I have some clarifying questions about crepant resolutions of singularities. The definition that I am aware of is that if $f:\tilde X \to X$ is a resolution of singularities, then the resolution is crepant if $K_{\tilde X} = f^*K_X$. My questions are as follows:

  1. Where can I find a detailed introduction to such resolutions? I have briefly looked at Birational Geometry of Algebraic Varieties by Kollar and Mori but was unable to find much. In particular I know that a crepant resolution of a surface is minimal, where can I find a proof of that (or is it straightforward enough that I should do this on my own)? What are any existence statements about them?
  2. How do you make sense of this definition if the variety is not normal? Do you resolve that normalization of $X$ and require the equality above between the normalization of $X$ and its resolution?
  3. Another definition (?) that I have come across (in, for example, Quiver Representation and Quiver Varieties by Kirillov, although it was not defined in this way there. It is defined this way in Nakajima's Lectures on Hilbert Schemes of Points on Surfaces) is that $K_{\tilde X} = \mathcal{O}_{\tilde X} = \pi^*(\mathcal{O}_X)$. What is the relation between this definition and the first? Is it a special case? Equivalent?

Any and all help is appreciated.


1 Answer 1


Your definition is the usual one. More or less. Probably you should assume that $K_X$ is Cartier or at least $\mathbb Q$-Cartier for the definition to even make sense.

  1. I don't think there is a detailed intro to crepant resolutions. Perhaps because they are rare. It is a strong condition on the singularity that it admits a crepant resolution, so I would be surprised if there is any general intro to them. There are some obvious things you can say. For instance, assuming $X$ is normal, see 2) below, either it must have strictly canonical singularities or the resolution be small at least locally where the singularities are better than canonical. But that's not enough. I don't think there is a simple criterion for when a crepant resolution exists, but if it admits a small resolution that's automatically crepant for trivial reasons.
  2. The issue with being normal starts much earlier. If $X$ is not normal, how do you define $K_X$? (This is possible, but not in the usual way). Even if you managed to define $K_X$ you run into the problem that resolutions are usually required to be an isom in codim 2 on the target, so if $X$ is not $R_1$, then you're in trouble. See this MO answer for more on these issues.
  3. I don't have that book handy so I'm just going to make a guess: Is it possible that it is talking about resolving a local scheme? In that case assuming that $K_X$ is Cartier means that it is (locally) trivial, so this definition is the same as the other one. Alternatively, I can imagine that the author is only making the definition for Calabi-Yau varieties.
  • $\begingroup$ Thanks very much. I suspected what the answer was to #1 but one can hope I guess. Thanks also for pointing me to your (very informative) answer with respect to #2. For #3, Nakajima was referencing the conjecture of Reid that for $\Gamma < \text{SL}(n,\mathbb{C})$ finite, the quotient $\mathbb{C}^n/\Gamma$ has a crepant resolution $X$. He then states parenthetically that this means $K_X = \mathcal{O}_X$. $\endgroup$
    – DKS
    Commented Jan 29, 2019 at 2:55

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