and effective divisor $E$ on $X'$. Pick irreducible divisor $ B \in |(n-1)A|$ for $n \gg 0$ such that $(X', \Delta')$ is klt, where $\Delta':=\Delta+\frac{1}{k}N + \frac{1}{n}E +\frac{1}{n}B$. Then we have, where $P' = ({\rm alb} \circ f)^*P$ with $P \in {\rm Pic}^0(A)$: $$K_{X'}+\lceil f^*L \rceil + P' \equiv K_{X'}+\frac{({\text{\rm alb}}\circ
f)^*M}{n}+\Delta'.$$ Let ${{\mathcal{F}}}:= {\text{\rm alb}}_*  f_* {{\mathcal{O}}}_{X'}(K_{X'} +\lceil
f^*L \rceil)$. By Kollár’s relative vanishing theorem (cf. [@Ko], 10.19.2), one sees that ${{\mathcal{F}}}$ is $IT^0$.

We claim that ${{\mathcal{F}}}\ne 0$.

Grant this claim for the time being. By the above lemma, it follows that $$h^0(X', K_{X'}+\lceil f^*L \rceil)=h^0(A, {{\mathcal{F}}})
\ne 0.$$ Since $K_{X'}+\lceil f^*L \rceil= f^*(K_X+\lceil L
\rceil) + \Gamma$, where $\Gamma$ is an exceptional divisor (possibly non-effective). It’s easy to see that $f_*
{{\mathcal{O}}}_{X'}(\Gamma) \subset {{\mathcal{O}}}_X$. By the projection formula, one has: $$0 \ne H^0(X',K_{X'}+\lceil f^*L \rceil)= H^0(X,{{\mathcal{O}}}_X(K_X+\lceil L
\rceil) {{\otimes}}f_* {{\mathcal{O}}}_{X'}(\Gamma) ) \subset H^0(X,K_X+\lceil L
\rceil).$$ This is the required non-vanishing.

To see the claim, if ${{\text{\rm dim}}}({\text{\rm alb}}(X))=2$, then ${\text{\rm alb}}\circ f$ is generically finite. Hence it is clear that ${{\mathcal{F}}}\ne 0$. If ${{\text{\rm dim}}}({\text{\rm alb}}(X))=1$. Let $F$ be a general fiber of ${\text{\rm alb}}\circ f$. Then we have: $${{\text{\rm rank}}}({{\mathcal{F}}}) =h^0(F, (K_{X'} +\lceil f^*L \rceil) |_F 