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Suppose I have a Gorenstein variety $X$ over $\mathbb{C}$ with trivial canonical bundle, and the action of a finite group $G$ on $X$, which acts trivially on the canonical bundle.

Question. When does the quotient $X/G$ have a crepant resolution?

I have some very specific examples in mind, namely when $X$ is one of the cyclic covers of $\mathbb{P}^n$ considered by Sheng-Xu-Zuo in https://arxiv.org/pdf/1211.3646.pdf, which admits (many) crepant resolutions, and $G$ being another cyclic group; I am also not sure whether the action of $G$ extends to any resolution $\tilde{X}$ of $X$, though I am leaning towards no, which is why the question is phrased as above. On the other hand I would also like to know what the general expectation/state of the art is for the general question, so any help would be much appreciated!

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  • $\begingroup$ This is a very old subject, which started probably even before : Akira Fujiki. On resolutions of cyclic quotient singularities.Publ. Res. Inst.Math. Sci., 10(1):293–328, 1974/75. On the other hand, if you are interested in more recent developments, it has been proved recently that all Gorenstein (finite) quotient singularities admit non-commutative crepant resolution of singularities. $\endgroup$
    – Libli
    Commented Apr 29, 2021 at 13:00
  • $\begingroup$ There are local obstructions to existence of crepant resolutions of quotient singularities, for example none exist for the Gorenstein quotient singularity $\mathbf{C}^4 / \pm 1$, see mathoverflow.net/questions/66657/…. $\endgroup$
    – Evgeny Shinder
    Commented May 6, 2021 at 22:06

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