I'm reading the paper "High direct image of dualizing sheaf" of professor Kollar. I summarizing my questions as follows:
Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. L be a very ample line bundle on $Y$. Suppose the torsion part of $R^if_*\omega_X$ is supported at a closed point $y\in Y$. We choose a section $y\in D'\in |L|$. Then the map $$H^0(L\otimes R^if_*\omega_X)\rightarrow H^0(\mathcal O(D')\otimes L\otimes R^if_*\omega_X)$$
induced by $\mathcal O\rightarrow \mathcal O(D')$ is not injective. (Actually it's false, hence we have torsion freeness ) I don't know why by choosing such a $D'$ then the injectivity of the above map fails.
Thanks advance, maybe I haven't summarizing the conditions in a very precise way! Experts's opinions will help me a lot.
