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:

- 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? - 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?
- 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.