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Let $f : X\to Y$ be a morphism of smooth projective varieties over a field $k$.

Assume $f^*\omega_{Y/k} \simeq \omega_{X/k}$.

I'd like to collect a bestiary of the properties $f$ has, or even criteria/characterizations.

Does the condition $f^*\omega_{Y/k}\simeq \omega_{X/k}$ implies $f$ is finite? (hope: no)

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  • $\begingroup$ If $E$ is an elliptic curve, then $f:E\to Spec\, k$ gives a counterexample.. $\endgroup$
    – byu
    Feb 18, 2018 at 0:11
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    $\begingroup$ If $f$ is smooth of relative dimension $d$, then this is saying that $\bigwedge^d \Omega_{X/Y} \cong \mathcal O_X$; in particular the fibres of $f$ have trivial canonical bundle (e.g. a family of elliptic curves, abelian varieties, K3 surfaces, etc.). I think a suitable version with $\bigwedge^d \Omega_{X/Y}$ replaced by the relative dualising sheaf or complex $\omega_{X/Y}$ might hold in bigger generality, but I'm not entirely sure of the precise and most general statement. $\endgroup$ Feb 18, 2018 at 2:44
  • $\begingroup$ In a slightly different setting, there is the notion of a crepant resolution for normal varieties whose definition is exactly what you're asking for, with $f$ the resolution of singularities for $Y$. These are also not finite. $\endgroup$ Feb 20, 2018 at 0:03
  • $\begingroup$ @EricCanton: If $Y$ (as assumed) is smooth, it cannot admit a crepant resolution. $\endgroup$ Feb 20, 2018 at 3:59
  • $\begingroup$ @SándorKovács: indeed, the slightly different setting is when $Y$ is not smooth. I should have made that explicit (and included a $Q$-Gorenstein hypothesis!). Thanks! $\endgroup$ Feb 20, 2018 at 5:15

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