High direct image of dualizing sheaf

I’m reading the paper "High direct image of dualizing sheaf" of professor Kollar. I summarizing my questions as follows:

Let $f:Xrightarrow Y$ 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 $yin Y$. We choose a section $yin D’in |L|$. Then the map
$$H^0(Lotimes R^if_*omega_X)rightarrow H^0(mathcal O(D’)otimes Lotimes R^if_*omega_X)$$

induced by $mathcal Orightarrow mathcal O(D’)$ 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.

MathOverflow Asked by xin fu on January 24, 2021

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