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)