The following appears in a paper by Hwang and Oguiso.
Let $V'$ be a complex variety with the property that its normalization is smooth and the dualizing sheaf $ω_{V'}$ is invertible. Denoting by $ν: V \to V'$ the normalization map, we have a natural injective sheaf map $ω_V \to ν^* ω_{V'}$.
My question: How is the sheaf map $\omega_V \to \nu^* \omega_{V'}$ defined?
I think this may be related to Hartshorne, Exercise III 7.2
Let $f: X \to Y$ be a finite morphism of projective schemes of the same dimension over a field $k$, and let $\omega_X$ be a dualizing sheaf for $Y$.
- Show that $f^! \omega_Y$ is a dualizing sheaf for $X$.
- If $X$ and $Y$ are both nonsingular, and $k$ algebraically closed, conclude that there is a natural trace map $t: f_* \omega_X \to \omega_Y$.
The trace map from 2. actually exists for the dualizing sheaves, even if $Y$ is not smooth, so I do get a morphism
$$ f_* \omega_{V} \to \omega_{V'},$$
but taking the pull-back yields $f^* f_* \omega_V \to f^* \omega_{V'}$, which is not the desired result.