Categorical crepant resolutions for quotient singularities

Abstract

We discuss some results on categorical crepant resolutions for varieties with quotient singularities. Namely, we prove that under appropriate hypotheses, the derived category of a smooth Deligne-Mumford stack is a strongly crepant non-commutative resolution of singularities of its coarse moduli space.

A proof of the Donaldson–Thomas crepant resolution conjecture

“…and so we denote a class in N ≤1 (X ) by (β, c), with β ∈ N 1 (X ) and c ∈ N 0 (X ). 2 In [4], Behrend defines for any finite type scheme M a constructible function ν : M → Z, and shows that if M is proper and carries a symmetric perfect obstruction theory, with associated virtual fundamental class [M] vir , then…”

“…The derived dualising functor D(−) := RH om(−, O X ) [2] induces an involution on N ≤1 (X ) which preserves N 0 (X ), and so induces an involution on N 1 (X ). Note that the splitting N ≤1 (X ) = N 1 (X ) ⊕ N 0 (X ) cannot always be chosen compatibly with this duality, so that in general D(β, c) =…”

“…It is given by a slope function ν : N eff ≤1 (X ) → R ∪ {+∞} and depends on the choice of an ample class on X and an auxiliary generating vector bundle (see Sect. 2…”

Homological Projective Duality for the Plücker embedding of the Grassmannian

“…Note that if S has rational singularities, then any geometric resolution π : S → S gives a categorical resolution D b ( S) and in this setting, weakly crepant, strongly crepant and π being a crepant geometric resolution are all equivalent. See [Abu16] for more details on crepancy for categorical resolutions.…”

“…is just the derived category of the twisted group ring, and we have an action by a finite group, so this twisted group ring is an NC(C)R. Therefore we can say that the Kuznetsov component of X d is matrix factorizations on a specific NC(C)R. It is crepant only when d divides n + 1. See [Abu16] for details on the NC(C)R part and [Bal+13] for more details on the rest, that paper is all written from the homological projective duality point of view, looking at the Veronese map for projective space, we will introduce homological projective duality in the next section. It is also these ideas that we will generalise to the Grassmannian, again see the next section for more details.…”