Perverse Schobers and Wall Crossing

Abstract

For a balanced wall crossing in geometric invariant theory, 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 characterise when this is an intersection cohomology complex of a local system on the punctured disk.

Perverse schobers on Riemann surfaces: constructions and examples

“…I review results from GIT sufficient to recall, from previous work [12], a construction of a spherical pair for certain GIT wall crossings: this is Theorem 5.9. For further details on GIT, I refer the reader to treatments of Halpern-Leistner [20], and Ballard, Favero, and Katzarkov [4].…”

“…Section 4 uses these notions to define schobers on Riemann surfaces, following ideas of Kapranov-Schechtman, and gives criteria for extending these objects from an open subset. In Section 5, I review my previous work [12] constructing spherical pairs from GIT wall crossings. Section 6 constructs schobers on C, singular at iZ, proving Theorem A and Proposition B.…”

“…Spherical functors. We will use the following sequence of spherical functors obtained by Halpern-Leistner and Shipman [21,Section 3.2], and shown to arise from the spherical pair of Theorem 5.9 by the author [12].…”

“…Bodzenta and Bondal have associated them to flops of curves [7]. In previous work [12], I constructed them for wall crossings in geometric invariant theory (GIT).…”

“…One of the simplest classes of schobers consists of perverse sheaf of categories on a disk, singular at a point, known as 'spherical pairs'. In previous work [12], I constructed spherical pairs for certain wall crossings in GIT stability. In this paper, under the same assumptions, I obtain a perverse sheaf on C, singular at iZ, in Theorem A.…”