy is to (1) choose resolutions around basepoints of minimal strata, (2) extend from a basepoint to the entire stratum, and (3) check compatibility of extensions across strata. The key lemma is a parallel transport type result to extend resolutions along simple exit paths. Then, for each s in S a stratum, parallel transport gives an action of the fundamental group of S on the set of germs of symplectic resolutions at s, which can be interpreted as an obstruction to (2). We prove that monodromy-free, compatible local resolutions extend and glue to a unique global resolution. In other language, the assignment of an open set U to the set of isomorphism classes of symplectic resolutions over of U is an S-constructible sheaf, where S is the stratification in symplectic leaves.
This talk will serve as a gentle introduction to these ideas, highlighting applications and working with small examples like the orbit space of a cyclic group action on a 2-dimensional complex torus.