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arXiv · 1701.01331

Derived categories of resolutions of cyclic quotient singularities

Andreas Krug·David Ploog·Pawel Sosna
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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.

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Andreas Krug, David Ploog, Pawel Sosna. 2017-01-05. Derived categories of resolutions of cyclic quotient singularities. https://arxiv.org/abs/1701.01331

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