In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, 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