A Parameterization of D equivalences of coherent sheaves of symplectic resolutions of a given symplectic singularity

TL;DR

The paper constructs a local system of derived categories of coherent sheaves on symplectic resolutions, parameterized by homotopy classes of paths, revealing a non-canonical but structured family of equivalences.

Contribution

It refines previous constructions to build a local system of categories over a topological space, connecting derived categories via homotopy classes of paths.

Findings

Established a local system of categories over V^0_C.

Parameterization of equivalences by homotopy classes.

No canonical equivalence between categories.

Abstract

Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical equivalence. By refining a construction from a previous paper, we construct a local system of categories over a topological space V^0_C, where these categories are assigned to different points in V^0_C. Natural equivalence functors between these categories are parameterized by homotopy classes of paths between the corresponding points.