Perverse Schobers and the McKay correspondence

Seaman, Christopher 2021. Perverse Schobers and the McKay correspondence. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

In [BKS18], Bondal, Kapranov and Schechtman gave the definition of a conjectural categorical analogue of perverse sheaves, known as perverse schobers. More accurately, due to the difficulties involved in categorifying the definition of perverse sheaves directly, they take a description of the category of perverse sheaves on a linear hyperplane arrangement H Rn in terms of a quiver representation due to Kapranov and Schechtman [KS16], and categorify this description. They call this notion an H-schober. In Chapter 1, we provide the background material for this thesis. In particular, we give the aforementioned quiver description of the category of perverse sheaves and the definition of an H-schober. In Chapter 2, we introduce the notion of geometric invariant theory quotients, which depend on a choice of stability parameter L; studying how this quotient changes as we vary the stability parameter is known as variation of geometric invariant theory (VGIT). For a given choice of stability parameter, we recount an iterative process for stratifying the unstable locus into a disjoint union of pieces, known as Kempf-Ness strata. An analysis of these KN strata leads to a method of constructing wall-crossing equivalences in VGIT via window subcategories. In Chapter 3, we describe a VGIT problem arising from the McKay correspondence. This naturally produces a hyperplane arrangement H and, for each cell in this hyperplane arrangement, the derived category of a quotient stack. In the remainder of this chapter we investigate the geometry of these quotient stacks for the particular case of G = Z3 := Z=3Z. In Chapter 4, we build an H-schober from the McKay correspondence as indicated. In particular, we are able to verify most of the schober conditions. The remaining conditions will be treated in future work.

Item Type:	Thesis (PhD)
Date Type:	Acceptance
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Date of First Compliant Deposit:	22 September 2021
Date of Acceptance:	February 2021
Last Modified:	04 Aug 2022 01:34
URI:	https://orca.cardiff.ac.uk/id/eprint/144325

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