SAG: Perverse schobers and the McKay correspondenceSeminar Algebraic Geometry (SAG)
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
I will talk about constructing a perverse schober, a poor man’s perverse sheaf of triangulated categories, in the context of the classical McKay correspondence for G \subset SL_2(C). The braid group of the corresponding ADE type acts on the derived category D(Y) of the minimal resolution Y of C2/G by spherical twists in the exceptional curves. This braid group is the fundamental group of the open stratum of \mathfrak{h}/W, the quotient of the ADE Cartan algebra by the Weil group action, so its action on D(Y) can be thought of as a local system of triangulated categories on this open stratum. A perverse schober extends this structure to the higher codimension stratas. We actually construct a W-equivariant schober on \mathfrak{h} by using an instance of the McKay correspondence – the root hyperplane arrangement in \mathfrak{h} coincides with the wall-and-chamber structure in the GIT stability space for the construction of Y as the moduli space of the McKay quiver representations. The schober we construct on the GIT stability space neatly packages up all the GIT wall-crossing equivalences and more. Our work is motivated by wanting to eventually tackle dim=3 case, where h/W picture no longer exists, however, it might still be possible to construct a similar schober on the GIT stability space. This is a joint work with Arman Sarikyan (LIMS).