$R^{\dim X-\dim Y}f_{\ast}\omega_X \simeq \omega_Y$ in positive characteristic? In Proposition 7.6 of his paper "Higher Direct Images of Dualizing Sheaves", Kollár shows that if $X,Y$ are smooth complex projective varieties and $f:X\rightarrow Y$ is a proper surjective morphism with connected fibers, then there is an isomorphism $R^df_{\ast}\omega_X \simeq \omega_Y$, where $d=\dim X-\dim Y$.
I was wondering whether this is true in positive characteristic. By Grothendieck duality one has $$Rf_{\ast}\mathcal{O}_X \simeq Rf_{\ast}R\mathcal{H}om(\omega_X^{\bullet},\omega_X^{\bullet}) \simeq R\mathcal{H}om(Rf_{\ast}\omega_X^{\bullet}, \omega_Y^{\bullet})[-d]$$ so taking 0-th cohomology we get $$f_{\ast}\mathcal{O}_X \simeq \mathcal{E}xt^{-d}(Rf_{\ast}\omega_X^{\bullet}, \omega_Y^{\bullet})$$
From the spectral sequence $$\mathcal{E}xt^p(\mathcal{H}^{-q}(\mathcal{F}^{\bullet}), \mathcal{G}^{\bullet}) \Longrightarrow \mathcal{E}xt^{p+q}(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet})$$ it follows that the RHS is isomorphic to $\mathcal{H}om(R^df_{\ast}\omega_X,\omega_Y)$ and the LHS is just $\mathcal{O}_Y$ by the connected fibers assumption so we have $$\mathcal{O}_Y \simeq \mathcal{H}om(R^df_{\ast}\omega_X,\omega_Y)$$
Finally since $R^df_{\ast}\omega_X$ is locally free we conclude that $R^df_{\ast}\omega_X \simeq \omega_Y$.
 A: So a paper of Blickle, myself and Tucker handles some questions really closely related to this, see the arXiv version here
Ok, throw away the connected fibers hypothesis, I'm going to assume my base field is $F$-finite (although we can also work with integral $F$-finite separated schemes instead of varieties).
Suppose that $f : X \to Y$ is a proper surjective morphism of varieties in characteristic $p > 0$.  Then we have a map $R^d \pi_* \omega_X \to \omega_Y$.  This map is never zero (see Prop 2.13 in the aforementioned paper).
The nice thing is that the image of this morphism in $\omega_Y$ always contains the parameter test submodule $\tau(\omega_Y) \subseteq \omega_Y$ (see Prop 2.21 and Definition 2.33).  For smooth varieties, $\tau(\omega_Y) = \omega_Y$ (and even for varieties with mild singularities).  It's worth remarking that this is in some sense optimal:  For non-smooth varieties, we also show that there always exists some $X \to Y$ such that $\tau(\omega_Y)$ is the image of $R^d \pi_* \omega_X \to \omega_Y$.
Anyway, the upshot of the above discussion is that if $Y$ is smooth, then $R^d \pi_* \omega_X \to \omega_Y$ is surjective.  Obviously without the connected fibers hypothesis, it can't be injective.
EDIT: Ok, so then we need the injectivity under the hypothesis of connected fibers (or maybe also $X$ smooth).  In an earlier version of this ansewr I was incorrectly speculating that maybe we could use something like the argument in the question to help here.  As  Sándor points out, that won't work...
