Abstract
AbstractFor a balanced wall crossing in geometric invariant theory (GIT), there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterize when this is an intersection cohomology complex of a local system on the punctured disk.
Highlights
Take embeddings of a pair of categories of interest E± into a single category E0. This data may be viewed as a perverse sheaf of categories on a disk as follows
For a perverse sheaf of vector spaces, possibly singular at 0, the local cohomology with support in K is concentrated in some fixed degree by ‘purity’: the categories E± and E0 should be seen as categorifications of the stalks of this sheaf of local cohomology at ±1 and 0 respectively, and the embeddings as categorifications of maps between them
Consider a GIT wall crossing given by the data of a projectiveover-affine variety X with an action of a connected reductive group G and a pair of linearizations M± in the sense of Section 3.5
Summary
Taking complexified Grothendieck groups of the left-hand diagram, we obtain a perverse sheaf of vector spaces P = KP This has a nice description as follows. By composing arrows in the diagram we obtain a flop–flop functor FF acting on D(X//+), and thence an endomorphism of the complexified Grothendieck group K(X//+) which determines a local system of vector spaces on the punctured disk ∆ − 0: the intersection cohomology complex of this local system recovers P.
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