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 $R^df_{\ast}\omega_X \simeq \omega_Y$.