Derived categories of resolutions of cyclic quotient singularities

TL;DR

This paper investigates the derived categories of resolutions of cyclic quotient singularities, revealing new equivalences and formulas, and proposing constructions for categorical crepant resolutions within derived categories.

Contribution

It generalizes known results about derived categories of orbifolds and resolutions, introduces a 'flop-flop=twist' formula, and proposes new methods for categorical crepant resolutions.

Findings

Derived equivalences between orbifolds and resolutions.

A 'flop-flop=twist' formula relating tensor products.

Candidates for categorical crepant resolutions tested on cyclic quotient singularities.

Abstract

For a cyclic group $G$ acting on a smooth variety $X$ with only one character occurring in the $G$-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold $[X/G]$ and the blow-up resolution $\widetilde Y \to X/G$. Some results generalise known facts about $X = A^n$ with diagonal $G$-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals $|G|$, we study the induced tensor products under the equivalence $D^b(\widetilde Y) \cong D^b([X/G])$ and give a 'flop-flop=twist' type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities.