Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. I am repeatedly encountering references to crepant morphisms $f:X\rightarrow Y$. I have found several definitions of such a morphism, one of which is the condition that $f^\*(K\_Y)=K\_X$. Is this generally accepted to be the meaning of a crepant morphism, or is this only the definition in some more restrictive context? Also, I would appreciate some perspective on why one naturally considers this class of morphisms.



Crepant Morphisms of Varieties
Crepant stands for *non-discrepant*. It's frequently applied to resolutions of singularities or birational maps (but can be applied more generally).


Let's start with the birational case, since that's where the history is.
If $f : X \to Y$ is birational, and $K\_Y$ is $\mathbb{Q}$-Cartier, then $f^\*(K\_Y)$ makes sense. In particular, if $nK\_Y$ is Cartier, then $f^\*(K\_Y) = \frac{1}{n} f^\*(nK\_Y)$ by definition.


Write $K\_X - f^\* K\_Y = \sum a\_i E\_i$ where we pick $K\_X$ and $K\_Y$ which agree where $f$ is an isomorphism. The $\sum a\_i E\_i$ is then independent of choices. 


Now, the numbers $a\_i$ are called *discrepancies*. If there are no discrepancies (ie, all the $a\_i$ are zero (for example, if $f$ is a small map), then the map is called *crepant*. Of course, all $a\_i = 0$ if and only if $K\_X = f^\* K\_Y$.


Of course, if the pullback of $K\_X$ is $K\_Y$, then this can be applied to many things. The existence of a crepant resolution of singularities also can be quite useful. Let me give a nonstandard example in characteristic $p > 0$, if $Y$ is Frobenius split and $f : X \to Y$ is crepant, then $X$ is also Frobenius split. Some variant of this appeared in the work of Mehta-van der Kallen and also Mehta-Srinivas.


