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.

Same as above, now suppose we have $R^if_*\omega_X$ is torsion free. Then the map above is injective. If we have flatness, then we have injectivity of the above map. It's also confusing to me why here torsion freeness will imply injectivity.

Thanks advance, maybe I haven't summarizing the conditions in a very precise way! Experts's opinions will help me a lot.

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.