In the following paper , in page 621 here 1 , I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of proof of inequality the authors first use of logarithmic transformation due to Kodaira. After they say
"Therefore $ω_{SF}$ is a smooth family of Ricci-flat metrics over $B$". I can not see this fact.
In the first part of proof they define $\tilde Y=\mathbb C\times \tilde B/L$, BUT I think we must define $\tilde Y=\mathbb C\times B/L$. For instance see this paper
In final The strategy of proof is not clear for me